From ca4c23c2ef63e98d0a7fd1b9b947f962b7932213 Mon Sep 17 00:00:00 2001 From: George Tsoukalas Date: Tue, 16 Jul 2024 18:04:52 +0000 Subject: [PATCH 01/12] Modified website main page. --- README.md | 6 +- docs/index.html | 144 +++++++++++++++++++++++++++++++++++++++++++----- 2 files changed, 132 insertions(+), 18 deletions(-) diff --git a/README.md b/README.md index 5e506112..e7435778 100644 --- a/README.md +++ b/README.md @@ -4,7 +4,7 @@ PutnamBench is a benchmark for evaluation of theorem-proving algorithms on compe PutnamBench aims to support research in automated mathematical reasoning by providing a multilingual benchmark for evaluating theorem-proving algorithms. It is released under permissive licenses (Apache 2.0 for Lean 4 and Isabelle, MIT for Coq). The informal statements are also available with permission from the MAA. -We encourage community contributions through pull requests, though we kindly ask not to include proofs of any problems to reduce testing contamination. (TODO: After initial release, as of 7/8 this is expected to be in about 1 week.) +We strongly encourage community feedback! Please let us know if you have any comments for improving PutnamBench. If you notice any mistakes, please raise an issue on the repository and we will address it. We kindly ask that you do not write formal proofs for any of the problems in an effort to reduce contamination. ## Citation The associated paper for PutnamBench is {TODO}. Please consider including the following citation if you find PutnamBench useful. @@ -34,5 +34,5 @@ We also report the number of problems in a certain category. Note that some prob ## Versioning -- Version: `v-1` -- Not yet officially released. Please refrain from opening issues or making PRs until the initial release. +- Version: `v0` +- In preliminary release to allow for initial community feedback. We seek to release an official first version following several weeks of discussion with the community. diff --git a/docs/index.html b/docs/index.html index 99cd1b14..e537b00a 100644 --- a/docs/index.html +++ b/docs/index.html @@ -91,38 +91,38 @@

PutnamBench: A Multilingual Competition-Mathematics Benchmark for Formal Theorem-Proving

- George Tsoukalas1, + George Tsoukalas, - Jasper Lee1, + Jasper Lee, - John Jennings1, + John Jennings, - Jimmy Xin1, + Jimmy Xin, - Michelle Ding1, + Michelle Ding, - Michael Jennings1, + Michael Jennings, - Amitayush Thakur1 + Amitayush Thakur, - Swarat Chaudhuri1 + Swarat Chaudhuri
- 1UT Austin, + University of Texas at Austin
- @@ -131,7 +131,7 @@

PutnamBench: A Multilingual Competition - @@ -149,6 +149,15 @@

PutnamBench: A Multilingual Competition Code + + + + + + Leaderboard + +

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PutnamBench: A Multilingual Competition

Abstract

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- TODO: This website is still under construction. -

We present PutnamBench, a new multilingual benchmark for evaluating the ability of neural theorem-provers to solve competition mathematics problems.

- PutnamBench consists of 1600+ hand-constructed formalizations of problems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. + PutnamBench consists of 1697 hand-constructed formalizations of problems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. All the theorems, of which there are 640, have formalizations in Lean 4 and Isabelle; a substantial subset also has Coq formalizations.

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Abstract

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Formal Theorem-Proving

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+ Formal theorem-proving is the task of proving mathematical theorems specified inside an interactive theorem prover such as Lean, Isabelle, or Coq. + These systems allow one to specify theorems and proofs inside a programming language. The correctness of a proof is automatically checked by the compiler. +

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+ Informal benchmarks such as MATH and GSM8K measure accuracy by exact match with the ground truth numerical answer. However, the reasoning path might be incorrect and cannot be checked automatically as the problem is specified in natural language. + By contrast, benchmarking formal theorem-proving through PutnamBench allows one to measure a model's capacity to produce precise logical arguments in a mathematical context. Furthermore, since no formal proofs have been written for any of our problems, direct contamination is not possible, whereas more measures must be taken to prevent contamination with informal reasoning benchmarks. +

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The William-Lowell Putnam Mathematical Competition

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+ The William Lowell Putnam Mathematical Competition, organized by the Mathematical Association of America (MAA), is the premier collegiate mathematics competition in North America. Thousands of undergraduate students from universities across the United States and Canada take the exam each year. The competition comprises two 3-hour-long sessions of six problems each, presented in approximately ascending order of difficulty within each session. +

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+ The Putnam Competition is known for its challenging problems that require a deep understanding of undergraduate-level mathematics and creative problem-solving ability. The competition includes problems involving concepts from a wide variety of topics in the undergraduate curriculum, including analysis, abstract algebra, and linear algebra. +

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+ The competition is scored out of 120 points, with the median score typically being zero. The top five individual scorers are named Putnam Fellows, and in recent years regularly include former International Mathematical Olympiad (IMO) medalists. Hence, we find that PutnamBench is a well-suited benchmark for measuring progress towards the Artificial Intelligence Mathematical Olympiad and the IMO Grand Challenge." +

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PutnamBench: Multilinguality & Factored Solutions

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+ Multilinguality. Most recent work on neural theorem-proving for competition mathematics has been focused in Lean and Isabelle. PutnamBench supports researchers using these tools by incorporating formalizations of all 640 problems in these languages. Previously without a benchmark for competition math, we also include formalizations in Coq, of which there are 417. Formalization obstacles specific to Coq, as mentioned in our paper, prevent us from formalizing all 640 problems. We actively support community feedback and contributions as we continue to iterate on PutnamBench-Coq. +

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+ We hope that the multilinguality of PutnamBench will allow the theorem-proving community to make progress towards general mathematical reasoning machines. +

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Factored Solutions. Roughly 60% of Putnam problems, in their natural language form, require exhibiting a (closed-form) solution along with a proof of its correctness. Such problems do not assert propositions, and hence are not immediately formalizable as they are not directly the statement of a theorem. Prior benchmarks have included the explicit solution in the theorem statement, but this can drastically reduce the problem's overall difficulty, as finding the answer can be the most challenging part. For such problems, we factor out the solution from the theorem statement, allowing two tasks: proving the theorem with the solution provided, and additionally determining the solution.

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Evaluations & Leaderboard

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+ We perform evaluations with some recent models, as described in the TODO: PAPER. + We find that PutnamBench is hard, with an aggregration of all models tried unable to successfully prove even 1% of theorems. + In particular, few-shot GPT-4o solves only a single problem in each language when allotted 10 attempts. + PutnamBench highlights several deficiencies of large language models (LLMs): weak ability to (1) perform precise logical reasoning, (2) operate in low-resource languages. +

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+ We are actively maintaining a TODO: Leaderboard to keep track of progress from upcoming approaches for neural-theorem proving. We will be incorporating results which are accompanied by a research publication (or arxiv preprint) detailing the general methodology. +

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Licensing & Maintenance

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+ PutnamBench is available under an Apache 2.0 License for Isabelle & Lean, and an MIT License for Coq. With a big thanks to the MAA, we also include the informal problem statements as sourced from the official archives. +

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+ PutnamBench will be actively maintained for the foreseeable future. We plan to incorporate problem formalizations from Putnam 2024 in December. We are actively encouraging community feedback and discussion. We are interested in hearing how you might plan to use PutnamBench! Your feedback may help us iterate on the structure of PutnamBench, such as incorporating a validation/training split. Please reach out to us if you have (1) suggestions to improve the benchmark, (2) comments on maintaining benchmark longevity and usefulness, (3) noticed issues in certain formalizations which require modification. +

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From d2b31ae33f5eedcc4bf8b63d176269b69f84f69a Mon Sep 17 00:00:00 2001 From: George Tsoukalas Date: Tue, 16 Jul 2024 18:07:29 +0000 Subject: [PATCH 02/12] Removed old informal batch files. --- informal/assignment.json | 122 ---------------- informal/batch_10.json | 50 ------- informal/batch_11.json | 44 ------ informal/batch_12.json | 44 ------ informal/batch_13.json | 46 ------ informal/batch_14.json | 44 ------ informal/batch_15.json | 56 ------- informal/batch_16.json | 56 ------- informal/batch_17.json | 56 ------- informal/batch_18.json | 44 ------ informal/batch_19.json | 38 ----- informal/batch_19621973hard.json | 152 ------------------- informal/batch_19741984hard.json | 56 ------- informal/batch_19851997hard.json | 80 ---------- informal/batch_19982010hard.json | 111 -------------- informal/batch_2.json | 80 ---------- informal/batch_20.json | 50 ------- informal/batch_20112023hard.json | 122 ---------------- informal/batch_21.json | 39 ----- informal/batch_22.json | 44 ------ informal/batch_23.json | 56 ------- informal/batch_24.json | 56 ------- informal/batch_25.json | 50 ------- informal/batch_26.json | 56 ------- informal/batch_27.json | 74 ---------- informal/batch_28.json | 68 --------- informal/batch_29.json | 64 -------- informal/batch_3.json | 56 ------- informal/batch_30.json | 62 -------- informal/batch_31.json | 68 --------- informal/batch_32.json | 53 ------- informal/batch_33.json | 56 ------- informal/batch_34.json | 41 ------ informal/batch_35.json | 60 -------- informal/batch_36.json | 60 -------- informal/batch_37.json | 62 -------- informal/batch_38.json | 58 -------- informal/batch_39.json | 68 --------- informal/batch_4.json | 44 ------ informal/batch_40.json | 56 ------- informal/batch_41.json | 62 -------- informal/batch_42.json | 55 ------- informal/batch_43.json | 59 -------- informal/batch_44.json | 62 -------- informal/batch_45.json | 50 ------- informal/batch_46.json | 242 ------------------------------- informal/batch_47.json | 38 ----- informal/batch_48.json | 62 -------- informal/batch_49.json | 38 ----- informal/batch_5.json | 49 ------- informal/batch_50.json | 56 ------- informal/batch_51.json | 62 -------- informal/batch_52.json | 58 -------- informal/batch_55.json | 50 ------- informal/batch_6.json | 56 ------- informal/batch_7.json | 50 ------- informal/batch_8.json | 62 -------- informal/batch_9.json | 55 ------- informal/batch_geo.json | 8 - informal/four_missing.json | 26 ---- informal/negativebatch.json | 139 ------------------ informal/putnam_1963.json | 44 ------ informal/putnam_1967.json | 20 --- 63 files changed, 3905 deletions(-) delete mode 100644 informal/assignment.json delete mode 100644 informal/batch_10.json delete mode 100644 informal/batch_11.json delete mode 100644 informal/batch_12.json delete mode 100644 informal/batch_13.json delete mode 100644 informal/batch_14.json delete mode 100644 informal/batch_15.json delete mode 100644 informal/batch_16.json delete mode 100644 informal/batch_17.json delete mode 100644 informal/batch_18.json delete mode 100644 informal/batch_19.json delete mode 100644 informal/batch_19621973hard.json delete mode 100644 informal/batch_19741984hard.json delete mode 100644 informal/batch_19851997hard.json delete mode 100644 informal/batch_19982010hard.json delete mode 100644 informal/batch_2.json delete mode 100644 informal/batch_20.json delete mode 100644 informal/batch_20112023hard.json delete mode 100644 informal/batch_21.json delete mode 100644 informal/batch_22.json delete mode 100644 informal/batch_23.json delete mode 100644 informal/batch_24.json delete mode 100644 informal/batch_25.json delete mode 100644 informal/batch_26.json delete mode 100644 informal/batch_27.json delete mode 100644 informal/batch_28.json delete mode 100644 informal/batch_29.json delete mode 100644 informal/batch_3.json delete mode 100644 informal/batch_30.json delete mode 100644 informal/batch_31.json delete mode 100644 informal/batch_32.json delete mode 100644 informal/batch_33.json delete mode 100644 informal/batch_34.json delete mode 100644 informal/batch_35.json delete mode 100644 informal/batch_36.json delete mode 100644 informal/batch_37.json delete mode 100644 informal/batch_38.json delete mode 100644 informal/batch_39.json delete mode 100644 informal/batch_4.json delete mode 100644 informal/batch_40.json delete mode 100644 informal/batch_41.json delete mode 100644 informal/batch_42.json delete mode 100644 informal/batch_43.json delete mode 100644 informal/batch_44.json delete mode 100644 informal/batch_45.json delete mode 100644 informal/batch_46.json delete mode 100644 informal/batch_47.json delete mode 100644 informal/batch_48.json delete mode 100644 informal/batch_49.json delete mode 100644 informal/batch_5.json delete mode 100644 informal/batch_50.json delete mode 100644 informal/batch_51.json delete mode 100644 informal/batch_52.json delete mode 100644 informal/batch_55.json delete mode 100644 informal/batch_6.json delete mode 100644 informal/batch_7.json delete mode 100644 informal/batch_8.json delete mode 100644 informal/batch_9.json delete mode 100644 informal/batch_geo.json delete mode 100644 informal/four_missing.json delete mode 100644 informal/negativebatch.json delete mode 100644 informal/putnam_1963.json delete mode 100644 informal/putnam_1967.json diff --git a/informal/assignment.json b/informal/assignment.json deleted file mode 100644 index 89544ca6..00000000 --- a/informal/assignment.json +++ /dev/null @@ -1,122 +0,0 @@ -[ - { - "problem_name" : "putnam_2023_a1", - "informal_statement" : "For a positive integer $n$, let $f_n(x) = \\cos(x) \\cos(2x) \\cos(3x) \\cdots \\cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.", - "informal_solution" : "Show that the solution is $n = 18$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2020_b5", - "informal_statement" : "For $j \\in \\{1, 2, 3, 4\\}$, let $z_j$ be a complex number with $|z_j| = 1$ and $z_j \\neq 1$. Prove that \\[ 3 - z_1 - z_2 - z_3 - z_4 + z_1 z_2 z_3 z_4 \\neq 0. \\]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2018_a5", - "informal_statement" : "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$, and $f(x) \\geq 0$ for all $x \\in \\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2018_b3", - "informal_statement" : "Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.", - "informal_solution" : "Show that the solution is the set $\\{2^2, 2^4, 2^8, 2^16\\}$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2018_b4", - "informal_statement" : "Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_n x_{n-1} - x_{n-2}$ for $n \\geq 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_b3", - "informal_statement" : "Suppose that $f(x) = \\sum_{i=0}^\\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2014_a1", - "informal_statement" : "Prove that every nonzero coefficient of the Taylor series of \\[(1 - x + x^2)e^x\\] about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.", - "informal_solution" : "None.", - "tags" : ["analysis", "number_theory"] - }, - { - "problem_name" : "putnam_2014_a5", - "informal_statement" : "Let \\[ P_n(x) = 1 + 2 x + 3 x^2 + \\cdots + n x^{n-1}.\\] Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j \\neq k$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2010_a4", - "informal_statement" : "Prove that for each positive integer $n$, the number $10^{10^{10^n}} + 10^{10^n} + 10^n - 1$ is not prime.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2010_a5", - "informal_statement" : "Let $G$ be a group, with operation $*$. Suppose that \\begin{enumerate} \\item[(i)] $G$ is a subset of $\\mathbb{R}^3$ (but $*$ need not be related to addition of vectors); \\item[(ii)] For each $\\mathbf{a},\\mathbf{b} \\in G$, either $\\mathbf{a}\\times \\mathbf{b} = \\mathbf{a}*\\mathbf{b}$ or $\\mathbf{a}\\times \\mathbf{b} = 0$ (or both), where $\\times$ is the usual cross product in $\\mathbb{R}^3$. \\end{enumerate} Prove that $\\mathbf{a} \\times \\mathbf{b} = 0$ for all $\\mathbf{a}, \\mathbf{b} \\in G$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra", "algebra"] - }, - { - "problem_name" : "putnam_2001_a1", - "informal_statement" : "Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\\in S$, $a*b\\in S$. Assume $(a*b)*a=b$ for all $a,b\\in S$. Prove that $a*(b*a)=b$ for all $a,b\\in S$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_2001_b4", - "informal_statement" : "Let $S$ denote the set of rational numbers different from $\\{-1,0,1\\}$. Define $f:S\\rightarrow S$ by $f(x)=x-1/x$. Prove or disprove that \\[\\bigcap_{n=1}^\\infty f^{(n)}(S) = \\emptyset,\\] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2000_a2", - "informal_statement" : "Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2000_b4", - "informal_statement" : "Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\\leq x\\leq 1$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1999_a2", - "informal_statement" : "Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\\dots,f_k(x$) such that \\[p(x) = \\sum_{j=1}^k (f_j(x))^2.\\]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1999_a5", - "informal_statement" : "Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree 1999, then \\[|p(0)|\\leq C \\int_{-1}^1 |p(x)|\\,dx.\\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1999_b4", - "informal_statement" : "Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1999_b6", - "informal_statement" : "Let $S$ be a finite set of integers, each greater than 1. Suppose that for each integer $n$ there is some $s\\in S$ such that $\\gcd(s,n)=1$ or $\\gcd(s,n)=s$. Show that there exist $s,t\\in S$ such that $\\gcd(s,t)$ is prime.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1998_a3", - "informal_statement" : "Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \\[f(a)\\cdot f'(a) \\cdot f''(a) \\cdot f'''(a)\\geq 0 .\\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1998_b6", - "informal_statement" : "Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\\sqrt{n^3+an^2+bn+c}$ is not an integer.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_10.json b/informal/batch_10.json deleted file mode 100644 index b573ae99..00000000 --- a/informal/batch_10.json +++ /dev/null @@ -1,50 +0,0 @@ -[ - { - "problem_name" : "putnam_2011_a3", - "informal_statement" : "Find a real number $c$ and a positive number $L$ for which $\\lim_{r \\to \\infty} \\frac{r^c \\int_0^{\\pi/2} x^r\\sin x\\,dx}{\\int_0^{\\pi/2} x^r\\cos x\\,dx}=L$.", - "informal_solution" : "Show that $(c,L)=(-1,2/\\pi)$ works.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2011_a4", - "informal_statement" : "For which positive integers $n$ is there an $n \\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?", - "informal_solution" : "Show that the answer is $n$ odd.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2011_b1", - "informal_statement" : "Let $h$ and $k$ be positive integers. Prove that for every $\\epsilon>0$, there are positive integers $m$ and $n$ such that $\\epsilon<|h\\sqrt{m}-k\\sqrt{n}|<2\\epsilon$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2011_b2", - "informal_statement" : "Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0$. Which primes appear in seven or more elements of $S$?", - "informal_solution" : "Show that only the primes $2$ and $5$ appear seven or more times.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2014_b3", - "informal_statement" : "Let $A$ be an $m \\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2.", - "informal_solution" : "None.", - "tags" : ["linear_algebra", "number_theory"] - }, - { - "problem_name" : "putnam_2015_b3", - "informal_statement" : "Let $S$ be the set of all $2 \\times 2$ real matrices $M=\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$.", - "informal_solution" : "Show that matrices of the form $\\alpha A$ or $\\beta B$, where $A=\\left(\\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix}\\right)$, $B=\\left(\\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix}\\right)$, and $\\alpha,\\beta \\in \\mathbb{R}$, are the only matrices in $S$ that satisfy the given condition.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2015_b6", - "informal_statement" : "For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\\sqrt{2k})$. Evaluate $\\sum_{k=1}^\\infty (-1)^{k-1}\\frac{A(k)}{k}$.", - "informal_solution" : "Show that the sum converges to $\\pi^2/16$.", - "tags" : ["analysis", "number_theory"] - }, - { - "problem_name" : "putnam_2019_a5", - "informal_statement" : "Let $p$ be an odd prime number, and let $\\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\\mathbb{F}_p[x]$ be the ring of polynomials over $\\mathbb{F}_p$, and let $q(x) \\in \\mathbb{F}_p[x]$ be given by $q(x)=\\sum_{k=1}^{p-1} a_kx^k$, where $a_k=k^{(p-1)/2}\\mod{p}$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\\mathbb{F}_p[x]$.", - "informal_solution" : "Show that the answer is $\\frac{p-1}{2}$.", - "tags" : ["abstract_algebra", "number_theory", "algebra"] - } -] diff --git a/informal/batch_11.json b/informal/batch_11.json deleted file mode 100644 index d5ab8896..00000000 --- a/informal/batch_11.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_2015_b4", - "informal_statement" : "Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express\n\\[\n\\sum_{(a,b,c) \\in T} \\frac{2^a}{3^b 5^c} \n\\]\nas a rational number in lowest terms.", - "informal_solution" : "The answer is $17/21$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2013_a2", - "informal_statement" : "Let $S$ be the set of all positive integers that are \\emph{not} perfect squares. For $n$ in $S$, consider choices of integers\n$a_1, a_2, \\dots, a_r$ such that $n < a_1< a_2 < \\cdots < a_r$\nand $n \\cdot a_1 \\cdot a_2 \\cdots a_r$ is a perfect square, and\nlet $f(n)$ be the minumum of $a_r$ over all such choices. For example,\n$2 \\cdot 3 \\cdot 6$ is a perfect square, while $2 \\cdot 3$, $2 \\cdot 4$, \n$2 \\cdot 5$, $2 \\cdot 3 \\cdot 4$, $2 \\cdot 3 \\cdot 5$, $2 \\cdot 4 \\cdot 5$, and $2 \\cdot 3 \\cdot 4 \\cdot 5$ are not, and so $f(2) = 6$.\nShow that the function $f$ from $S$ to the integers is one-to-one.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2013_b2", - "informal_statement" : "Let $C = \\bigcup_{N=1}^\\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form\n\\[\nf(x) = 1 + \\sum_{n=1}^N a_n \\cos(2 \\pi n x)\n\\]\nfor which:\n\\begin{enumerate}\n\\item[(i)]\n$f(x) \\geq 0$ for all real $x$, and\n\\item[(ii)]\n$a_n = 0$ whenever $n$ is a multiple of $3$.\n\\end{enumerate}\nDetermine the maximum value of $f(0)$ as $f$ ranges through $C$, and \nprove that this maximum is attained.", - "informal_solution" : "The maximum value of $f(0)$ is $3$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2012_a3", - "informal_statement" : "Let $f: [-1, 1] \\to \\mathbb{R}$ be a continuous function such that\n\\begin{itemize}\n\\item[(i)]\n$f(x) = \\frac{2-x^2}{2} f \\left( \\frac{x^2}{2-x^2} \\right)$ for every $x$ in $[-1, 1]$,\n\\item[(ii)]\n$f(0) = 1$, and\n\\item[(iii)]\n$\\lim_{x \\to 1^-} \\frac{f(x)}{\\sqrt{1-x}}$ exists and is finite.\n\\end{itemize}\nProve that $f$ is unique, and express $f(x)$ in closed form.", - "informal_solution" : "$f(x) = \\sqrt{1-x^2}$ for all $x \\in [-1,1]$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_2011_b6", - "informal_statement" : "Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\\{0,1,2,\\dots,p-1\\}$,\n\\[\n\\sum_{k=0}^{p-1} k! n^k \\qquad \\mbox{is not divisible by $p$.}\n\\]", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2010_a2", - "informal_statement" : "Find all differentiable functions $f:\\mathbb{R} \\to \\mathbb{R}$ such that\n\\[\nf'(x) = \\frac{f(x+n)-f(x)}{n}\n\\]\nfor all real numbers $x$ and all positive integers $n$.", - "informal_solution" : "The solution consists of all functions of the form $f(x) = cx+d$ for some real numbers $c,d$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2010_a6", - "informal_statement" : "Let $f:[0,\\infty)\\to \\mathbb{R}$ be a strictly decreasing continuous function\nsuch that $\\lim_{x\\to\\infty} f(x) = 0$. Prove that\n$\\int_0^\\infty \\frac{f(x)-f(x+1)}{f(x)}\\,dx$ diverges.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_12.json b/informal/batch_12.json deleted file mode 100644 index fdb60255..00000000 --- a/informal/batch_12.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_2008_a1", - "informal_statement" : "Let $f:\\mathbb{R}^2 \\to \\mathbb{R}$ be a function such that $f(x,y)+f(y,z)+f(z,x)=0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g:\\mathbb{R} \\to \\mathbb{R}$ such that $f(x,y)=g(x)-g(y)$ for all real numbers $x$ and $y$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2008_b2", - "informal_statement" : "Let $F_0(x)=\\ln x$. For $n \\geq 0$ and $x>0$, let $F_{n+1}(x)=\\int_0^x F_n(t)\\,dt$. Evaluate $\\lim_{n \\to \\infty} \\frac{n!F_n(1)}{\\ln n}$.", - "informal_solution" : "Show that the desired limit is $-1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2010_a1", - "informal_statement" : "Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\\{1,2,3,6\\},\\{4,8\\},\\{5,7\\}$ shows that the largest $k$ is \\emph{at least} $3$.]", - "informal_solution" : "Show that the largest such $k$ is $\\lceil \\frac{n}{2} \\rceil$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2010_b4", - "informal_statement" : "Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which $p(x)q(x+1)-p(x+1)q(x)=1$.", - "informal_solution" : "Show that the pairs $(p,q)$ satisfying the given equation are those of the form $p(x)=ax+b,q(x)=cx+d$ for $a,b,c,d \\in \\mathbb{R}$ such that $bc-ad=1$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2010_b6", - "informal_statement" : "Let $A$ be an $n \\times n$ matrix of real numbers for some $n \\geq 1$. For each positive integer $k$, let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\\dots,n+1$, then $A^k=A^{[k]}$ for all $k \\geq 1$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2012_a4", - "informal_statement" : "Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2013_a4", - "informal_statement" : "A finite collection of digits $0$ and $1$ is written around a circle. An \\emph{arc} of length $L \\geq 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w$, respectively. Assume that $|Z(w)-Z(w')| \\leq 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\\dots,w_k$ have the property that $Z=\\frac{1}{k}\\sum_{j=1}^k Z(w_j)$ and $N=\\frac{1}{k}\\sum_{j=1}^k N(w_j)$ are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - } -] diff --git a/informal/batch_13.json b/informal/batch_13.json deleted file mode 100644 index 7bf221e1..00000000 --- a/informal/batch_13.json +++ /dev/null @@ -1,46 +0,0 @@ -[ - { - "problem_name" : "putnam_2014_b6", - "informal_statement" : "Let $f: [0,1] \\to \\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $\\left| f(x) - f(y) \\right| \\leq K \\left| x - y \\right|$ for all $x,y \\in [0,1]$. Suppose also that for each rational number $r \\in [0,1]$, there exist integers $a$ and $b$ such that $f(r) = a + br$. Prove that there exist finitely many intervals $I_1, \\dots, I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1] = \\bigcup_{i=1}^n I_i$.", - "tags" : ["analysis", "number_theory"] - }, - { - "problem_name" : "putnam_2011_b3", - "informal_statement" : "Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at $0$?", - "informal_solution" : "Prove that $f$ is differentiable.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2011_b5", - "informal_statement" : "Let $a_1, a_2, \\dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$,\n\\[\n\\int_{-\\infty}^\\infty \\left( \\sum_{i=1}^n \\frac{1}{1 + (x-a_i)^2} \\right)^2\\,dx \\leq An.\n\\]\nProve there is a constant $B>0$ such that for all $n$,\n\\[\n\\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \\geq Bn^3.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2009_a1", - "informal_statement" : "Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?", - "informal_solution" : "Prove that $f$ is identically $0$.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_2009_a4", - "informal_statement" : "Let $S$ be a set of rational numbers such that\n\\begin{enumerate}\n\\item[(a)] $0 \\in S$;\n\\item[(b)] If $x \\in S$ then $x+1\\in S$ and $x-1\\in S$; and\n\\item[(c)] If $x\\in S$ and $x\\not\\in\\{0,1\\}$, then $\\frac{1}{x(x-1)}\\in S$.\n\\end{enumerate}\nMust $S$ contain all rational numbers?", - "informal_solution" : "Prove that $S$ need not contain all rationals.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2009_b3", - "informal_statement" : "Call a subset $S$ of $\\{1, 2, \\dots, n\\}$ \\emph{mediocre} if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\\{1,2,\\dots,n\\}$. [For instance, every subset of $\\{1,2,3\\}$ except $\\{1,3\\}$ is mediocre, so $A(3) = 7$.] Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.", - "informal_solution" : "Show that the answer is $n = 2^k - 1$ for some integer $k$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2009_b5", - "informal_statement" : "Let $f: (1, \\infty) \\to \\mathbb{R}$ be a differentiable function such that\n\\[\n f'(x) = \\frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}\n\\qquad \\mbox{for all $x>1$.}\n\\]\nProve that $\\lim_{x \\to \\infty} f(x) = \\infty$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2009_b6", - "informal_statement" : "Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \\dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\\,\\mathrm{mod}\\,c$ for some earlier positive terms $b$ and $c$. [Here $b\\,\\mathrm{mod}\\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \\leq (b\\,\\mathrm{mod}\\,c) < c$.]", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_14.json b/informal/batch_14.json deleted file mode 100644 index 2a299cfb..00000000 --- a/informal/batch_14.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_2009_a5", - "informal_statement" : "Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}?", - "informal_solution" : "Show that the answer is no such finite abelian group exists.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_2009_b1", - "informal_statement" : "Show that every positive rational number can be written as a quotient of products of factorails of (not necessarily distinct) primes. For example, 10/9 = (2! * 5!)/(3! * 3! * 3!).", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2008_a4", - "informal_statement" : "Define $f : \\mathbb{R} \\to \\mathbb{R} by $f(x) = x$ if $x \\leq e$ and $f(x) = x * f(\\ln(x))$ if $x > e$. Does $\\sum_{n=1}^{\\infty} 1/(f(n))$ converge?", - "informal_solution" : "Show that the sum does not converge.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2008_b5", - "informal_statement" : "Find all continuously differentiable functions f : \\mathbb{R} \\to \\mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$.", - "informal_solution" : "Show that the solution is the set of all functions of the form n + x, n - x where n is any integer.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2008_b6", - "informal_statement" : "Let $n$ and $k$ be positive integers. Say that a permutation $\\sigma$ of $\\{1,2,\\dots,n\\} is $k-limited$ if \\|\\sigma(i) - i\\| \\leq k$ for all $i$. Prove that the number of $k-limited$ permutations $\\{1,2,\\dots,n\\}$ is odd if and only if $n \\equiv 0$ or $1 (mod 2k+1)$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2007_a1", - "informal_statement" : "Find all values of $\\alpha$ for which the curves $y = \\alpha*x^2 + \\alpha*x + 1/24$ and $x = \\alpha*y^2 + \\alpha*y + 1/24$ are tangent to each other.", - "informal_solution" : "Show that the solution is the set \\{2/3, 3/2, (13 + \\sqrt{601})/12, (13 - \\sqrt{601})/12}.", - "tags" : ["algebra", "geometry"] - }, - { - "problem_name" : "putnam_2007_b1", - "informal_statement" : "Let $f$ be a nonconstant polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n) + 1)$ if and only if $n = 1$", - "informal_solution" : "None.", - "tags" : ["algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_15.json b/informal/batch_15.json deleted file mode 100644 index 2886cbc3..00000000 --- a/informal/batch_15.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_2008_a6", - "informal_statement" : "Prove that there exists a constant $c>0$ such that in every nontrivial finite group $G$ there exists a sequence of length at most $c \\log |G|$ with the property that each element of $G$ equals the product of some subsequence. (The elements of $G$ in the sequence are not required to be distinct. A \\emph{subsequence} of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $4, 4, 2$ is a subsequence of $2, 4, 6, 4, 2$, but $2, 2, 4$ is not.)", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_2008_b4", - "informal_statement" : "Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients such that $h(0), h(1), \\dots, h(p^2-1)$ are distinct modulo $p^2$. Show that $h(0), h(1), \\dots, h(p^3-1)$ are distinct modulo $p^3$.", - "tags" : ["algebra", "number_theory"] - }, - { - "problem_name" : "putnam_2007_a4", - "informal_statement" : "A \\emph{repunit} is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is a repunit, then so is $f(n)$.", - "informal_solution" : "Show that the desired polynomials $f$ are those of the form\n\\[\nf(n) = \\frac{1}{9}(10^c (9n+1)^d - 1)\n\\]\nfor integers $d \\geq 0$ and $c \\geq 1-d$.", - "tags" : ["analysis", "algebra", "number_theory"] - }, - { - "problem_name" : "putnam_2007_b2", - "informal_statement" : "Suppose that $f: [0,1] \\to \\mathbb{R}$ has a continuous derivative and that $\\int_0^1 f(x)\\,dx = 0$. Prove that for every $\\alpha \\in (0,1)$,\n\\[\n\\left| \\int_0^\\alpha f(x)\\,dx \\right| \\leq \\frac{1}{8} \\max_{0 \\leq x\n\\leq 1} |f'(x)|.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2007_b3", - "informal_statement" : "Let $x_0 = 1$ and for $n \\geq 0$, let $x_{n+1} = 3x_n + \\lfloor x_n \\sqrt{5} \\rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\\lfloor a \\rfloor$ means the largest integer $\\leq a$.)", - "informal_solution" : "Prove that $x_{2007} = \\frac{2^{2006}}{\\sqrt{5}}(\\alpha^{3997}-\\alpha^{-3997})$, where $\\alpha = \\frac{1+\\sqrt{5}}{2}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2007_b4", - "informal_statement" : "Let $n$ be a positive integer. Find the number of pairs $P, Q$ of polynomials with real coefficients such that\n\\[\n(P(X))^2 + (Q(X))^2 = X^{2n} + 1\n\\]\nand $\\deg P > \\deg Q$.", - "informal_solution" : "Show that the number of pairs is $2^{n+1}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2007_b5", - "informal_statement" : "Let $k$ be a positive integer. Prove that there exist polynomials $P_0(n), P_1(n), \\dots, P_{k-1}(n)$ (which may depend on $k$) such that for any integer $n$,\n\\[\n\\left\\lfloor \\frac{n}{k} \\right\\rfloor^k = P_0(n) + P_1(n) \\left\\lfloor\n\\frac{n}{k} \\right\\rfloor + \\cdots + P_{k-1}(n) \\left\\lfloor \\frac{n}{k}\n\\right\\rfloor^{k-1}.\n\\]\n($\\lfloor a \\rfloor$ means the largest integer $\\leq a$.)", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2006_a3", - "informal_statement" : "Let $1, 2, 3, \\dots, 2005, 2006, 2007, 2009, 2012, 2016, \\dots$ be a sequence defined by $x_k = k$ for $k=1, 2, \\dots, 2006$ and $x_{k+1} = x_k + x_{k-2005}$ for $k \\geq 2006$. Show that the sequence has $2005$ consecutive terms each divisible by $2006$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2006_b2", - "informal_statement" : "Prove that, for every set $X = \\{x_1, x_2, \\dots, x_n\\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that\n\\[\n\\left| m + \\sum_{s \\in S} s \\right| \\leq \\frac{1}{n+1}.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2006_b5", - "informal_statement" : "For each continuous function $f: [0,1] \\to \\mathbb{R}$, let $I(f) = \\int_0^1 x^2 f(x)\\,dx$ and $J(x) = \\int_0^1 x \\left(f(x)\\right)^2\\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$.", - "informal_solution" : "Show that the answer is \\frac{1}{16}.", - "tags" : ["analysis", "algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_16.json b/informal/batch_16.json deleted file mode 100644 index abe0e3fe..00000000 --- a/informal/batch_16.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_2005_a3", - "informal_statement" : "Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=p(z)/z^{n/2}$. Show that all zeros of $g'(z)=0$ have absolute value $1$.", - "informal_solution" : "None.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_2005_a4", - "informal_statement" : "Let $H$ be an $n \\times n$ matrix all of whose entries are $\\pm 1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a \\times b$ submatrix whose entries are all $1$. Show that $ab \\leq n$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2005_b1", - "informal_statement" : "Find a nonzero polynomial $P(x,y)$ such that $P(\\lfloor a \\rfloor,\\lfloor 2a \\rfloor)=0$ for all real numbers $a$. (Note: $\\lfloor \\nu \\rfloor$ is the greatest integer less than or equal to $\\nu$.)", - "informal_solution" : "Show that $P(x,y)=(y-2x)(y-2x-1)$ works.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2005_b2", - "informal_statement" : "Find all positive integers $n,k_1,\\dots,k_n$ such that $k_1+\\cdots+k_n=5n-4$ and $\\frac{1}{k_1}+\\cdots+\\frac{1}{k_n}=1$.", - "informal_solution" : "Show that the solutions are $n=1$ and $k_1=1$, $n=3$ and $(k_1,k_2,k_3)$ is a permutation of $(2,3,6)$, and $n=4$ and $(k_1,k_2,k_3,k_4)=(4,4,4,4)$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2005_b3", - "informal_statement" : "Find all differentiable functions $f:(0,\\infty) \\to (0,\\infty)$ for which there is a positive real number $a$ such that $f'(\\frac{a}{x})=\\frac{x}{f(x)}$ for all $x>0$.", - "informal_solution" : "Show that the functions are precisely $f(x)=cx^d$ for $c,d>0$ arbitrary except that we must take $c=1$ in case $d=1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2006_a4", - "informal_statement" : "Let $S=\\{1,2,\\dots,n\\}$ for some integer $n>1$. Say a permutation $\\pi$ of $S$ has a \\emph{local maximum} at $k \\in S$ if\n\\begin{enumerate}\n\\item[(i)] $\\pi(k)>\\pi(k+1)$ for $k=1$;\n\\item[(ii)] $\\pi(k-1)<\\pi(k)$ and $\\pi(k)>\\pi(k+1)$ for $1 1$,\n\\[\n\\frac{1}{2ne} < \\frac{1}{e} - \\left( 1 - \\frac{1}{n} \\right)^n\n< \\frac{1}{ne}.\n\\]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2002_b5", - "informal_statement" : "A palindrome in base $b$ is a positive integer whose base-$b$\ndigits read the same backwards and forwards; for example,\n$2002$ is a 4-digit palindrome in base 10. Note that 200 is not\na palindrome in base 10, but it is the 3-digit palindrome\n242 in base 9, and 404 in base 7. Prove that there is an integer\nwhich is a 3-digit palindrome in base $b$ for at least 2002\ndifferent values of $b$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2002_b6", - "informal_statement" : "Let $p$ be a prime number. Prove that the determinant of the matrix\n\\[\n\\begin{pmatrix}\nx & y & z \\\\\nx^p & y^p & z^p \\\\\nx^{p^2} & y^{p^2} & z^{p^2}\n\\end{pmatrix}\n\\]\nis congruent modulo $p$ to a product of polynomials of the form\n$ax+by+cz$, where $a,b,c$ are integers. (We say two integer\npolynomials are congruent modulo $p$ if corresponding coefficients\nare congruent modulo $p$.)", - "informal_solution" : "None.", - "tags" : ["linear_algebra", "number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2001_a3", - "informal_statement" : "For each integer $m$, consider the polynomial\n\\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\\] For what values of $m$ is $P_m(x)$\nthe product of two non-constant polynomials with integer coefficients?", - "informal_solution" : "$P_m(x)$ factors into two nonconstant polynomials over\nthe integers if and only if $m$ is either a square or twice a square.", - "tags" : ["algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_19621973hard.json b/informal/batch_19621973hard.json deleted file mode 100644 index 978751a2..00000000 --- a/informal/batch_19621973hard.json +++ /dev/null @@ -1,152 +0,0 @@ -[ - { - "problem_name" : "putnam_1962_a3", - "informal_statement" : "Let $\\triangle ABC$ be a triangle in the Euclidean plane, with points $P$, $Q$, and $R$ lying on segments $\\overline{BC}$, $\\overline{CA}$, $\\overline{AB}$ respectively such that $$\\frac{AQ}{QC} = \\frac{BR}{RA} = \\frac{CP}{PB} = k$$ for some positive constant $k$. If $\\triangle UVW$ is the triangle formed by parts of segments $\\overline{AP}$, $\\overline{BQ}$, and $\\overline{CR}$, prove that $$\\frac{[\\triangle UVW]}{[\\triangle ABC]} = \\frac{(k - 1)^2}{k^2 + k + 1},$$ where $[\\triangle]$ denotes the area of the triangle $\\triangle$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1962_b2", - "informal_statement" : "Let $\\mathbb{S}$ be the set of all subsets of the natural numbers. Prove the existence of a function $f : \\mathbb{R} \\to \\mathbb{S}$ such that $f(a) \\subset f(b)$ whenever $a < b$.", - "informal_solution" : "None.", - "tags" : ["set_theory"] - }, - { - "problem_name" : "putnam_1962_b3", - "informal_statement" : "Let $S$ be a convex region in the Euclidean plane, containing the origin, for which every ray from the origin has at least one point outside $S$. Assuming that either the origin is an interior point of $S$ or $S$ is topologically closed, prove that $S$ is bounded.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_a3", - "informal_statement" : "Find an integral formula (i.e., a function $z$ such that $y(x) = \\int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\\delta (\\delta - 1) (\\delta - 2) \\cdots (\\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \\cdots = y^{(n-1)}(1) = 0$, where $n \\in \\mathbb{N}$, $f$ is continuous for all $x \\ge 1$, and $\\delta$ denotes $x\\frac{d}{dx}$.", - "informal_solution" : "Show that the solution is $$y(x) = \\int_{1}^{x} \\frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_a6", - "informal_statement" : "Let $U$ and $V$ be distinct points on an ellipse, with $M$ the midpoint of chord $\\overline{UV}$, and let $\\overline{AB}$ and $\\overline{CD}$ be any two other chords through $M$. If line $UV$ intersects line $AC$ at $P$ and line $BD$ at $Q$, prove that $M$ is the midpoint of segment $\\overline{PQ}$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1965_a1", - "informal_statement" : "Let $\\triangle ABC$ satisfy $\\angle CAB < \\angle BCA < \\frac{\\pi}{2} < \\angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\\angle CAB$.", - "informal_solution" : "Show that the solution is $\\angle CAB = \\frac{\\pi}{15}$.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1965_b1", - "informal_statement" : "Find $$\\lim_{n \\to \\infty} \\int_{0}^{1} \\int_{0}^{1} \\cdots \\int_{0}^{1} \\cos^2\\left(\\frac{\\pi}{2n}(x_1 + x_2 + \\cdots + x_n)\\right) dx_1 dx_2 \\cdots dx_n.$$", - "informal_solution" : "Show that the limit is $\frac{1}{2}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1966_a2", - "informal_statement" : "Let $a$, $b$, and $c$ be the side lengths of a triangle with inradius $r$. If $p = \\frac{a + b + c}{2}$, show that $$\\frac{1}{(p - a)^2} + \\frac{1}{(p - b)^2} + \\frac{1}{(p - c)^2} \\ge \\frac{1}{r^2}.$$", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1966_b1", - "informal_statement" : "If a convex polygon $L$ is contained entirely within a square of side length $1$, prove that the sum of the squares of the side lengths of $L$ is no greater than $4$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1966_b5", - "informal_statement" : "Prove that any set of $n \\ge 3$ distinct points in the Euclidean plane, no three of which are collinear, forms the vertex set of some simple (non-self-intersecting) closed polygon.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1967_a5", - "informal_statement" : "Prove that any convex region in the Euclidean plane with area greater than $\\pi/4$ contains a pair of points exactly $1$ unit apart.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1967_b1", - "informal_statement" : "Let $\\hexagon ABCDEF$ be a hexagon inscribed in a circle of radius $r$. If $AB = CD = EF = r$, prove that the midpoints of $\\overline{BC}$, $\\overline{DE}$, and $\\overline{FA}$ form the vertices of an equilateral triangle.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1967_b5", - "informal_statement" : "For any positive integer $n$, prove that the sum of the first $n$ terms of the bimonial expansion of $(2 - 1)^{-n}$ (starting with the maximal exponent of $2$) is $\\frac{1}{2}.$", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1969_a1", - "informal_statement" : "What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?", - "informal_solution" : "Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers.", - "tags" : ["algebra", "set_theory"] - }, - { - "problem_name" : "putnam_1969_a5", - "informal_statement" : "Consider the system of differential equations $$\\frac{dx}{dt} = -2y + u(t), \\frac{dy}{dt} = -2x + u(t)$$ for some continuous function $u(t)$. Prove that, if $x(0) \\ne y(0)$, the solution will never pass through $(0, 0)$ regardless of the choice of $u(t)$, and if $x(0) = y(0)$, a suitable $u(t)$ can be chosen for any $T > 0$ so that $(x(T), y(T)) = (0, 0)$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_a2", - "informal_statement" : "Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \\delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$", - "informal_solution" : "None.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1970_b2", - "informal_statement" : "Let $H$ be a polynomial of degree at most $3$ and $T$ be a positive real number. Show that the average value of $H(t)$ over the interval $[-T, T]$ equals the average of $H\\left(-\\frac{T}{\\sqrt{3}}\\right)$ and $H\\left(\\frac{T}{\\sqrt{3}}\\right)$.", - "informal_solution" : "None.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1970_b4", - "informal_statement" : "Let $x : \\mathbb{R} \\to \\mathbb{R}$ be a twice differentiable function satisfying $x(1) - x(0) = 1$, $x'(0) = x'(1) = 0$, and $|x'(t)| \\le \\frac{3}{2}$ for all $t \\in (0, 1)$. Prove that there exists some $t \\in [0, 1]$ such that $|x''(t)| \\ge \\frac{9}{2}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_b6", - "informal_statement" : "Prove that if a quadrilateral with side lengths $a$, $b$, $c$, and $d$ and area $\\sqrt{abcd}$ can be circumscribed to a circle (i.e., a circle can be inscribed in it), then it must be cyclic (i.e., it can be inscribed in a circle).", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1971_a5", - "informal_statement" : "After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.", - "informal_solution" : "Show that the solution is $a = 11$ and $b = 8$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1971_b3", - "informal_statement" : "If two cars travel around a track at constant speeds of one lap per hour, starting from the same point but at different times, prove that the total amount of time for which the first car has completed exactly twice as many laps as the second is exactly $1$ hour.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1972_b2", - "informal_statement" : "Let $x : \\mathbb{R} \\to \\mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$.", - "informal_solution" : "Show that the maximum possible time is $t = \frac{2s}{v}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1972_b5", - "informal_statement" : "Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\\angle ABC = \\angle CDA$ and $\\angle BCD = \\angle DAB$, then $AB = CD$ and $AD = BC$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1973_a1", - "informal_statement" : "Let $\\triangle ABC$ be any triangle in the Euclidean plane, and let points $X$, $Y$, and $Z$ lie on sides $\\overline{BC}$, $\\overline{CA}$, and $\\overline{AB}$ respectively. If $BX \\le XC$, $CY \\le YA$, and $AZ \\le ZB$, prove that $[\\triangle XYZ] \\ge \\frac{1}{4} [\\triangle ABC]$. Regardless of this constraint on $X$, $Y$, and $Z$, prove that one of $[\\triangle AZY]$, $[\\triangle BXZ]$, or $[\\triangle CYX]$ is less than or equal to $[\\triangle XYZ]$. (Here, $[\\triangle]$ denotes the area of triangle $\\triangle$.)", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1973_a2", - "informal_statement" : "Consider an infinite series whose $n$th term is given by $\\pm \\frac{1}{n}$, where the actual values of the $\\pm$ signs repeat in blocks of $8$ (so the $\\frac{1}{9}$ term has the same sign as the $\\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that being balanced is a sufficient condition for the sequence to converge. Is being balanced also necessary for the sequence to converge?", - "informal_solution" : "Show that the condition is necessary.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_19741984hard.json b/informal/batch_19741984hard.json deleted file mode 100644 index f970bffd..00000000 --- a/informal/batch_19741984hard.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1974_b1", - "informal_statement" : "Prove that the optimal configuration of 5 (not necessarily distinct) points $p_1, \\dots, p_5$ on the unit circle which maximizes the sum of the ten distances \\[\\Sigma_{i < j}, d(p_i, p_j) \\] is the one which evenly spaces the points like a regular pentagon.", - "informal_solution" : "None.", - "tags" : ["algebra", "geometry"] - }, - { - "problem_name" : "putnam_1974_b2", - "informal_statement" : "Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \\to 0$ as $x \\to +\\infty$, then $y(x)$ and $y'(x) \\to 0$ as $x \\to +\\infty$.", - "informal_solution" : "None", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1975_a4", - "informal_statement" : "Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\\theta = e^{2\\pi i/n}$. Expression $(1 - \\theta)^{-1}$ explicitly as a polynomial in $\\theta$ \\[ a_k \\theta^k + a_{k-1}\\theta^{k-1} + \\dots + a_1\\theta + a_0\\], with integer coefficients $a_i$.", - "informal_solution" : "Show that the solution is the polynomial $0 + \\theta + \\theta^3 + \\dots + \\theta^{m-2}$, alternating consecutive coefficients between 0 and 1.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1975_a5", - "informal_statement" : "On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \\[y'' = f(x)y\\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \\[z(x) = c \\sqrt{y_1(x)y_2(x)}\\] satisfies the equation \\[z'' + 1/z^3 = f(x)z.\\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1975_b2", - "informal_statement" : "In three-dimensional Euclidean space, define a \\emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \\emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \\dots$ of slabs of thicknesses $d_1, d_2, \\dots,$ respectively, such that $\\Sigma_{i=1}^{\\infty} d_i$ converges, prove that there is some point in the space which is not contained in any of the slabs. ", - "informal_solution" : "None.", - "tags" : ["analysis", "geometry"] - }, - { - "problem_name" : "putnam_1975_b6", - "informal_statement" : "Show that if $s_n = 1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1976_b3", - "informal_statement" : "Suppose that we have $n$ events $A_1, \\dots, A_n$, each of which has probability at least $1 - a$ of occufring, where $a < 1/4$. Further suppose that $A_i$ and $A_j$ are mutually independent if $|i-j| > 1$, although $A_i$ and $A_{i+1}$ may be dependent. Assume as known that the recurrence $u_{k+1} = u_k - au_{k-1}$, $u_0 = 1, u_1 = 1-a$ defines positive real numbers $u_k$ for $k = 0,1,\\dots$. Show that the probability of all $A_1, \\dots, A_n$ occurring is at least $u_n$.", - "informal_solution" : "None.", - "tags" : ["probability"] - }, - { - "problem_name" : "putnam_1979_b5", - "informal_statement" : "In the plane, let $C$ be a closed convex set that contains $(0,0) but no other point with integer coordinations. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C) \\leq 4$.", - "informal_solution" : "", - "tags" : ["geometry", "analysis"] - }, - { - "problem_name" : "putnam_1982_a4", - "informal_statement" : "Assume that the system of simultaneous differentiable equations \\[y' = -z^3, z' = y^3\\] with the initial conditions $y(0) = 1, z(0) = 0$ has a unique solution $y = f(x), z = g(x)$ defined for all real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, \\[f(x) + L = f(x), g(x + L) = g(x).\\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_19851997hard.json b/informal/batch_19851997hard.json deleted file mode 100644 index ddf98056..00000000 --- a/informal/batch_19851997hard.json +++ /dev/null @@ -1,80 +0,0 @@ -[ - { - "problem_name" : "putnam_1986_a5", - "informal_statement" : "Suppose $f_1(x),f_2(x),\\dots,f_n(x)$ are functions of $n$ real variables $x=(x_1,\\dots,x_n)$ with continuous second-order partial derivatives everywhere on $\\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that $\\frac{\\partial f_i}{\\partial x_j}-\\frac{\\partial f_j}{\\partial x_i}=c_{ij}$ for all $i$ and $j$, $1 \\leq i \\leq n$, $1 \\leq j \\leq n$. Prove that there is a function $g(x)$ on $\\mathbb{R}^n$ such that $f_i+\\partial g/\\partial x_i$ is linear for all $i$, $1 \\leq i \\leq n$. (A linear function is one of the form $a_0+a_1x_1+a_2x_2+\\dots+a_nx_n$.)", - "informal_solution" : "None.", - "tags" : ["analysis", "linear_algebra"] - }, - { - "problem_name" : "putnam_1986_b1", - "informal_statement" : "Inscribe a rectangle of base $b$ and height $h$ and an isosceles triangle of base $b$ (against a corresponding side of the rectangle and pointed in the other direction) in a circle of radius one. For what value of $h$ do the rectangle and triangle have the same area?", - "informal_solution" : "Show that the only such value of $h$ is $2/5$.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_1987_a5", - "informal_statement" : "Let $\\vec{G}(x,y)=\\left(\\frac{-y}{x^2+4y^2},\\frac{x}{x^2+4y^2},0\\right)$. Prove or disprove that there is a vector-valued function $\\vec{F}(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z))$ with the following properties:\n\\begin{enumerate}\n\\item[(i)] $M$, $N$, $P$ have continuous partial derivatives for all $(x,y,z) \\neq (0,0,0)$;\n\\item[(ii)] $\\text{Curl}\\,\\vec{F}=\\vec{0}$ for all $(x,y,z) \\neq (0,0,0)$;\n\\item[(iii)] $\\vec{F}(x,y,0)=\\vec{G}(x,y)$.\n\\end{enumerate}", - "informal_solution" : "Show that there is no such $\\vec{F}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1987_b3", - "informal_statement" : "Let $F$ be a field in which $1+1 \\neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\\left(\\frac{r^2-1}{r^2+1},\\frac{2r}{r^2+1}\\right)$, where $r$ runs through the elements of $F$ such that $r^2 \\neq -1$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1988_a1", - "informal_statement" : "Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \\leq 1$ and $|y| \\leq 1$. Find the area of $R$.", - "informal_solution" : "Show that the area of $R$ is $6$.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1989_b1", - "informal_statement" : "A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $(a\\sqrt{b}+c)/d$, where $a$, $b$, $c$, $d$ are integers and $b$, $d$ are positive.", - "informal_solution" : "Show that the probability is $(4\\sqrt{2}-5)/3$.", - "tags" : ["probability", "geometry"] - }, - { - "problem_name" : "putnam_1989_b6", - "informal_statement" : "Let $(x_1,x_2,\\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0 a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?", - "informal_solution" : "Prove that the possible costs are $1/3 < c \\leq 1.$", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_2010_b3", - "informal_statement" : "There are $2010$ boxes labeled $B_1, B_2, \\dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \\emph{exactly} $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?", - "informal_solution" : "Prove that it is possible if and only if $n \\geq 1005$.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_2.json b/informal/batch_2.json deleted file mode 100644 index b7d710ee..00000000 --- a/informal/batch_2.json +++ /dev/null @@ -1,80 +0,0 @@ -[ - { - "problem_name" : "putnam_2020_b6", - "informal_statement" : "Let $n$ be a positive integer. Prove that $\\sum_{k=1}^n(-1)^{\\lfloor k(\\sqrt{2}-1) \\rfloor} \\geq 0$.", - "informal_solution" : "None", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2021_a2", - "informal_statement" : "For every positive real number $x$, let $g(x)=\\lim_{r \\to 0}((x+1)^{r+1}-x^{r+1})^\\frac{1}{r}$. Find $\\lim_{x \\to \\infty}\\frac{g(x)}{x}$.", - "informal_solution" : "Show that the limit is $e$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2021_a5", - "informal_statement" : "Let $A$ be the set of all integers $n$ such that $1 \\leq n \\leq 2021$ and $\\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\\sum_{n \\in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.", - "informal_solution" : "Show that the values of $j$ in question are those not divisible by either $42$ or $46$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2021_a6", - "informal_statement" : "Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?", - "informal_solution" : "Show that it does follow that $P(2)$ is a composite integer.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2021_b2", - "informal_statement" : "Determine the maximum value of the sum $S = \\sum_{n=1}^\\infty \\frac{n}{2^n}(a_1a_2 \\dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\\dots$ of nonnegative real numbers satisfying $\\sum_{k=1}^\\infty a_k=1$.", - "informal_solution" : "Show that the answer is $2/3$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2022_a6", - "informal_statement" : "Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\\dots,x_{2n}$ with $-10$.", - "informal_solution" : "Show that the only such functions are the functions $f(x)=\\frac{1}{1+cx}$ for some $c \\geq 0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2023_b4", - "informal_statement" : "For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \\geq t_0$ by the following properties:\n\\begin{enumerate}\n\\item[(a)] $f(t)$ is continuous for $t \\geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\\dots,t_n$;\n\\item[(b)] $f(t_0)=1/2$;\n\\item[(c)] $\\lim_{t \\to t_k^+} f'(t)=0$ for $0 \\leq k \\leq n$;\n\\item[(d)] For $0 \\leq k \\leq n-1$, we have $f''(t)=k+1$ when $t_kt_n$.\n\\end{enumerate}\nConsidering all choices of $n$ and $t_0,t_1,\\dots,t_n$ such that $t_k \\geq t_{k-1}+1$ for $1 \\leq k \\leq n$, what is the least possible value of $T$ for which $f(t_0+T)=2023$?", - "informal_solution" : "Show that the minimum value of $T$ is $29$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2023_b5", - "informal_statement" : "Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\\pi:\\{1,2,\\dots,n\\} \\to \\{1,2,\\dots,n\\}$ such that $\\pi(\\pi(k)) \\equiv mk \\pmod{n}$ for all $k \\in \\{1,2,\\dots,n\\}$.", - "informal_solution" : "Show that the desired property holds if and only if $n=1$ or $n \\equiv 2 \\pmod{4}$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2023_b6", - "informal_statement" : "Let $n$ be a positive integer. For $i$ and $j$ in $\\{1,2,\\dots,n\\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \\begin{bmatrix}\n6 & 3 & 2 & 2 & 2 \\\\\n3 & 0 & 1 & 0 & 1 \\\\\n2 & 1 & 0 & 0 & 1 \\\\\n2 & 0 & 0 & 0 & 1 \\\\\n2 & 1 & 1 & 1 & 2\n\\end{bmatrix}$. Compute the determinant of $S$.", - "informal_solution" : "Show that the determinant equals $(-1)^{\\lceil n/2 \\rceil-1}2\\lceil\\frac{n}{2}\\rceil$.", - "tags" : ["linear_algebra"] - } -] diff --git a/informal/batch_20.json b/informal/batch_20.json deleted file mode 100644 index 65d993c2..00000000 --- a/informal/batch_20.json +++ /dev/null @@ -1,50 +0,0 @@ -[ - { - "problem_name" : "putnam_2000_a1", - "informal_statement" : "Let $A$ be a positive real number. What are the possible values of $\\sum_{j=0}^\\infty x_j^2$, given that $x_0,x_1,\\ldots$ are positive numbers for which $\\sum_{j=0}^\\infty x_j=A$?", - "informal_solution" : "Show that the possible values comprise the interval $(0,A^2)$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2000_a4", - "informal_statement" : "Show that the improper integral $\\lim_{B \\to \\infty} \\int_0^B \\sin(x)\\sin(x^2)\\,dx$ converges.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2000_a5", - "informal_statement" : "Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2001_b1", - "informal_statement" : "Let $n$ be an even positive integer. Write the numbers $1,2,\\ldots,n^2$ in the squares of an $n \\times n$ grid so that the $k$-th row, from left to right, is $(k-1)n+1,(k-1)n+2,\\ldots,(k-1)n+n$. Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2001_b2", - "informal_statement" : "Find all pairs of real numbers $(x,y)$ satisfying the system of equations\n\\begin{align*}\n\\frac{1}{x}+\\frac{1}{2y}&=(x^2+3y^2)(3x^2+y^2) \\\\\n\\frac{1}{x}-\\frac{1}{2y}&=2(y^4-x^4).\n\\end{align*}", - "informal_solution" : "Show that $x=(3^{1/5}+1)/2$ and $y=(3^{1/5}-1)/2$ is the unique solution satisfying the given equations.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2001_b3", - "informal_statement" : "For any positive integer $n$, let $\\langle n \\rangle$ denote the closest integer to $\\sqrt{n}$. Evaluate $\\sum_{n=1}^\\infty \\frac{2^{\\langle n \\rangle}+2^{-\\langle n \\rangle}}{2^n}$.", - "informal_solution" : "Show that the sum is $3$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2001_b5", - "informal_statement" : "Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2001_b6", - "informal_statement" : "Assume that $(a_n)_{n \\geq 1}$ is an increasing sequence of positive real numbers such that $\\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\\ldots,n-1$?", - "informal_solution" : "Show that the answer is yes, there must exist infinitely many such $n$.", - "tags" : ["analysis"] - } -] diff --git a/informal/batch_20112023hard.json b/informal/batch_20112023hard.json deleted file mode 100644 index 105d21bd..00000000 --- a/informal/batch_20112023hard.json +++ /dev/null @@ -1,122 +0,0 @@ -[ - { - "problem_name" : "putnam_2011_a1", - "informal_statement" : "Define a \\emph{growing spiral} in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\\dots,P_n$ such that $n \\geq 2$ and:\n\\begin{itemize}\n\\item the directed line segments $P_0P_1,P_1P_2,\\dots,P_{n-1}P_n$ are in the successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc.;\n\\item the lengths of these line segments are positive and strictly increasing.\n\\end{itemize}\nHow many of the points $(x,y)$ with integer coordinates $0 \\leq x \\leq 2011,0 \\leq y \\leq 2011$ \\emph{cannot} be the last point, $P_n$ of any growing spiral?", - "informal_solution" : "Show that there are $10053$ excluded points.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_2011_a5", - "informal_statement" : "Let $F:\\mathbb{R}^2 \\to \\mathbb{R}$ and $g:\\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable functions with the following properties:\n\\begin{itemize}\n\\item $F(u,u)=0$ for every $u \\in \\mathbb{R}$;\n\\item for every $x \\in \\mathbb{R}$, $g(x)>0$ and $x^2g(x) \\leq 1$;\n\\item for every $(u,v) \\in \\mathbb{R}^2$, the vector $\\nabla F(u,v)$ is either $\\mathbf{0}$ or parallel to the vector $\\langle g(u),-g(v) \\rangle$.\n\\end{itemize}\nProve that there exists a constant $C$ such that for every $n \\geq 2$ and any $x_1,\\dots,x_{n+1} \\in \\mathbb{R}$, we have $\\min_{i \\neq j} |F(x_i,x_j)| \\leq \\frac{C}{n}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2011_a6", - "informal_statement" : "Let $G$ be an abelian group with $n$ elements, and let $\\{g_1=e,g_2,\\dots,g_k\\} \\subsetneq G$ be a (not necessarily minimal) set of distinct generators of $G$. A special die, which randomly selects one of the elements $g_1,g_2,\\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g \\in G$. Prove that there exists a real number $b \\in (0,1)$ such that $\\lim_{m \\to \\infty} \\frac{1}{b^{2m}} \\sum_{x \\in G} (\\text{Prob}(g=x)-\\frac{1}{n})^2$ is positive and finite.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra", "linear_algebra"] - }, - { - "problem_name" : "putnam_2011_b4", - "informal_statement" : "In a tournament, $2011$ players meet $2011$ times to play a multiplayer game. Every game is played by all $2011$ players together and ends with each of the players either winning or losing. The standings are kept in two $2011 \\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk})$. Initially, $T=W=0$. After every game, for every $(h,k)$ (including for $h=k$), if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1$, while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1$. Prove that at the end of the tournament, $\\det(T+iW)$ is a non-negative integer divisible by $2^{2010}$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2012_a1", - "informal_statement" : "Let $d_1,d_2,\\dots,d_{12}$ be real numbers in the open interval $(1,12)$. Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.", - "informal_solution" : "None.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_2012_b3", - "informal_statement" : "A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?", - "informal_solution" : "Show that the answer is yes.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_2014_a4", - "informal_statement" : "Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\\left[X\\right]=1$, $E\\left[X^2\\right]=2$, and $E\\left[X^3\\right]=5$. (Here $E\\left[Y\\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$.", - "informal_solution" : "Show that the answer is $\\frac{1}{3}$.", - "tags" : ["probability", "analysis"] - }, - { - "problem_name" : "putnam_2015_a1", - "informal_statement" : "Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1$. Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB$.", - "informal_solution" : "None.", - "tags" : ["geometry", "analysis"] - }, - { - "problem_name" : "putnam_2015_b2", - "informal_statement" : "Given a list of the positive integers $1,2,3,4,\\dots$, take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6,16,27,36,\\dots$. Prove or disprove that there is some number in the sequence whose base $10$ representation ends with $2015$.", - "informal_solution" : "Show that such a number does exist.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2016_b3", - "informal_statement" : "Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\\triangle ABC$ is at most $1$ whenever $A$, $B$, and $C$ are in $S$. Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_2016_b4", - "informal_statement" : "Let $A$ be a $2n \\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.", - "informal_solution" : "Show that the expected value equals $\\frac{(2n)!}{4^nn!}$.", - "tags" : ["linear_algebra", "probability"] - }, - { - "problem_name" : "putnam_2018_a6", - "informal_statement" : "Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient $\\frac{\\text{area}(\\triangle ABC)}{\\text{area}(\\triangle ABD)}$ is a rational number.", - "informal_solution" : "None.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_2018_b5", - "informal_statement" : "Let $f=(f_1,f_2)$ be a function from $\\mathbb{R}^2$ to $\\mathbb{R}^2$ with continuous partial derivatives $\\frac{\\partial f_i}{\\partial x_j}$ that are positive everywhere. Suppose that $\\frac{\\partial f_1}{\\partial x_1} \\frac{\\partial f_2}{\\partial x_2}-\\frac{1}{4}\\left(\\frac{\\partial f_1}{\\partial x_2}+\\frac{\\partial f_2}{\\partial x_1}\\right)^2>0$ everywhere. Prove that $f$ is one-to-one.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2019_b4", - "informal_statement" : "Let $\\mathcal{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x \\geq 1,y \\geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):\n\\begin{gather*}\nxf_x+yf_y=xy\\ln(xy), \\\\\nx^2f_{xx}+y^2f_{yy}=xy.\n\\end{gather*}\nFor each $f \\in \\mathcal{F}$, let $m(f)=\\min_{s \\geq 1} (f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s))$. Determine $m(f)$, and show that it is independent of the choice of $f$.", - "informal_solution" : "Show that $m(f)=2\\ln 2-\\frac{1}{2}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2021_a3", - "informal_statement" : "Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates.", - "informal_solution" : "Show that the integers $N$ with this property are those of the form $3m^2$ for some positive integer $m$.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_2021_b3", - "informal_statement" : "Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\\mathbb{R}^2$, and define $\\rho(x,y)=yh_x-xh_y$. Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\\rho$ over the interior of $\\mathcal{S}$ is zero.", - "informal_solution" : "Show that the given statement is true.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2021_b5", - "informal_statement" : "Say that an $n$-by-$n$ matrix $A=(a_{ij})_{1 \\leq i,j \\leq n}$ with integer entries is \\emph{very odd} if, for every nonempty subset $S$ of $\\{1,2,\\dots,n\\}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \\in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \\geq 1$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra", "number_theory"] - }, - { - "problem_name" : "putnam_2022_b3", - "informal_statement" : "Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate this recoloring process, will we always end up with all the numbers red after a finite number of steps?", - "informal_solution" : "Show that the answer is yes.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2022_b5", - "informal_statement" : "For $0 \\leq p \\leq 1/2$, let $X_1,X_2,\\dots$ be independent random variables such that\n$X_i=\\begin{cases}\n1 & \\text{with probability $p$,} \\\\\n-1 & \\text{with probability $p$,} \\\\\n0 & \\text{with probability $1-2p$,}\n\\end{cases}$\nfor all $i \\geq 1$. Given a positive integer $n$ and integers $b,a_1,\\dots,a_n$, let $P(b,a_1,\\dots,a_n)$ denote the probability that $a_1X_1+\\dots+a_nX_n=b$. For which values of $p$ is it the case that $P(0,a_1,\\dots,a_n) \\geq P(b,a_1,\\dots,a_n)$ for all positive integers $n$ and all integers $b,a_1,\\dots,a_n$?", - "informal_solution" : "Show that the answer is $p \\leq 1/4$.", - "tags" : ["probability", "algebra"] - }, - { - "problem_name" : "putnam_2023_b1", - "informal_statement" : "Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \\leq i \\leq m$ and $1 \\leq j \\leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \\leq i \\leq m-1$ and $1 \\leq j \\leq n-1$. If a coin occupies the square $(i,j)$ with $i \\leq m-1$ and $j \\leq n-1$ and the squares $(i+1,j)$, $(i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?", - "informal_solution" : "Show that the number of such configurations is $\\binom{m+n-2}{m-1}$.", - "tags" : ["combinatorics"] - } -] diff --git a/informal/batch_21.json b/informal/batch_21.json deleted file mode 100644 index 4e540591..00000000 --- a/informal/batch_21.json +++ /dev/null @@ -1,39 +0,0 @@ -[ - { - "problem_name" : "putnam_2014_b1", - "informal_statement" : "A \\emph{base $10$ over-expansion} of a positive integer $N$ is an expression of the form\n\\[\nN = d_k 10^k + d_{k-1} 10^{k-1} + \\cdots + d_0 10^0\n\\]\nwith $d_k \\neq 0$ and $d_i \\in \\{0,1,2,\\dots,10\\}$ for all $i$. For instance, the integer $N = 10$ has two base $10$ over-expansions: $10 = 10 \\cdot 10^0$ and the usual base $10$ expansion $10 = 1 \\cdot 10^1 + 0 \\cdot 10^0$. Which positive integers have a unique base $10$ over-expansion?", - "informal_solution" : "Prove that the answer is the integers with no $0$'s in their usual base $10$ expansion.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2009_b4", - "informal_statement" : "Say that a polynomial with real coefficients in two variables, $x,y$, is \\emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\\mathbb{R}$. Find the dimension of $V$.", - "informal_solution" : "Prove that the dimension of $V$ is $2020050$.", - "tags" : ["algebra", "linear_algebra"] - }, - { - "problem_name" : "putnam_2000_a6", - "informal_statement" : "Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2000_b1", - "informal_statement" : "Let $a_j,b_j,c_j$ be integers for $1\\leq j\\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\\leq j\\leq N$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2000_b2", - "informal_statement" : "Prove that the expression\n\\[\n\\frac{gcd(m,n)}{n}\\binom{n}{m}\n\\]\nis an integer for all pairs of integers $n\\geq m\\geq 1$.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2000_b3", - "informal_statement" : "Let $f(t)=\\sum_{j=1}^N a_j \\sin(2\\pi jt)$, where each $a_j$ is real and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicities) of $\\frac{d^k f}{dt^k}$. Prove that\n\\[\nN_0\\leq N_1\\leq N_2\\leq \\cdots \\mbox{ and } \\lim_{k\\to\\infty} N_k = 2N.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2000_b5", - "informal_statement" : "Let $S_0$ be a finite set of positive integers. We define finite sets $S_1,S_2,\\ldots$ of positive integers as follows: the integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N=S_0\\cup\\{N+a: a\\in S_0\\}$.", - "tags" : ["algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_22.json b/informal/batch_22.json deleted file mode 100644 index ca09d526..00000000 --- a/informal/batch_22.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_1999_a1", - "informal_statement" : "Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \\[|f(x)|-|g(x)|+h(x) = \\begin{cases} -1 & \\mbox{if $x<-1$} \\\\3x+2 & \\mbox{if $-1 \\leq x \\leq 0$} \\\\-2x+2 & \\mbox{if $x>0$.}\\end{cases}\\]?", - "informal_solution" : "Show that the answer is such functions do exist.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1999_a3", - "informal_statement" : "Consider the power series expansion \\[\\frac{1}{1-2x-x^2} = \\sum_{n=0}^\\infty a_n x^n.\\] Prove that, for each integer $n\\geq 0$, there is an integer $m$ such that \\[a_n^2 + a_{n+1}^2 = a_m .\\]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1999_a4", - "informal_statement" : "Sum the series \\[\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{m^2 n}{3^m(n3^m+m3^n)}.\\]", - "informal_solution" : "Show that the solution is 9/32.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1999_a6", - "informal_statement" : "The sequence $(a_n)_{n\\geq 1}$ is defined by $a_1=1, a_2=2, a_3=24,$ and, for $n\\geq 4$, \\[a_n = \\frac{6a_{n-1}^2a_{n-3} - 8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\\] Show that, for all n, $a_n$ is an integer multiple of $n$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1999_b2", - "informal_statement" : "Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1999_b3", - "informal_statement" : "Let $A=\\{(x,y):0\\leq x,y<1\\}$. For $(x,y)\\in A$, let \\[S(x,y) = \\sum_{\\frac{1}{2}\\leq \\frac{m}{n}\\leq 2} x^m y^n,\\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \\[\\lim_{(x,y)\\rightarrow (1,1), (x,y)\\in A} (1-xy^2)(1-x^2y)S(x,y).\\]", - "informal_solution" : "Show that the answer is 3.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1999_b5", - "informal_statement" : "For an integer $n\\geq 3$, let $\\theta=2\\pi/n$. Evaluate the determinant of the $n\\times n$ matrix $I+A$, where $I$ is the $n\\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\\cos(j\\theta+k\\theta)$ for all $j,k$.", - "informal_solution" : "Show that the answer is $(1 - n^2)/4$.", - "tags" : ["linear_algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_23.json b/informal/batch_23.json deleted file mode 100644 index ec956eeb..00000000 --- a/informal/batch_23.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1998_a4", - "informal_statement" : "Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$, $A_5=A_4 A_3 = 10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.", - "informal_solution" : "Show that the solution is those n for which n can be written as 6k+1 for some integer k.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1998_a5", - "informal_statement" : "Let $\\mathcal F$ be a finite collection of open discs in $\\mathbb R^2$ whose union contains a set $E\\subseteq \\mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\\ldots, D_n$ in $\\mathcal F$ such that \\[E\\subseteq \\cup_{j=1}^n 3D_j.\\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1998_b1", - "informal_statement" : "Find the minimum value of \\[\\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\\] for $x>0$.", - "informal_solution" : "Show that the minimum value is 6.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1998_b2", - "informal_statement" : "Given a point $(a,b)$ with $00$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\\mathbb{R} \\to \\mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \\in \\mathbb{R}$.", - "informal_solution" : "Show that if $c \\leq 1/4$ then $f$ must be constant, and if $c>1/4$ then $f$ can be defined on $[0,c]$ as any continuous function with equal values on the endpoints, then extended to $x>c$ by the relation $f(x)=f(x^2+c)$, then extended further to $x<0$ by the relation $f(x)=f(-x)$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1996_b1", - "informal_statement" : "Define a \\emph{selfish} set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of $\\{1,2,\\ldots,n\\}$ which are \\emph{minimal} selfish sets, that is, selfish sets none of whose proper subsets is selfish.", - "informal_solution" : "Show that the number of subsets is $F_n$, the $n$th Fibonacci number.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1996_b2", - "informal_statement" : "Show that for every positive integer $n$, $(\\frac{2n-1}{e})^{\\frac{2n-1}{2}}<1 \\cdot 3 \\cdot 5 \\cdots (2n-1)<(\\frac{2n+1}{e})^{\\frac{2n+1}{2}}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1996_b3", - "informal_statement" : "Given that $\\{x_1,x_2,\\ldots,x_n\\}=\\{1,2,\\ldots,n\\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \\geq 2$), of $x_1x_2+x_2x_3+\\cdots+x_{n-1}x_n+x_nx_1$.", - "informal_solution" : "Show that the maximum is $(2n^3+3n^2-11n+18)/6$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1996_b4", - "informal_statement" : "For any square matrix $A$, we can define $\\sin A$ by the usual power series: $\\sin A=\\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \\times 2$ matrix $A$ with real entries such that $\\sin A=\\begin{pmatrix} 1 & 1996 \\\\ 0 & 1 \\end{pmatrix}$.", - "informal_solution" : "Show that there does not exist such a matrix $A$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1996_b5", - "informal_statement" : "Given a finite string $S$ of symbols $X$ and $O$, we write $\\Delta(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. For example, $\\Delta(XOOXOOX)=-1$. We call a string $S$ \\emph{balanced} if every substring $T$ of (consecutive symbols of) $S$ has $-2 \\leq \\Delta(T) \\leq 2$. Thus, $XOOXOOX$ is not balanced, since it contains the substring $OOXOO$. Find, with proof, the number of balanced strings of length $n$.", - "informal_solution" : "Show that the number of balanced strings of length $n$ is $3 \\cdot 2^{n/2}-2$ if $n$ is even, and $2^{(n+1)/2}-2$ if $n$ is odd.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_b1", - "informal_statement" : "Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\\lambda \\neq 0$ and every point $P$ not on $L_1$ or $L_2$, there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\\overrightarrow{PA_2}=\\lambda \\overrightarrow{PA_1}$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_2019_b1", - "informal_statement" : "Denote by $\\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \\geq 0$, let $P_n$ be the subset of $\\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \\leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.", - "informal_solution" : "Show that the answer is $5n+1$.", - "tags" : ["geometry"] - } -] diff --git a/informal/batch_26.json b/informal/batch_26.json deleted file mode 100644 index 4aef7ffc..00000000 --- a/informal/batch_26.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1995_a1", - "informal_statement" : "Let $S$ be a set of real numbers which is closed under multiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$). Let $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given that the product of any {\\em three} (not necessarily distinct) elements of $T$ is in $T$ and that the product of any three elements of $U$ is in $U$, show that at least one of the two subsets $T,U$ is closed under multiplication.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1995_a2", - "informal_statement" : "For what pairs $(a,b)$ of positive real numbers does the improper integral \\[ \\int_{b}^{\\infty} \\left( \\sqrt{\\sqrt{x+a}-\\sqrt{x}} - \\sqrt{\\sqrt{x}-\\sqrt{x-b}} \\right)\\,dx \\] converge?", - "informal_solution" : "Show that the solution is those pairs $(a,b)$ where $a = b$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1995_a3", - "informal_statement" : "The number $d_{1}d_{2}\\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ is $d_{1}d_{2}\\dots d_{9}$ by the corresponding digit $e_{i}$ ($1 \\leq i \\leq 9$) is divisible by 7. The number $f_{1}f_{2}\\dots f_{9}$ is related to $e_{1}e_{2}\\dots e_{9}$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_{i}$ by the corresponding $f_{i}$ is divisible by 7. Show that, for each $i$, $d_{i}-f_{i}$ is divisible by 7. [For example, if $d_{1}d_{2}\\dots d_{9} = 199501996$, then $e_{6}$ may be 2 or 9, since $199502996$ and $199509996$ are multiples of 7.]", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1995_a4", - "informal_statement" : "Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\\_{2},\\dots,x_{n}$ satisfy \\[\\sum_{i=1}^{k} x_{i} \\leq k-1 \\qquad \\mbox{for} \\quad k=1,2,\\dots,n.\\]", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1995_a5", - "informal_statement" : "Let $x_{1},x_{2},\\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \\begin{align*} \\frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \\cdots + a_{1n}x_{n} \\ \\frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \\cdots + a_{2n}x_{n} \\ \\vdots && \\vdots \\ \\frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \\cdots + a_{nn}x_{n} \\end{align*} for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \\to 0$ as $t \\to \\infty$. Are the functions $x_{1},x_{2},\\dots,x_{n}$ necessarily linearly dependent?", - "informal_solution" : "Show that the answer is yes, the functions must be linearly dependent.", - "tags" : ["linear_algebra", "analysis"] - }, - { - "problem_name" : "putnam_1995_b1", - "informal_statement" : "For a partition $\\pi$ of $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$, let $\\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\\pi$ and $\\pi'$, there are two distinct numbers $x$ and $y$ in $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$ such that $\\pi(x) = \\pi(y)$ and $\\pi'(x) = \\pi'(y)$. [A {\\em partition} of a set $S$ is a collection of disjoint subsets (parts) whose union is $S$.]", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1995_b3", - "informal_statement" : "To each positive integer with $n^{2}$ decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for $n=2$, to the integer 8617 we associate $\\det \\left( \\begin{array}{cc} 8 & 6 \\ 1 & 7 \\end{array} \\right) = 50$. Find, as a function of $n$, the sum of all the determinants associated with $n^{2}$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n=2$, there are 9000 determinants.)", - "informal_solution" : "Show that the solution is $45$ if $n = 1$, $45^2*10$ if $n = 2$, and $0$ if $n$ is greater than 3.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1995_b4", - "informal_statement" : "Evaluate \\[ \\sqrt[8]{2207 - \\frac{1}{2207-\\frac{1}{2207-\\dots}}}. \\] Express your answer in the form $\\frac{a+b\\sqrt{c}}{d}$, where $a,b,c,d$ are integers.", - "informal_solution" : "Show that the solution is $(3 + 1*\\sqrt{5})/2.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1995_b6", - "informal_statement" : "For a positive real number $\\alpha$, define \\[ S(\\alpha) = \\{ \\lfloor n\\alpha \\rfloor : n = 1,2,3,\\dots \\}. \\] Prove that $\\{1,2,3,\\dots\\}$ cannot be expressed as the disjoint union of three sets $S(\\alpha), S(\\beta)$ and $S(\\gamma)$. [As usual, $\\lfloor x \\rfloor$ is the greatest integer $\\leq x$.]", - "informal_solution" : "None.", - "tags" : ["algebra", "number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_27.json b/informal/batch_27.json deleted file mode 100644 index dbe88630..00000000 --- a/informal/batch_27.json +++ /dev/null @@ -1,74 +0,0 @@ -[ - { - "problem_name" : "putnam_1994_a1", - "informal_statement" : "Suppose that a sequence $a_1,a_2,a_3,\\dots$ satisfies $0f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$.", - "informal_solution" : "Show that the desired set is $(-\\infty,1)$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1994_b4", - "informal_statement" : "For $n \\geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\\begin{pmatrix} 3 & 2 \\\\ 4 & 3 \\end{pmatrix}$ and $I=\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}$. Show that $\\lim_{n \\to \\infty} d_n=\\infty$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra", "number_theory", "analysis"] - }, - { - "problem_name" : "putnam_1994_b5", - "informal_statement" : "For any real number $\\alpha$, define the function $f_\\alpha(x)=\\lfloor \\alpha x \\rfloor$. Let $n$ be a positive integer. Show that there exists an $\\alpha$ such that for $1 \\leq k \\leq n$, $f_\\alpha^k(n^2)=n^2-k=f_{\\alpha^k}(n^2)$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1994_b6", - "informal_statement" : "For any integer $a$, set $n_a=101a-100 \\cdot 2^a$. Show that for $0 \\leq a,b,c,d \\leq 99$, $n_a+n_b \\equiv n_c+n_d \\pmod{10100}$ implies $\\{a,b\\}=\\{c,d\\}$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2004_a1", - "informal_statement" : "Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N)$, of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than $80\\%$ of $N$, but by the end of the season, $S(N)$ was more than $80\\%$ of $N$. Was there necessarily a moment in between when $S(N)$ was exactly $80\\%$ of $N$?", - "informal_solution" : "Show that the answer is yes.", - "tags" : ["probability"] - }, - { - "problem_name" : "putnam_2013_a5", - "informal_statement" : "For $m \\geq 3$, a list of $\\binom{m}{3}$ real numbers $a_{ijk}$ ($1 \\leq ia_2+a_3,a_2>a_3+a_4,\\dots,a_{r-2}>a_{r-1}+a_r,a_{r-1}>a_r$. Let $B(n)$ denote the number of $b_1+b_2+\\cdots+b_s$ which add up to $n$, with\n\\begin{enumerate}\n\\item $b_1 \\geq b_2 \\geq \\dots \\geq b_s$,\n\\item each $b_i$ is in the sequence $1,2,4,\\dots,g_j,\\dots$ defined by $g_1=1$, $g_2=2$, and $g_j=g_{j-1}+g_{j-2}+1$, and\n\\item if $b_1=g_k$ then every element in $\\{1,2,4,\\dots,g_k\\}$ appears at least once as a $b_i$.\n\\end{enumerate}\nProve that $A(n)=B(n)$ for each $n \\geq 1$. (For example, $A(7)=5$ because the relevant sums are $7,6+1,5+2,4+3,4+2+1$, and $B(7)=5$ because the relevant sums are $4+2+1,2+2+2+1,2+2+1+1+1,2+1+1+1+1+1,1+1+1+1+1+1+1$.)", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1991_b1", - "informal_statement" : "For each integer $n \\geq 0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2 \\leq n$. Define a sequence $(a_k)_{k=0}^\\infty$ by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k \\geq 0$. For what positive integers $A$ is this sequence eventually constant?", - "informal_solution" : "Show that this sequence is eventually constant if and only if $A$ is a perfect square.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1991_b2", - "informal_statement" : "Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\\infty,\\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,\n\\begin{align*}\nf(x+y)&=f(x)f(y)-g(x)g(y), \\\\\ng(x+y)&=f(x)g(y)+g(x)f(y).\n\\end{align*}\nIf $f'(0)=0$, prove that $(f(x))^2+(g(x))^2=1$ for all $x$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1991_b4", - "informal_statement" : "Suppose $p$ is an odd prime. Prove that $\\sum_{j=0}^p \\binom{p}{j}\\binom{p+j}{j} \\equiv 2^p+1 \\pmod{p^2}$.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1991_b5", - "informal_statement" : "Let $p$ be an odd prime and let $\\mathbb{Z}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $\\{x^2:x \\in \\mathbb{Z}_p\\} \\cap \\{y^2+1:y \\in \\mathbb{Z}_p\\}$?", - "informal_solution" : "Show that the number of elements in the intersection is $\\lceil p/4 \\rceil$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1991_b6", - "informal_statement" : "Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $a^xb^{1-x} \\leq a\\frac{\\sinh ux}{\\sinh u}+b\\frac{\\sinh u(1-x)}{\\sinh u}$ for all $u$ with $0<|u| \\leq c$ and for all $x$, $0|T|$ for each $s \\in S$, and $t>|S|$ for each $t \\in T$. How many admissible ordered pairs of subsets of $\\{1,2,\\dots,10\\}$ are there? Prove your answer.", - "informal_solution" : "Show that the number of admissible ordered pairs of subsets of $\\{1,2,\\dots,10\\}$ equals the $22$nd Fibonacci number $F_{22}=17711$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1990_b1", - "informal_statement" : "Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\\int_0^x [(f(t))^2+(f'(t))^2]\\,dt+1990$.", - "informal_solution" : "Show that there are two such functions, namely $f(x)=\\sqrt{1990}e^x$, and $f(x)=-\\sqrt{1990}e^x$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1990_b2", - "informal_statement" : "Prove that for $|x|<1$, $|z|>1$, $1+\\sum_{j=1}^\\infty (1+x^j)P_j=0$, where $P_j$ is $\\frac{(1-z)(1-zx)(1-zx^2) \\cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \\cdots (z-x^j)}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1990_b3", - "informal_statement" : "Let $S$ be a set of $2 \\times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \\leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1990_b4", - "informal_statement" : "Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence $g_1,g_2,g_3,\\dots,g_{2n}$ such that\n\\begin{itemize}\n\\item[(1)] every element of $G$ occurs exactly twice, and\n\\item[(2)] $g_{i+1}$ equals $g_ia$ or $g_ib$ for $i=1,2,\\dots,2n$. (Interpret $g_{2n+1}$ as $g_1$.)\n\\end{itemize}", - "informal_solution" : "Show that such a sequence does exist.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1990_b5", - "informal_statement" : "Is there an infinite sequence $a_0,a_1,a_2,\\dots$ of nonzero real numbers such that for $n=1,2,3,\\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\\cdots+a_nx^n$ has exactly $n$ distinct real roots?", - "informal_solution" : "Show that the answer is yes, such an infinite sequence exists.", - "tags" : ["algebra", "analysis"] - } -] diff --git a/informal/batch_32.json b/informal/batch_32.json deleted file mode 100644 index 243aa0da..00000000 --- a/informal/batch_32.json +++ /dev/null @@ -1,53 +0,0 @@ -[ - { - "problem_name" : "putnam_1989_a3", - "informal_statement" : "Prove that if\n\\[\n11z^{10}+10iz^9+10iz-11=0,\n\\]\nthen $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1989_a6", - "informal_statement" : "Let $\\alpha=1+a_1x+a_2x^2+\\cdots$ be a formal power series with coefficients in the field of two elements. Let\n\\[\na_n =\n\\begin{cases}\n1 & \\parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\\\[.3in]\n0 & \\text{otherwise.}\n\\end{cases}\n\\]\n(For example, $a_{36}=1$ because $36=100100_2$ and $a_{20}=0$ because $20=10100_2.$) Prove that $\\alpha^3+x\\alpha+1=0.$", - "tags" : ["algebra", "abstract_algebra"] - }, - { - "problem_name" : "putnam_1989_b2", - "informal_statement" : "Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\\{a^n:\\,n=1, 2, 3, \\ldots\\}$ is finite. Must $S$ be a group?", - "informal_solution" : "Prove that $S$ must be a group.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1989_b3", - "informal_statement" : "Let $f$ be a function on $[0,\\infty)$, differentiable and satisfying\n\\[\nf'(x)=-3f(x)+6f(2x)\n\\]\nfor $x>0$. Assume that $|f(x)|\\le e^{-\\sqrt{x}}$ for $x\\ge 0$ (so that $f(x)$ tends rapidly to $0$ as $x$ increases). For $n$ a non-negative integer, define\n\\[\n\\mu_n=\\int_0^\\infty x^n f(x)\\,dx\n\\]\n(sometimes called the $n$th moment of $f$).\n\\begin{enumerate}\n\\item[a)] Express $\\mu_n$ in terms of $\\mu_0$.\n\\item[b)] Prove that the sequence $\\{\\mu_n \\frac{3^n}{n!}\\}$ always converges, and that the limit is $0$ only if $\\mu_0=0$.\n\\end{enumerate}", - "informal_solution" : "Show that for each $n \\geq 0$, $\\mu_n = \\frac{n!}{3^n} \\left( \\prod_{m=1}^{n}(1 - 2^{-m}) \\right)^{-1} \\mu_0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1989_b4", - "informal_statement" : "Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?", - "informal_solution" : "Prove that such a collection exists.", - "tags" : ["set_theory"] - }, - { - "problem_name" : "putnam_1988_a2", - "informal_statement" : "A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$.", - "informal_solution" : "Show that such $(a,b)$ and $g$ exist.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1988_a3", - "informal_statement" : "Determine, with proof, the set of real numbers $x$ for which\n\\[\n\\sum_{n=1}^\\infty \\left( \\frac{1}{n} \\csc \\frac{1}{n} - 1 \\right)^x\n\\]\nconverges.", - "informal_solution" : "Show that the series converges if and only if $x > \\frac{1}{2}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1988_a4", - "informal_statement" : "\\begin{enumerate}\n\\item[(a)] If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?\n\\item[(b)] What if ``three'' is replaced by ``nine''?\n\\end{enumerate}", - "informal_solution" : "Prove that the points must exist with three colors, but not necessarily with nine.", - "tags" : ["geometry", "combinatorics"] - }, - { - "problem_name" : "putnam_1988_a5", - "informal_statement" : "Prove that there exists a \\emph{unique} function $f$ from the set $\\mathrm{R}^+$ of positive real numbers to $\\mathrm{R}^+$ such that\n\\[\nf(f(x)) = 6x-f(x)\n\\]\nand\n\\[\nf(x)>0\n\\]\nfor all $x>0$.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_33.json b/informal/batch_33.json deleted file mode 100644 index d8d72de1..00000000 --- a/informal/batch_33.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1987_a1", - "informal_statement" : "Curves $A$, $B$, $C$, and $D$ are defined in the plane as follows:\n\\begin{align*}\nA&=\\left\\{ (x,y):x^2-y^2=\\frac{x}{x^2+y^2} \\right\\}, \\\\\nB&=\\left\\{ (x,y):2xy+\\frac{y}{x^2+y^2}=3 \\right\\}, \\\\\nC&=\\left\\{ (x,y):x^3-3xy^2+3y=1 \\right\\}, \\\\\nD&=\\left\\{ (x,y):3x^2y-3x-y^3=0 \\right\\}.\n\\end{align*}\nProve that $A \\cap B=C \\cap D$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1987_a2", - "informal_statement" : "The sequence of digits $123456789101112131415161718192021 \\dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters the sequence in the placement of the two-digit integer $55$. Find, with proof, $f(1987)$.", - "informal_solution" : "Show that the value of $f(1987)$ is $1984$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1988_a6", - "informal_statement" : "If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.", - "informal_solution" : "Show that the answer is yes, $A$ must be a scalar multiple of the identity.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1988_b1", - "informal_statement" : "A \\emph{composite} (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1988_b2", - "informal_statement" : "Prove or disprove: If $x$ and $y$ are real numbers with $y \\geq 0$ and $y(y+1) \\leq (x+1)^2$, then $y(y-1) \\leq x^2$.", - "informal_solution" : "Show that this is true.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1988_b3", - "informal_statement" : "For every $n$ in the set $N=\\{1,2,\\dots\\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \\leq g$ for all $n \\in N$.", - "informal_solution" : "Show that the smallest such $g$ is $(1+\\sqrt{3})/2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1988_b4", - "informal_statement" : "Prove that if $\\sum_{n=1}^\\infty a_n$ is a convergent series of positive real numbers, then so is $\\sum_{n=1}^\\infty (a_n)^{n/(n+1)}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1988_b5", - "informal_statement" : "For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix is the largest $k$ such that there is a $k \\times k$ submatrix with nonzero determinant.) One may note that\n\\begin{align*}\nM_1&=\\begin{pmatrix} 0 & -1 & 1 \\\\ 1 & 0 & -1 \\\\ -1 & 1 & 0 \\end{pmatrix} \\\\\nM_2&=\\begin{pmatrix} 0 & -1 & -1 & 1 & 1 \\\\ 1 & 0 & -1 & -1 & 1 \\\\ 1 & 1 & 0 & -1 & -1 \\\\ -1 & 1 & 1 & 0 & -1 \\\\ -1 & -1 & 1 & 1 & 0 \\end{pmatrix}.\n\\end{align*}", - "informal_solution" : "Show that the rank of $M_n$ equals $2n$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1988_b6", - "informal_statement" : "Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n=n(n+1)/2$ with $n$ in $\\{0,1,2,\\dots\\}$.)", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - } -] diff --git a/informal/batch_34.json b/informal/batch_34.json deleted file mode 100644 index ec732b4e..00000000 --- a/informal/batch_34.json +++ /dev/null @@ -1,41 +0,0 @@ -[ - { - "problem_name" : "putnam_1987_a4", - "informal_statement" : "Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that\n\\[\nP(ux, uy, uz) = u^2 F(y-x,z-x) \\quad \\mbox{for all real $x,y,z,u$},\n\\]\nand such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.", - "informal_solution" : "Prove that $|C - A| = \\frac{5}{3}\\sqrt{30}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1987_a6", - "informal_statement" : "For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series\n\\[\n\\sum_{n=1}^\\infty \\frac{x^{a(n)}}{n^3}\n\\]\nconverge?", - "informal_solution" : "Show that for positive $x$, the series converges if and only if $x < 25$.", - "tags" : ["algebra", "analysis"] - }, - { - "problem_name" : "putnam_1987_b1", - "informal_statement" : "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-x)}\\,dx}{\\sqrt{\\ln(9-x)}+\\sqrt{\\ln(x+3)}}.\n\\]", - "informal_solution" : "Prove that the integral evaluates to $1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1987_b2", - "informal_statement" : "Let $r, s$ and $t$ be integers with $0 \\leq r$, $0 \\leq s$ and $r+s \\leq t$. Prove that\n\\[\n\\frac{\\binom s0}{\\binom tr}\n+ \\frac{\\binom s1}{\\binom{t}{r+1}} + \\cdots\n+ \\frac{\\binom ss}{\\binom{t}{r+s}}\n= \\frac{t+1}{(t+1-s)\\binom{t-s}{r}}.\n\\]", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1987_b4", - "informal_statement" : "Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \\cos y_n - y_n \\sin y_n$ and $y_{n+1}= x_n \\sin y_n + y_n \\cos y_n$ for $n=1,2,3,\\dots$. For each of $\\lim_{n\\to \\infty} x_n$ and $\\lim_{n \\to \\infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.", - "informal_solution" : "Show that $\\lim_{n \\to \\infty} x_n = -1$ and $\\lim_{n \\to \\infty} y_n = 0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1987_b5", - "informal_statement" : "Let $O_n$ be the $n$-dimensional vector $(0,0,\\cdots, 0)$. Let $M$ be a $2n \\times n$ matrix of complex numbers such that whenever $(z_1, z_2, \\dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \\dots, r_{2n}$, there are complex numbers $w_1, w_2, \\dots, w_n$ such that\n\\[\n\\mathrm{re}\\left[ M \\left( \\begin{array}{c} w_1 \\\\ \\vdots \\\\ w_n \\end{array} \\right) \\right] = \\left( \\begin{array}{c} r_1 \\\\ \\vdots \\\\ r_{2n} \\end{array} \\right).\n\\]\n(Note: if $C$ is a matrix of complex numbers, $\\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1987_b6", - "informal_statement" : "Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \\cap \\{2a: a \\in S\\}$. Prove that $N$ is even.", - "tags" : ["abstract_algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_35.json b/informal/batch_35.json deleted file mode 100644 index bb9f73f4..00000000 --- a/informal/batch_35.json +++ /dev/null @@ -1,60 +0,0 @@ -[ - { - "problem_name" : "putnam_1986_a1", - "informal_statement" : "Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \\leq 13x^2$.", - "informal_solution" : "Show that the maximum value is $18$.", - "tags" : ["algebra", "analysis"] - }, - { - "problem_name" : "putnam_1986_a2", - "informal_statement" : "What is the units (i.e., rightmost) digit of\n\\[\n\\left\\lfloor \\frac{10^{20000}}{10^{100}+3}\\right\\rfloor ?\n\\]", - "informal_solution" : "Show that the answer is $3$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1986_a3", - "informal_statement" : "Evaluate $\\sum_{n=0}^\\infty \\mathrm{Arccot}(n^2+n+1)$, where $\\mathrm{Arccot}\\,t$ for $t \\geq 0$ denotes the number $\\theta$ in the interval $0 < \\theta \\leq \\pi/2$ with $\\cot \\theta = t$.", - "informal_solution" : "Show that the sum equals $\\pi/2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1986_a4", - "informal_statement" : "A \\emph{transversal} of an $n\\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \\times n$ matrices $A$ satisfying the following two conditions:\n\\begin{enumerate}\n\\item[(a)] Each entry $\\alpha_{i,j}$ of $A$ is in the set\n$\\{-1,0,1\\}$.\n\\item[(b)] The sum of the $n$ entries of a transversal is the same for all transversals of $A$.\n\\end{enumerate}\nAn example of such a matrix $A$ is\n\\[\nA = \\left( \\begin{array}{ccc} -1 & 0 & -1 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 0\n\\end{array}\n\\right).\n\\]\nDetermine with proof a formula for $f(n)$ of the form\n\\[\nf(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,\n\\]\nwhere the $a_i$'s and $b_i$'s are rational numbers.", - "informal_solution" : "Prove that $f(n) = 4^n + 2 \\cdot 3^n - 4 \\cdot 2^n + 1$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1986_a6", - "informal_statement" : "Let $a_1, a_2, \\dots, a_n$ be real numbers, and let $b_1, b_2, \\dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity\n\\[\n(1-x)^n f(x) = 1 + \\sum_{i=1}^n a_i x^{b_i}.\n\\]\nFind a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \\dots, b_n$ and $n$ (but independent of $a_1, a_2, \\dots, a_n$).", - "informal_solution" : "Show that $f(1) = b_1 b_2 \\dots b_n / n!$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1986_b2", - "informal_statement" : "Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations\n\\[\nx(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,\n\\]\nand list all such triples $T$.", - "informal_solution" : "Show that the possibilities for $T$ are $(0, 0, 0), \\, (0, -1, 1), \\, (1, 0, -1), \\, (-1, 1, 0)$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1986_b3", - "informal_statement" : "Let $\\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\\Gamma$ and $m$ a positive integer, let $f \\equiv g \\pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\\Gamma$ with $rf+sg\\equiv 1 \\pmod{p}$ and $fg \\equiv h \\pmod{p}$, prove that there exist $F$ and $G$ in $\\Gamma$ with $F \\equiv f \\pmod{p}$, $G \\equiv g \\pmod{p}$, and $FG \\equiv h \\pmod{p^n}$.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1986_b4", - "informal_statement" : "For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \\sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\\lim_{r\\to \\infty}G(r)$ exists and equals $0$.", - "informal_solution" : "Show that the limit exists and equals $0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1986_b5", - "informal_statement" : "Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying\n\\[\nf(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).\n\\]\nProve or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\\pm x, \\pm y, \\pm z$, where the number of minus signs is $0$ or $2$.", - "informal_solution" : "Prove that the assertion is false.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1986_b6", - "informal_statement" : "Suppose $A,B,C,D$ are $n \\times n$ matrices with entries in a field $F$, satisfying the conditions that $AB^T$ and $CD^T$ are symmetric and $AD^T - BC^T = I$. Here $I$ is the $n \\times n$ identity matrix, and if $M$ is an $n \\times n$ matrix, $M^T$ is its transpose. Prove that $A^T D - C^T B = I$.", - "tags" : ["linear_algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_36.json b/informal/batch_36.json deleted file mode 100644 index 8bcb543f..00000000 --- a/informal/batch_36.json +++ /dev/null @@ -1,60 +0,0 @@ -[ - { - "problem_name" : "putnam_1985_a1", - "informal_statement" : "Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $A_1 \\cup A_2 \\cup A_3 = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $A_1 \\cap A_2 \\cap A_3 = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.", - "informal_solution" : "Prove that the number of such triples is $2^{10}3^{10}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1985_a3", - "informal_statement" : "Let $d$ be a real number. For each integer $m \\geq 0$, define a sequence $\\{a_m(j)\\}$, $j=0,1,2,\\dots$ by the condition\n\\begin{align*}\na_m(0) &= d/2^m, \\\\\na_m(j+1) &= (a_m(j))^2 + 2a_m(j), \\qquad j \\geq 0.\n\\end{align*}\nEvaluate $\\lim_{n \\to \\infty} a_n(n)$.", - "informal_solution" : "Show that the limit equals $e^d - 1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1985_a4", - "informal_statement" : "Define a sequence $\\{a_i\\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?", - "informal_solution" : "Prove that the only number that occurs infinitely often is $87$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1985_a5", - "informal_statement" : "Let $I_m = \\int_0^{2\\pi} \\cos(x)\\cos(2x)\\cdots \\cos(mx)\\,dx$. For which integers $m$, $1 \\leq m \\leq 10$ is $I_m \\neq 0$?", - "informal_solution" : "Prove that the integers $m$ with $1 \\leq m \\leq 10$ and $I_m \\neq 0$ are $m = 3, 4, 7, 8$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1985_a6", - "informal_statement" : "If $p(x)= a_0 + a_1 x + \\cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set\n\\[\n\\Gamma(p(x)) = a_0^2 + a_1^2 + \\cdots + a_m^2.\n\\]\nLet $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that\n\\begin{enumerate}\n\\item[(i)] $g(0)=1$, and\n\\item[(ii)] $\\Gamma(f(x)^n) = \\Gamma(g(x)^n)$\n\\end{enumerate}\nfor every integer $n \\geq 1$.", - "informal_solution" : "Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1985_b1", - "informal_statement" : "Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial\n\\[\np(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)\n\\]\nhas exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.", - "informal_solution" : "Show that the minimum $k = 3$ is obtained for $\\{m_1, m_2, m_3, m_4, m_5\\} = \\{-2, -1, 0, 1, 2\\}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1985_b2", - "informal_statement" : "Define polynomials $f_n(x)$ for $n \\geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \\geq 1$, and\n\\[\n\\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)\n\\]\nfor $n \\geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.", - "informal_solution" : "Show that $f_{100}(1) = 101^{99}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1985_b3", - "informal_statement" : "Let\n\\[\n\\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \\dots \\\\\na_{2,1} & a_{2,2} & a_{2,3} & \\dots \\\\\na_{3,1} & a_{3,2} & a_{3,3} & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{array}\n\\]\nbe a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m,n} > mn$ for some pair of positive integers $(m,n)$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1985_b5", - "informal_statement" : "Evaluate $\\int_0^\\infty t^{-1/2}e^{-1985(t+t^{-1})}\\,dt$. You may assume that $\\int_{-\\infty}^\\infty e^{-x^2}\\,dx = \\sqrt{\\pi}$.", - "informal_solution" : "Show that the integral evaluates to $\\sqrt{\\frac{\\pi}{1985}}e^{-3970}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1985_b6", - "informal_statement" : "Let $G$ be a finite set of real $n\\times n$ matrices $\\{M_i\\}$, $1 \\leq i \\leq r$, which form a group under matrix\nmultiplication. Suppose that $\\sum_{i=1}^r \\mathrm{tr}(M_i)=0$, where $\\mathrm{tr}(A)$ denotes the trace of the matrix $A$. Prove that $\\sum_{i=1}^r M_i$ is the $n \\times n$ zero matrix.", - "tags" : ["abstract_algebra", "linear_algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_37.json b/informal/batch_37.json deleted file mode 100644 index 23b422f7..00000000 --- a/informal/batch_37.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_1983_a1", - "informal_statement" : "How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?", - "informal_solution" : "Show that the desired count is $2301$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1983_a3", - "informal_statement" : "Let $p$ be in the set $\\{3,5,7,11,\\dots\\}$ of odd primes and let $F(n)=1+2n+3n^2+\\dots+(p-1)n^{p-2}$. Prove that if $a$ and $b$ are distinct integers in $\\{0,1,2,\\dots,p-1\\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$, that is, $F(a)-F(b)$ is not exactly divisible by $p$.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1984_a2", - "informal_statement" : "Express $\\sum_{k=1}^\\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.", - "informal_solution" : "Show that the sum converges to $2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1984_a3", - "informal_statement" : "Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \\neq b$, and let $M_n$ denote the $2n \\times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by\n\\[\nm_{ij}=\\begin{cases}\nx & \\text{if }i=j, \\\\\na & \\text{if }i \\neq j\\text{ and }i+j\\text{ is even}, \\\\\nb & \\text{if }i \\neq j\\text{ and }i+j\\text{ is odd}.\n\\end{cases}\n\\]\nThus, for example, $M_2=\\begin{pmatrix} x & b & a & b \\\\ b & x & b & a \\\\ a & b & x & b \\\\ b & a & b & x \\end{pmatrix}$. Express $\\lim_{x \\to a} \\det M_n/(x-a)^{2n-2}$ as a polynomial in $a$, $b$, and $n$, where $\\det M_n$ denotes the determinant of $M_n$.", - "informal_solution" : "Show that $\\lim_{x \\to a} \\frac{\\det M_n}{(x-a)^{2n-2}}=n^2(a^2-b^2)$.", - "tags" : ["linear_algebra", "analysis"] - }, - { - "problem_name" : "putnam_1984_a5", - "informal_statement" : "Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \\leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\\iiint_R x^1y^9z^8w^4\\,dx\\,dy\\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.", - "informal_solution" : "Show that the integral we desire is $1!9!8!4!/25!$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1984_a6", - "informal_statement" : "Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$.\n\\begin{enumerate}\n\\item[(a)] Show that if $a_1,a_2,\\dots,a_k$ are \\emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\\dots+a_k$.\n\\item[(b)] Assuming part (a), we can define $g(s)=f(5^{a_1}+5^{a_2}+\\dots+5^{a_k})$, where $s=a_1+a_2+\\dots+a_k$. Find the least positive integer $p$ for which $g(s)=g(s + p)$, for all $s \\geq 1$, or else show that no such $p$ exists.\n\\end{enumerate}", - "informal_solution" : "Show that the least such $p$ is $p=4$.", - "tags" : ["algebra", "number_theory"] - }, - { - "problem_name" : "putnam_1984_b1", - "informal_statement" : "Let $n$ be a positive integer, and define $f(n)=1!+2!+\\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \\geq 1$.", - "informal_solution" : "Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1984_b2", - "informal_statement" : "Find the minimum value of $(u-v)^2+(\\sqrt{2-u^2}-\\frac{9}{v})^2$ for $00$.", - "informal_solution" : "Show that the minimum value is $8$.", - "tags" : ["geometry", "analysis"] - }, - { - "problem_name" : "putnam_1984_b3", - "informal_statement" : "Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$,\n\\begin{enumerate}\n\\item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and\n\\item[(ii)] $x*(y*z) \\neq (x*y)*z$ (\\emph{no} case of associativity holds).\n\\end{enumerate}", - "informal_solution" : "Show that the statement is true.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1984_b5", - "informal_statement" : "For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.", - "informal_solution" : "Show that $\\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$.", - "tags" : ["algebra", "analysis"] - } -] diff --git a/informal/batch_38.json b/informal/batch_38.json deleted file mode 100644 index 4d406dd5..00000000 --- a/informal/batch_38.json +++ /dev/null @@ -1,58 +0,0 @@ -[ - { - "problem_name" : "putnam_1983_a4", - "informal_statement" : "Prove that for $m = 5 \\pmod 6$, \n\\[\n\\binom{m}{2} - \\binom{m}{5} + \\binom{m}{8} - \\binom{m}{11} + ... - \\binom{m}{m-6} + \\binom{m}{m-3} \\neq 0.\n\\]", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1983_a5", - "informal_statement" : "Prove or disprove that there exists a positive real number $\\alpha$ such that $[\\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)", - "informal_solution" : "Prove that such an $\\alpha$ exists.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1983_a6", - "informal_statement" : "Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\\lim_{a \\to \\infty} a^4 \\exp(-a^3) \\int_T \\exp(x^3+y^3) \\, dx \\, dy$.", - "informal_solution" : "Show that the integral evaluates to $\\frac{2}{9}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1983_b2", - "informal_statement" : "Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$?", - "informal_solution" : "Prove that such a polynomial exists.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1983_b4", - "informal_statement" : "Let $f(n) = n + [\\sqrt n]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Define the sequence $a_i$ by $a_0 = m$, $a_{n+1} = f(a_n)$. Prove that it contains at least one square.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1983_b5", - "informal_statement" : "Define $\\left\\lVert x \\right\\rVert$ as the distance from $x$ to the nearest integer. Find $\\lim_{n \\to \\infty} \\frac{1}{n} \\int_{1}^{n} \\left\\lVert \\frac{n}{x} \\right\\rVert \\, dx$. You may assume that $\\prod_{n=1}^{\\infty} \\frac{2n}{(2n-1)} \\cdot \\frac{2n}{(2n+1)} = \\frac{\\pi}{2}$.", - "informal_solution" : "Show that the limit equals $\\ln \\left( \\frac{4}{\\pi} \\right)$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1983_b6", - "informal_statement" : "Let $n$ be a positive integer and let $\\alpha \\neq 1$ be a complex $(2n + 1)\\textsuperscript{th}$ root of unity. Prove that there always exist polynomials $p(x)$, $q(x)$ with integer coefficients such that $p(\\alpha)^2 + q(\\alpha)^2 = -1$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1982_a2", - "informal_statement" : "Let $B_n(x) = 1^x + 2^x + \\dots + n^x$ and let $f(n) = \\frac{B_n(\\log_n 2)}{(n \\log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \\dots$ converge?", - "informal_solution" : "Prove that the series converges.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1982_a3", - "informal_statement" : "Evaluate $\\int_0^{\\infty} \\frac{\\tan^{-1}(\\pi x) - \\tan^{-1} x}{x} \\, dx$.", - "informal_solution" : "Show that the integral evaluates to $\\frac{\\pi}{2} \\ln \\pi$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1982_a5", - "informal_statement" : "Let $a, b, c, d$ be positive integers satisfying $a + c \\leq 1982$ and $\\frac{a}{b} + \\frac{c}{d} < 1$. Prove that $1 - \\frac{a}{b} - \\frac{c}{d} > \\frac{1}{1983^3}$.", - "tags" : ["algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_39.json b/informal/batch_39.json deleted file mode 100644 index c991eb93..00000000 --- a/informal/batch_39.json +++ /dev/null @@ -1,68 +0,0 @@ -[ - { - "problem_name" : "putnam_1982_a6", - "informal_statement" : "Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \\dots$ be an infinite sequence of real numbers with the following properties:\n\\begin{enumerate}\n\\item\n$|x_n|$ is a strictly decreasing function of $n$;\n\\item\n$\\lim_{n \\rightarrow \\infty} |b(n) - n| \\cdot |x_n| = 0$;\n\\item\n$\\lim_{n \\rightarrow \\infty}\\sum_{k = 1}^{n} x_k = 1$.\n\\end{enumerate}\nProve or disprove: these conditions imply that $$\\lim_{n \\rightarrow \\infty} \\sum_{k = 1}^{n} x_{b(k)} = 1.$$", - "informal_solution" : "The limit need not equal $1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1982_b2", - "informal_statement" : "Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \\le x^2 + y^2$. Also, let $g = \\sum_{k = 0}^{\\infty} e^{-k^2}$. Express the value $$\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$.", - "informal_solution" : "The desired polynomial is $\\pi(2g - 1)^2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1982_b3", - "informal_statement" : "Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\\{1, 2, 3, \\dots, n\\}$. Express $\\lim_{n \\rightarrow \\infty} p_n \\sqrt{n}$ in the form $r(\\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$.", - "informal_solution" : "The limit equals $\\frac{4}{3}(\\sqrt{2} - 1)$.", - "tags" : ["analysis", "number_theory", "probability"] - }, - { - "problem_name" : "putnam_1982_b4", - "informal_statement" : "Let $n_1, n_2, \\dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\\cdots n_s$ divides $(n_1 + k)(n_2 + k) \\cdots (n_s + k)$. Prove or provide a counterexample to the following claims:\n\\begin{enumerate}\n\\item\nFor some $i$, $|n_i| = 1$.\n\\item\nIf all $n_i$ are positive, then $\\{n_1, n_2, \\dots, n_s\\} = \\{1, 2, \\dots, s\\}$.\n\\end{enumerate}", - "informal_solution" : "Both claims are true.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1982_b5", - "informal_statement" : "For all $x > e^e$, let $S = u_0, u_1, \\dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \\log_{u_n} x$ for all $n \\ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1981_a1", - "informal_statement" : "Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \\cdots n^n$. Find $\\lim_{n \\rightarrow \\infty} \\frac{E(n)}{n^2}$.", - "informal_solution" : "The limit equals $\\frac{1}{8}$.", - "tags" : ["analysis", "number_theory"] - }, - { - "problem_name" : "putnam_1981_a3", - "informal_statement" : "Does the limit $$lim_{t \\rightarrow \\infty}e^{-t}\\int_{0}^{t}\\int_{0}^{t}\\frac{e^x - e^y}{x - y} dx dy$$exist?", - "informal_solution" : "The limit does not exist.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1981_a5", - "informal_statement" : "Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$\nGiven that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots.", - "informal_solution" : "$Q(x)$ must have at least $2n - 1$ distinct real roots.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1981_b1", - "informal_statement" : "Find the value of $$\\lim_{n \\rightarrow \\infty} \\frac{1}{n^5}\\sum_{h=1}^{n}\\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$", - "informal_solution" : "The limit equals $-1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1981_b2", - "informal_statement" : "Determine the minimum value attained by $$(r - 1)^2 + (\\frac{s}{r} - 1)^2 + (\\frac{t}{s} - 1)^2 + (\\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \\le r \\le s \\le t \\le 4$.", - "informal_solution" : "The minimum is $12 - 8\\sqrt{2}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1981_b3", - "informal_statement" : "Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_4.json b/informal/batch_4.json deleted file mode 100644 index 047fd62a..00000000 --- a/informal/batch_4.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_2021_b4", - "informal_statement" : "Let $F_0, F_1, \\ldots$ be the sequence of Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \\geq 2$. For $m > 2$, let $R_m$ be the remainder when the product $\\prod_{k=1}^{F_{m-1}} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2020_a1", - "informal_statement" : "Find the number of positive integers $N$ satisfying: (i) $N$ is divisible by $2020$, (ii) $N$ has at most $2020$ decimal digits, (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.", - "informal_solution" : "Show that the solution is $508536$.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2019_b5", - "informal_statement" : "Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \\geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n + 1) = F_{2n+1}$ for $n = 0,1,2,\\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.", - "informal_solution" : "Show that the solution takes the form of $(j, k) = (2019, 1010)$.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2018_a1", - "informal_statement" : "Find all ordered pairs $(a,b)$ of positive integers for which $\\frac{1}{a} + \\frac{1}{b} = \\frac{3}{2018}$.", - "informal_solution" : "Show that all solutions are in the set of ${(673,1358114), (674,340033), (1009,2018), (2018,1009), (340033,674), (1358114,673)}$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2018_a3", - "informal_statement" : "Determine the greatest possible value of $\\sum_{i=1}^{10} \\cos(3x_i)$ for real numbers $x_1, x_2, \\ldots, x_{10}$ satisfying $\\sum_{i=1}^{10} \\cos(x_i) = 0$.", - "informal_solution" : "Show that the solution is $\\frac{480}{49}$", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2018_b2", - "informal_statement" : "Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \\cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\\{z \\in \\mathbb{C}: |z| \\leq 1\\}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2017_a1", - "informal_statement" : "Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?.", - "informal_solution" : "Show that all solutions are in the set $\\{x \\in \\mathbb{Z}\\, |\\, x > 0 \\land (x = 1 \\lor 5 \\mid x)\\}", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_40.json b/informal/batch_40.json deleted file mode 100644 index 84ccd709..00000000 --- a/informal/batch_40.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1980_a2", - "informal_statement" : "Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \\cdot 7^s=\\text{lcm}[a,b,c]=\\text{lcm}[a,b,d]=\\text{lcm}[a,c,d]=\\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\\text{lcm}[x,y,z]$ denotes the least common multiple of $x,y,z$.)", - "informal_solution" : "Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1980_a3", - "informal_statement" : "Evaluate $\\int_0^{\\pi/2}\\frac{dx}{1+(\\tan x)^{\\sqrt{2}}}$.", - "informal_solution" : "Show that the integral is $\\pi/4$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1980_a4", - "informal_statement" : "\\begin{enumerate}\n\\item[(a)] Prove that there exist integers $a,b,c$, not all zero and each of absolute value less than one million, such that $|a+b\\sqrt{2}+c\\sqrt{3}|<10^{-11}$.\n\\item[(b)] Let $a,b,c$ be integers, not all zero and each of absolute value less than one million. Prove that $|a+b\\sqrt{2}+c\\sqrt{3}|>10^{-21}$.\n\\end{enumerate}", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1980_a5", - "informal_statement" : "Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $0=\\int_0^xP(t)\\sin t\\,dt=\\int_0^xP(t)\\cos t\\,dt$ has only finitely many real solutions $x$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1980_a6", - "informal_statement" : "Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0 \\leq x \\leq 1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $u \\leq \\int_0^1|f'(x)-f(x)|\\,dx$ for all $f$ in $C$.", - "informal_solution" : "Show that $u=1/e$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1980_b1", - "informal_statement" : "For which real numbers $c$ is $(e^x+e^{-x})/2 \\leq e^{cx^2}$ for all real $x$?", - "informal_solution" : "Show that the inequality holds if and only if $c \\geq 1/2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1980_b3", - "informal_statement" : "For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \\geq 0$? (Express the answer in the simplest form.)", - "informal_solution" : "Show that $u_n>0$ for all $n \\geq 0$ if and only if $a \\geq 3$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1981_b4", - "informal_statement" : "Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \\in V$ whenever $A,B \\in V$ and $r$ and $s$ are scalars (i.e., real numbers). \\emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of rank $3$. [The rank of a nonzero matrix $M$ is the largest $k$ such that the entries of some $k$ rows and some $k$ columns form a $k$ by $k$ matrix with a nonzero determinant.]", - "informal_solution" : "Show that the assertion is false.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1981_b5", - "informal_statement" : "Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\\exp \\left(\\sum_{n=1}^\\infty \\frac{B(n)}{n(n+1)}\\right)$ is a rational number. Here $\\exp(x)$ denotes $e^x$.", - "informal_solution" : "Show that the expression is a rational number.", - "tags" : ["analysis", "algebra"] - } -] diff --git a/informal/batch_41.json b/informal/batch_41.json deleted file mode 100644 index cb2369b3..00000000 --- a/informal/batch_41.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_1980_b4", - "informal_statement" : "Let $X$ be a finite set with at least $10$ elements; for each $i \\in \\{0, 1, ..., 1065\\}$, let $A_i \\subseteq X$ satisfy $|A_i| > \\frac{1}{2}|X|$. Prove that there exist $10$ elements $x_1, x_2, \\dots, x_{10} \\in X$ such that each $A_i$ contains at least one of $x_1, x_2, \\dots, x_{10}$.", - "informal_solution" : "None.", - "tags" : ["set_theory", "combinatorics"] - }, - { - "problem_name" : "putnam_1980_b5", - "informal_statement" : "A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \\le sf(u) + (1 - s)f(v)$$ for all $s \\in [0, 1]$.\nLet $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \\rightarrow \\mathbb{R}$ such that $$f(1) - 2f\\left(\\frac{2}{3}\\right) + f\\left(\\frac{1}{3}\\right) \\ge t\\left(f\\left(\\frac{2}{3}\\right) - 2f\\left(\\frac{1}{3}\\right) + f(0)\\right).$$\nFor which real numbers $t \\ge 0$ is $S_t$ closed under multiplication?", - "informal_solution" : "$S_t$ is closed under multiplication if and only if $1 \\ge t$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1980_b6", - "informal_statement" : "For integers $d, n$ with $1 \\le d \\le n$, let $G(1, n) = \\frac{1}{n}$ and $G(d, n) = \\frac{d}{n}\\sum_{i=d}^{n}G(d - 1, i - 1)$ for all $d > 1$. If $1 < d \\le p$ for some prime $p$, prove that the reduced denominator of $G(d, p)$ is not divisible by $p$.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1979_a1", - "informal_statement" : "For which positive integers $n$ and $a_1, a_2, \\dots, a_n$ with $\\sum_{i = 1}^{n} a_i = 1979$ does $\\prod_{i = 1}^{n} a_i$ attain the greatest value?", - "informal_solution" : "$n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1979_a2", - "informal_statement" : "For which real numbers $k$ does there exist a continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?", - "informal_solution" : "Such a function exists if and only if $k \\ge 0$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1979_a3", - "informal_statement" : "Let $x_1, x_2, x_3, \\dots$ be a sequence of nonzero real numbers such that $$x_n = \\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \\ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$?", - "informal_solution" : "We must have $x_1 = x_2 = m$ for some integer $m$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1979_a4", - "informal_statement" : "Let $A$ be a set of $2n$ points in the plane, $n$ colored red and $n$ colored blue, such that no three points in $A$ are collinear. Must there exist $n$ closed straight line segments, each connecting one red and one blue point in $A$, such that no two of the $n$ line segments intersect?", - "informal_solution" : "Such line segments must exist.", - "tags" : ["geometry", "combinatorics"] - }, - { - "problem_name" : "putnam_1979_a5", - "informal_statement" : "Let $S(x)$ denote the sequence $\\lfloor 0 \\rfloor, \\lfloor x \\rfloor, \\lfloor 2x \\rfloor, \\lfloor 3x \\rfloor, \\dots$, where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$. Prove that there exist distinct real roots $\\alpha$ and $\\beta$ of $x^3 - 10x^2 + 29x - 25$ such that infinitely many positive integers appear in both $S(\\alpha)$ and $S(\\beta)$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1979_a6", - "informal_statement" : "For all $i \\in \\{0, 1, \\dots, n - 1\\}$, let $p_i \\in [0, 1]$. Prove that there exists some $x \\in [0, 1]$ such that $$\\sum_{i = 0}^{n - 1} \\frac{1}{|x - p_i|} \\le 8n\\left(\\sum_{i = 0}^{n-1} \\frac{1}{2i + 1}\\right).$$", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1979_b2", - "informal_statement" : "If $0 < a < b$, find $$\\lim_{t \\to 0} \\left( \\int_{0}^{1}(bx + a(1-x))^t dx \\right)^{\\frac{1}{t}}$$ in terms of $a$ and $b$.", - "informal_solution" : "The limit equals $$e^{-1}\\left(\\frac{b^b}{a^a}\\right)^{\\frac{1}{b-a}}.$$", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_42.json b/informal/batch_42.json deleted file mode 100644 index a1424282..00000000 --- a/informal/batch_42.json +++ /dev/null @@ -1,55 +0,0 @@ -[ - { - "problem_name" : "putnam_1979_b3", - "informal_statement" : "Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \\in F$ is $x^2 + bx + c + d$ irreducible?", - "informal_solution" : "Show that there are $\\frac{n - 1}{2}$ such elements $d$.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1979_b6", - "informal_statement" : "Let $z_i$ be complex numbers for $i = 1, 2, \\dots, n$. Show that \n\\[\n\\left \\lvert \\mathrm{Re} \\, [(z_1^2 + z_2^2 + \\dots + z_n^2)^{1/2} ] \\right \\rvert \\leq \\lvert \\mathrm{Re} \\, z_1 \\rvert + \\lvert \\mathrm{Re} \\, z_2 \\rvert + \\dots + \\lvert \\mathrm{Re} \\, z_n \\rvert.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1978_a1", - "informal_statement" : "Let $S = \\{1, 4, 7, 10, 13, 16, \\dots , 100\\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1978_a2", - "informal_statement" : "Let $A$ be the real $n \\times n$ matrix $(a_{ij})$ where $a_{ij} = a$ for $i < j$, $b \\; (\\neq a)$ for $i > j$, and $c_i$ for $i = j$. Show that $\\det A = \\frac{b p(a) - a p(b)}{b - a}$, where $p(x) = \\prod_{i=1}^n (c_i - x)$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1978_a3", - "informal_statement" : "Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let\n\\[\nI_k = \\int_0^{\\infty} \\frac{x^k}{p(x)} \\, dx.\n\\]\nFor which $k$ is $I_k$ smallest?", - "informal_solution" : "Show that $I_k$ is smallest for $k = 2$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1978_a4", - "informal_statement" : "A binary operation (represented by multiplication) on $S$ has the property that $(ab)(cd) = ad$ for all $a, b, c, d$. Show that: \n\\begin{itemize}\n\\item[(1)] if $ab = c$, then $cc = c$; \n\\item[(2)] if $ab = c$, then $ad = cd$ for all $d$. \n\\end{itemize}\nFind a set $S$, and such a binary operation, which also satisfies:\n\\begin{itemize}\n\\item[(A)] $a a = a$ for all $a$; \n\\item[(B)] $ab = a \\neq b$ for some $a, b$; \n\\item[(C)] $ab \\neq a$ for some $a, b$.\n\\end{itemize}", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1978_a5", - "informal_statement" : "Let $a_1, a_2, \\dots , a_n$ be reals in the interval $(0, \\pi)$ with arithmetic mean $\\mu$. Show that \n\\[\n\\prod_{i=1}^n \\left( \\frac{\\sin a_i}{a_i} \\right) \\leq \\left( \\frac{\\sin \\mu}{\\mu} \\right)^n.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1978_a6", - "informal_statement" : "Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart.", - "tags" : ["geometry", "combinatorics"] - }, - { - "problem_name" : "putnam_1978_b2", - "informal_statement" : "Find\n\\[\n\\sum_{i=1}^{\\infty} \\sum_{j=1}^{\\infty} \\frac{1}{i^2j + 2ij + ij^2}.\n\\]", - "informal_solution" : "Prove that the sum evaluates to $\\frac{7}{4}$.", - "tags" : ["algebra", "analysis"] - }, - { - "problem_name" : "putnam_1978_b3", - "informal_statement" : "The polynomials $P_n(x)$ are defined by \n\\begin{align*}\nP_1(x) &= 1 + x, \\\\\nP_2(x) &= 1 + 2x, \\\\\nP_{2n+1}(x) &= P_{2n}(x) + (n + 1) x P_{2n-1}(x), \\\\\nP_{2n+2}(x) &= P_{2n+1}(x) + (n + 1) x P_{2n}(x). \n\\end{align*}\nLet $a_n$ be the largest real root of $P_n(x)$. Prove that $a_n$ is strictly monotonically increasing and tends to zero.", - "tags" : ["algebra", "analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_43.json b/informal/batch_43.json deleted file mode 100644 index 15ba0269..00000000 --- a/informal/batch_43.json +++ /dev/null @@ -1,59 +0,0 @@ -[ - { - "problem_name" : "putnam_1978_b4", - "informal_statement" : "Show that we can find integers $a, b, c, d$ such that $a^2 + b^2 + c^2 + d^2 = abc + abd + acd + bcd$, and the smallest of $a, b, c, d$ is arbitrarily large.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1978_b5", - "informal_statement" : "Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \\subseteq [0, 1]$.", - "informal_solution" : "Prove that $p(x) = 4x^4 - 4x^2 + 1$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1978_b6", - "informal_statement" : "Let $a_{ij}$ be real numbers in $[0, 1]$. Show that\n\\[\n\\left( \\sum_{i=1}^n \\sum_{j=1}^{mi} \\frac{a_{ij}}{i} \\right) ^2 \\leq 2m \\sum_{i=1}^n \\sum_{j=1}^{mi} a_{ij}.\n\\]", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_a1", - "informal_statement" : "Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.", - "informal_solution" : "Prove that $k = -\\frac{7}{8}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_a2", - "informal_statement" : "Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = \\frac{1}{d}$.", - "informal_solution" : "Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_a3", - "informal_statement" : "Let $f, g, h$ be functions $\\mathbb{R} \\to \\mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \\frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \\frac{h(x + 4) + h(x - 4)}{2}$.", - "informal_solution" : "Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_a4", - "informal_statement" : "Find $\\sum_{n=0}^{\\infty} \\frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \\in (0, 1)$.", - "informal_solution" : "Prove that the sum equals $\\frac{x}{1 - x}$.", - "tags" : ["algebra", "analysis"] - }, - { - "problem_name" : "putnam_1977_a5", - "informal_statement" : "Let $p$ be a prime and $m \\geq n$ be non-negative integers. Show that $\\binom{pm}{pn} = \\binom{m}{n} \\pmod p$, where $\\binom{m}{n}$ is the binomial coefficient.", - "tags" : ["algebra", "number_theory"] - }, - { - "problem_name" : "putnam_1977_a6", - "informal_statement" : "Let $X$ be the square $[0, 1] \\times [0, 1]$, and let $f : X \\to \\mathbb{R}$ be continuous. If $\\int_Y f(x, y) \\, dx \\, dy = 0$ for all squares $Y$ such that\n\\begin{itemize}\n\\item[(1)] $Y \\subseteq X$, \n\\item[(2)] $Y$ has sides parallel to those of $X$, \n\\item[(3)] at least one of $Y$'s sides is contained in the boundary of $X$, \n\\end{itemize}\nis it true that $f(x, y) = 0$ for all $x, y$?", - "informal_solution" : "Prove that $f(x,y)$ must be identically zero.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1977_b1", - "informal_statement" : "Find $\\prod_{n=2}^{\\infty} \\frac{(n^3 - 1)}{(n^3 + 1)}$.", - "informal_solution" : "Prove that the product equals $\\frac{2}{3}$.", - "tags" : ["algebra", "analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_44.json b/informal/batch_44.json deleted file mode 100644 index 22f176a1..00000000 --- a/informal/batch_44.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_1977_b3", - "informal_statement" : "An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\\textit{balanced}$ if all three elements are less than $\\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and $f(x) = 2x$ otherwise. Will finitely many iterations of this balancing act always eventually produce a balanced triple?", - "informal_solution" : "Not necessarily.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_b5", - "informal_statement" : "If $a_1, a_2, \\dots, a_n$ are real numbers with $n > 1$ and $A$ satisfies $$A + \\sum_{i = 1}^{n} a_i^2 < \\frac{1}{n-1}\\left(\\sum_{i=1}^{n}a_i\\right)^2,$$ prove that $A < 2a_{i}a_{j}$ for all $i, j$ with $1 \\le i < j \\le n$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1977_b6", - "informal_statement" : "Let $G$ be a group and $H$ be a subgroup of $G$ with $h$ elements. Suppose that $G$ contains some element $a$ such that $(xa)^3 = 1$ for all $x \\in H$ (here $1$ represents the identity element of $G$). Let $P$ be the subset of $G$ containing all products of the form $x_1 a x_2 a \\cdots x_n a$ with $n \\ge 1$ and $x_i \\in H$ for all $i \\in \\{1, 2, \\dots, n\\}$. Prove that $P$ is a finite set and contains no more than $3h^2$ elements.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1976_a2", - "informal_statement" : "Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1976_a3", - "informal_statement" : "Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.", - "informal_solution" : "The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1976_a4", - "informal_statement" : "Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$.", - "informal_solution" : "The possible answers are $s = -\\frac{1}{r + 1}$ and $s = -\\frac{r + 1}{r}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1976_a6", - "informal_statement" : "Suppose that $f : \\mathbb{R} \\to \\mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \\le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$.", - "informal_solution" : "None.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1976_b1", - "informal_statement" : "Find $$\\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^{n}\\left(\\left\\lfloor \\frac{2n}{k} \\right\\rfloor - 2\\left\\lfloor \\frac{n}{k} \\right\\rfloor\\right).$$ Your answer should be in the form $\\ln(a) - b$, where $a$ and $b$ are positive integers.", - "informal_solution" : "The limit equals $\\ln(4) - 1$, so $a = 4$ and $b = 1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1976_b2", - "informal_statement" : "Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \\cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \\ne 1$ and $B \\ne 1$. Find the number of elements of $G$ than can be written as $C^2$ for some $C \\in G$ and express each such square as a word in $A$ and $B$.", - "informal_solution" : "There are $8$ such squares: $1$, $A^2$, $B$, $B^2$, $B^3$, $B^4$, $B^5$, and $B^6$.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1976_b6", - "informal_statement" : "Let $\\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \\textit{quasiperfect} if $\\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/batch_45.json b/informal/batch_45.json deleted file mode 100644 index 62e238be..00000000 --- a/informal/batch_45.json +++ /dev/null @@ -1,50 +0,0 @@ -[ - { - "problem_name" : "putnam_1976_b5", - "informal_statement" : "Find $$\\sum_{k=0}^{n} (-1)^k {n \\choose k} (x - k)^n.$$", - "informal_solution" : "The sum equals $n!$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1975_a1", - "informal_statement" : "If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \\frac{a^2 + a}{2} + \\frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.", - "informal_solution" : "$x = a + b + 1$ and $y = a - b$ (or vice versa).", - "tags" : ["algebra", "number_theory"] - }, - { - "problem_name" : "putnam_1975_a2", - "informal_statement" : "For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\\{|z| < 1\\}$) in the complex plane?", - "informal_solution" : "The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1975_a3", - "informal_statement" : "If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\\{(x, y, z) \\in \\mathbb{R}^3 : x^b + y^b + z^b = 1, x \\ge 0, y \\ge 0, z \\ge 0\\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?", - "informal_solution" : "$f$ attains its maximum at $\\left(x_0, (1 - x_0^b)^{\\frac{1}{b}}, 0\\right)$ and its minimum at $\\left(0, (1 - z_0^b)^{\\frac{1}{b}}, z_0\\right)$, where $x_0 = \\left(\\frac{a}{b}\\right)^{\\frac{1}{b-a}}$ and $z_0 = \\left(\\frac{b}{c}\\right)^{\\frac{1}{c-b}}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1975_b1", - "informal_statement" : "Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.", - "informal_solution" : "$a$ must equal $7$.", - "tags" : ["abstract_algebra", "number_theory"] - }, - { - "problem_name" : "putnam_1975_b3", - "informal_statement" : "Let $s_k (a_1, a_2, \\dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \\dots, a_n) = \\sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\\frac{s_k (a_1, a_2, \\dots, a_n)}{(s_1 (a_1, a_2, \\dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \\dots, a_n)$ of positive real numbers with $n \\ge k$.", - "informal_solution" : "The supremum $M_k$ is $\frac{1}{k!}$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1975_b4", - "informal_statement" : "Let $C = \\{(x, y) \\in \\mathbb{R}^2 : x^2 + y^2 = 1\\}$ denote the unit circle. Does there exist $B \\subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$?", - "informal_solution" : "Such $B$ does not exist.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1975_b5", - "informal_statement" : "Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \\ge 0$. Prove that $$\\sum_{n=0}^{\\infty} \\frac{f_n(1)}{n!} = e^e.$$", - "informal_solution" : "None.", - "tags" : ["analysis", "algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_46.json b/informal/batch_46.json deleted file mode 100644 index 3a8beb9c..00000000 --- a/informal/batch_46.json +++ /dev/null @@ -1,242 +0,0 @@ -[ - { - "problem_name" : "putnam_1974_a1", - "informal_statement" : "Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?", - "informal_solution" : "Show that the answer is 11.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1974_a3", - "informal_statement" : "A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \\equiv 1 \\bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$.", - "informal_solution" : "Show that that the answer to (a) is the set of primes which are $1 \\bmod 8$, and the solution to (b) is the set of primes which are $5 \\bmod 8$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1974_a4", - "informal_statement" : "Evaluate in closed form: $\\frac{1}{2^{n-1}} \\sum_{k < n/2} (n-2k)*{n \\choose k}$.", - "informal_solution" : "Show that the solution is $\\frac{n}{2^{n-1}} * {(n-1) \\choose \\left[ (n-1)/2 \\right]}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1974_a6", - "informal_statement" : "Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.", - "informal_solution" : "Show that the answer is 25.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1974_b3", - "informal_statement" : "Prove that if $\\alpha$ is a real number such that $\\cos (\\pi \\alpha) = 1/3$, the $\\alpha$ is irrational.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1974_b4", - "informal_statement" : "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous in each variable seperately. Show that there exists a sequence of continuous functions $g_n : \\mathbb{R}^2 \\to \\mathbb{R}$ such that $f(x,y) = \\lim_{n \\to \\infty} g_n(x,y)$ for all $(x,y) \\in \\mathbb{R}^2$." , - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1974_b5", - "informal_statement" : "Show that $1 + (n/1!) + (n^2/2!) + \\dots + (n^n/n!) > e^n/2$ for every integer $n \\geq 0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1974_b6", - "informal_statement" : "For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\\equiv 0 \\bmod 3, \\equiv 1 \\bmod 3, \\equiv 2 \\bmod 3$?", - "informal_solution" : "Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\\equiv 0 \\bmod 3$ and $\\equiv 1 \\bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\\equiv 2 \\bmod 3$.", - "tags" : ["set_theory"] - }, - { - "problem_name" : "putnam_1973_a3", - "informal_statement" : "Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $k + \\frac{n}{k}$ as $k$ is allowed to range through all positive integers. Prove that $b(n)$ and $\\sqrt{4n + 1}$ have the same integer part.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1973_a4", - "informal_statement" : "How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?", - "informal_solution" : "Show that the solution is 3.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1973_a6", - "informal_statement" : "Prove that it is impossible for seven distinct straight lines to be situated in the Euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines interest.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1973_b1", - "informal_statement" : "Let $a_1, \\dots, a_{2n + 1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \\dots = a_{2n+1}$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1973_b2", - "informal_statement" : "Let $z = x+iy$ be a complex number with $x$ and $y$ rational and with $\\| z \\| = 1$. Show thaat the number $\\| z^{2n} - 1 \\|$ is rational for every integer $n$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1973_b3", - "informal_statement" : "Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \\leq c$, and $-a \\leq b < a$.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1973_b4", - "informal_statement" : "Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \\leq 1$ and $f 0 = 0$. Prove that $\\left[\\int_0^1 f(x) dx\\right]]^2 \\geq \\int_0^1 (f(x))^3 dx$, and find an example where equality holds.", - "informal_solution" : "Show that one such example where equality holds is the identity function.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1972_a1", - "informal_statement" : "Show that there are no four consecutive binomial coefficients ${n \\choose r}, {n \\choose (r+1)}, {n \\choose (r+2)}, {n \\choose (r+3)}$ where $n,r$ are positive integers and $r+3 \\leq n$, which are in arithmetic progression.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1972_a2", - "informal_statement" : "Let $S$ be a set and $\\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \\cdot (x \\cdot y) = y$ (2) for all $x,y$ in $S$, $(y \\cdot x) \\cdot x = y$. Show that $\\cdot$ is commutative but not necessarily associative.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1972_a3", - "informal_statement" : "We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \\dots$ whenever it does for the sequence $x_1, x_2, x_3 \\dots$. Find all supercontinuous functions on $[0,1]$.", - "informal_solution" : "Show that the solution is the set of affine functions.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1972_a5", - "informal_statement" : "Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1972_a6", - "informal_statement" : "Let $f$ be an integrable function in $0 \\leq x \\leq 1$ and suppose for all $0 \\leq i \\leq n-1, \\int_0^1 x^i f(x) dx = 0$. Further suppose that $\\int_0^1 x^n f(x) dx = 1$. Show that $|f(x)| \\geq 2^n(n+1)$ on a set of positive measure.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1972_b3", - "informal_statement" : "Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1972_b4", - "informal_statement" : "Let $n \\geq 2$ be an integer. Show that there exists a polynomial $P(x,y,z)$ with integral coefficients such that $x \\equiv P(x^n, x^{n+1}, x + x^{n+2})$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1972_b6", - "informal_statement" : "Let $n_1 < n_2 < \\dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \\dots + z^{n_k}$ has not roots inside the circle $|z| < (\\frac{\\sqrt{5}-1}{2}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1971_a2", - "informal_statement" : "Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.", - "informal_solution" : "Show that the only such polynomial is the identity function.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1971_a3", - "informal_statement" : "The three vertices of a triangle of sides $a,b,c$ are lattice points and lie on a circle of radius $R$. Show that $abc \\geq 2R$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1971_a4", - "informal_statement" : "Show that for $\\epsilon \\in (0,1)$, the expression $(x + y)^n (x^2 - 2-\\epsilon)xy + y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large, where $n$ is an integer. ", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1971_a6", - "informal_statement" : "Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1971_b1", - "informal_statement" : "Let $S$ be a set and let $\\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \\cdot x$, (2) for all $x,y,z$ in $S$, $(x \\cdot y) \\cdot z) = (y \\cdot z) \\cdot x$. Show that $\\cdot$ is associative and commutative. ", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1971_b6", - "informal_statement" : "Let $\\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\\sum_{n = 1}^x \\delta(n)/n - 2x/3| < 1$ for all positive integers $x$." , - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1970_a4", - "informal_statement" : "Suppose $(x_n)$ is a sequence such that $\\lim_{n \\to \\infty} (x_n - x_{n-2} = 0$. Prove that $\\lim_{n \\to \\infty} \\frac{x_n - x_{n-1}}{n} = 0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_b1", - "informal_statement" : "Evaluate the infinite product $\\lim_{n \\to \\infty} \\frac{1}{n^4} \\prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.", - "informal_solution" : "Show that the solution is $e^{2 \\log(5) - 4 + 2 arctan(2)}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_b3", - "informal_statement" : "A closed subset $S$ of $\\mathbb{R}^2$ lies in $a < x < b$. Show that its projection on the y-axis is closed.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_b5", - "informal_statement" : "Let $u_n$ denote the function $u_n(x) = -n$ if $x \\leq -n$, $x$ if $-n < x \\leq n$, and $n$ otherwise. Let $F$ be a function on the reals. Show that $F$ is continuous if and only if $u_n \\circ F$ is continuous for all natural numbers $n$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1969_a2", - "informal_statement" : "Let $D_n$ be the determinant of the $n$ by $n$ matrix whose value in the $i$th row and $j$th column is $|i-j|$. Show that $D_n = (-1)^{n-1} * (n-1) * (2^{n-2}).$", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_1969_a4", - "informal_statement" : "Show that $\\int_0^1 x^x dx = \\sum_{n=1}^{\\infty} (-1)^{n+1}n^{-n}$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1969_a6", - "informal_statement" : "Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \\geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1969_b1", - "informal_statement" : "Let $n$ be a positive integer such that $n+1$ is divisible by $24$. Prove that the sum of all the divisors of $n$ is divisible by $24$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1969_b2", - "informal_statement" : "Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?", - "informal_solution" : "Show that the statement is no longer true if 'two' is replaced by 'three'.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1969_b3", - "informal_statement" : "Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \\geq 1$, and also $\\lim_{n \\to \\infty} \\frac{T_n}{T_{n+1}} = 1. Show that $\\pi * T_1^2 = 2$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1969_b5", - "informal_statement" : "Let $a_1 < a_2 < a_3 < \\dots$ be an increasing sequence of positive integers. Assume that the sequences $\\sum_{i = 1}^{\\infty} 1/(a n)$ is convergent. For any number $x$, let $k(x)$ be the number of $a_n$'s which do not exceed $x$. Show that $\\lim_{x \\to \\infty} k(x)/x = 0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_47.json b/informal/batch_47.json deleted file mode 100644 index 81b343e7..00000000 --- a/informal/batch_47.json +++ /dev/null @@ -1,38 +0,0 @@ -[ - { - "problem_name" : "putnam_1972_b1", - "informal_statement" : "Prove that no three consecutive coefficients of the power series of $$\\sum_{n = 0}^{\\infty} \\frac{x^n(x - 1)^{2n}}{n!}$$ all equal $0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1971_a1", - "informal_statement" : "Let $S$ be a set of $9$ lattice points (points with integer coordinates) in $3$-dimensional Euclidean space. Prove that there exists a lattice point along the interior of some line segment that joins two distinct points in $S$.", - "informal_solution" : "None.", - "tags" : ["geometry", "combinatorics"] - }, - { - "problem_name" : "putnam_1971_b2", - "informal_statement" : "Find all functions $F : \\mathbb{R} \\setminus \\{0, 1\\} \\to \\mathbb{R}$ that satisfy $F(x) + F\\left(\\frac{x - 1}{x}\\right) = 1 + x$ for all $x \\in \\mathbb{R} \\setminus \\{0, 1\\}$.", - "informal_solution" : "The only such function is $F(x) = \\frac{x^3 - x^2 - 1}{2x(x - 1)}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1970_a1", - "informal_statement" : "Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \\cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1970_a3", - "informal_statement" : "Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.", - "informal_solution" : "The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1969_b6", - "informal_statement" : "Let $A$ be a $3 \\times 2$ matrix and $B$ be a $2 \\times 3$ matrix such that $$AB = \n\\begin{pmatrix}\n8 & 2 & -2 \\\\\n2 & 5 & 4 \\\\\n-2 & 4 & 5\n\\end{pmatrix}.\n$$ Prove that $$BA = \n\\begin{pmatrix}\n9 & 0 \\\\\n0 & 9\n\\end{pmatrix}.$$", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_48.json b/informal/batch_48.json deleted file mode 100644 index 615bc91f..00000000 --- a/informal/batch_48.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_1968_a1", - "informal_statement" : "Prove that $$\\frac{22}{7} - \\pi = \\int_{0}^{1} \\frac{x^4(1 - x)^4}{1 + x^2} dx$$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1968_a2", - "informal_statement" : "For all integers $a$, $b$, $c$, $d$, $e$, and $f$ such that $ad \\neq bc$ and any real number $\\epsilon > 0$, prove that there exist rational numbers $r$ and $s$ such that $$0 < |ra + sb - e| < \\varepsilon$$ and $$0 < |rc + sd - f| < \\varepsilon.$$", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1968_a3", - "informal_statement" : "Let $S$ be a finite set. Prove that there exists a list of subsets of $S$ such that\n\\begin{enumerate}\n\\item The first element of the list is the empty set,\n\\item Each subset of $S$ occurs exactly once in the list, and\n\\item Each successive element in the list is formed by adding or removing one element from the previous subset in the list.\n\\end{enumerate}", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1968_a4", - "informal_statement" : "Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\\{(x, y, z) \\mid x^2 + y^2 + z^2 = 1\\}$ is at most $n^2$.", - "informal_solution" : "None.", - "tags" : ["geometry", "algebra"] - }, - { - "problem_name" : "putnam_1968_a5", - "informal_statement" : "Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \\le 1$ for all $x \\in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \\in V$.", - "informal_solution" : "The supremum is $8$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1968_a6", - "informal_statement" : "Find all polynomials of the form $\\sum_{0}^{n} a_{i} x^{n-i}$ with $n \\ge 1$ and $a_i = \\pm 1$ for all $0 \\le i \\le n$ whose roots are all real.", - "informal_solution" : "The set of such polynomials is $$\\{\\pm (x - 1), \\pm (x + 1), \\pm (x^2 + x - 1), \\pm (x^2 - x - 1), \\pm (x^3 + x^2 - x - 1), \\pm (x^3 - x^2 - x + 1)\\}.$$", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1968_b2", - "informal_statement" : "Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_1968_b4", - "informal_statement" : "Suppose that $f : \\mathbb{R} \\to \\mathbb{R}$ is continuous on $(-\\infty, \\infty)$ and that $\\int_{-\\infty}^{\\infty} f(x) dx$ exists. Prove that $$\\int_{-\\infty}^{\\infty} f\\left(x - \\frac{1}{x}\\right) dx = \\int_{-\\infty}^{\\infty} f(x) dx.$$", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1968_b5", - "informal_statement" : "Let $p$ be a prime number. Find the number of distinct $2 \\times 2$ matrices $$\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$$ such that $a, b, c, d \\in \\{0, 1, ..., p - 1\\}$, $a + d \\equiv 1 \\pmod p$, and $ad - bc \\equiv 0 \\pmod p$.", - "informal_solution" : "There are $p^2 + p$ such matrices.", - "tags" : ["linear_algebra", "number_theory", "combinatorics"] - }, - { - "problem_name" : "putnam_1968_b6", - "informal_statement" : "Prove that no sequence $\\{K_n\\}_{n=0}^{\\infty}$ of compact (closed and bounded) sets of rational numbers has the property that every compact set of rational numbers is contained by at least one $K_n$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_49.json b/informal/batch_49.json deleted file mode 100644 index f755916f..00000000 --- a/informal/batch_49.json +++ /dev/null @@ -1,38 +0,0 @@ -[ - { - "problem_name" : "putnam_1967_a1", - "informal_statement" : "Let $f(x)=a_1\\sin x+a_2\\sin 2x+\\dots+a_n\\sin nx$, where $a_1,a_2,\\dots,a_n$ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \\leq |\\sin x|$ for all real $x$, prove that $|a_1|+|2a_2|+\\dots+|na_n| \\leq 1$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1967_a2", - "informal_statement" : "Define $S_0$ to be $1$. For $n \\geq 1$, let $S_n$ be the number of $n \\times n$ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$, ($i,j=1,2,\\dots,n$) and where $\\sum_{i=1}^n a_{ij}=1$, ($j=1,2,\\dots,n$). Prove\n\\begin{enumerate}\n\\item[(a)] $S_{n+1}=S_n+nS_{n-1}$\n\\item[(b)] $\\sum_{n=0}^\\infty S_n\\frac{x^n}{n!} = \\exp(x+x^2/2)$, where $\\exp(x)=e^x$.\n\\end{enumerate}", - "informal_solution" : "None.", - "tags" : ["linear_algebra", "analysis"] - }, - { - "problem_name" : "putnam_1967_a3", - "informal_statement" : "Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0 \\frac{1}{2}$ there does not exist a real-valued function $u$ such that for all $x$ in the closed interval $0 \\leq x \\leq 1$, $u(x)=1+\\lambda\\int_x^1 u(y)u(y-x)\\,dy$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1967_b2", - "informal_statement" : "Let $0 \\leq p \\leq 1$ and $0 \\leq r \\leq 1$ and consider the identities\n\\begin{enumerate}\n\\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,\n\\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\\alpha x^2+\\beta xy+\\gamma y^2$.\n\\end{enumerate}\nShow that (with respect to $p$ and $r$)\n\\begin{enumerate}\n\\item[(a)] $\\max\\{A,B,C\\} \\geq 4/9$,\n\\item[(b)] $\\max\\{\\alpha,\\beta,\\gamma\\} \\geq 4/9$.\n\\end{enumerate}", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1967_b3", - "informal_statement" : "If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then $\\lim_{n \\to \\infty} \\int_0^1 f(x)g(nx)\\,dx=(\\int_0^1 f(x)\\,dx)(\\int_0^1 g(x)\\,dx)$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] diff --git a/informal/batch_5.json b/informal/batch_5.json deleted file mode 100644 index 942648ab..00000000 --- a/informal/batch_5.json +++ /dev/null @@ -1,49 +0,0 @@ -[ - { - "problem_name" : "putnam_2016_a2", - "informal_statement" : "Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that\n\\[\n\binom{m}{n-1} > \\binom{m-1}{n}.\n\\]\nEvaluate\n\\[\n\\lim_{n \to \\infty} \\frac{M(n)}{n}.\n\\]", - "informal_solution" : "Show that the answer is $\\frac{3 + \\sqrt{5}}{2}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2016_a6", - "informal_statement" : "Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1]$,\n\\[\n\\int_0^1 \\left| P(x) \\right|\\,dx \\leq C \\max_{x \\in [0,1]} \\left| P(x) \\right|.\n\\]", - "informal_solution" : "Prove that the smallest such value of $C$ is $\\frac{5}{6}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2016_b1", - "informal_statement" : "Let $x_0,x_1,x_2,\\dots$ be the sequence such that $x_0=1$ and for $n \\geq 0$,\n\\[\nx_{n+1} = \\ln(e^{x_n} - x_n)\n\\]\n(as usual, the function $\\ln$ is the natural logarithm). Show that the infinite series\n\\[\nx_0 + x_1 + x_2 + \\cdots\n\\]\nconverges and find its sum.", - "informal_solution" : "The sum converges to $e - 1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2017_a4", - "informal_statement" : "A class with $2N$ students took a quiz, on which the possible scores were $0,1,\\dots,10$. Each of these scores occurred at least once, and the average score was exactly $7.4$. Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_b2", - "informal_statement" : "Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers\n\\[\nN = a + (a+1) +(a+2) + \\cdots + (a+k-1)\n\\]\nfor $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?", - "informal_solution" : "Prove that the smallest value of $a$ is $16$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_b4", - "informal_statement" : "Evaluate the sum \\begin{gather*} \\sum_{k=0}^\\infty \\left( 3 \\cdot \\frac{\\ln(4k+2)}{4k+2} - \\frac{\\ln(4k+3)}{4k+3} - \\frac{\\ln(4k+4)}{4k+4} - \\frac{\\ln(4k+5)}{4k+5} \right) \\ = 3 \\cdot \\frac{\\ln 2}{2} - \\frac{\\ln 3}{3} - \\frac{\\ln 4}{4} - \\frac{\\ln 5}{5} + 3 \\cdot \\frac{\\ln 6}{6} - \\frac{\\ln 7}{7} \\ - \\frac{\\ln 8}{8} - \\frac{\\ln 9}{9} + 3 \\cdot \\frac{\\ln 10}{10} - \\cdots . \\end{gather*} (As usual, $\\ln x$ denotes the natural logarithm of $x$.)", - "informal_solution" : "Prove that the sum equals $(\\ln 2)^2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2020_a2", - "informal_statement" : "Let $k$ be a nonnegative integer. Evaluate\n\\[\n\\sum_{j=0}^k 2^{k-j} \\binom{k+j}{j}.\n\\]\n", - "informal_solution" : "Show that the answer is $4^k$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2020_b4", - "informal_statement" : "Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\\mathbf{v} = (s_0, s_1, \\cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\\cdots,2n$. Define \\[ q(\\mathbf{v}) = 1 + \\sum_{j=1}^{2n-1} 3^{s_j}, \\] and let $M(n)$ be the average of $\\frac{1}{q(\\mathbf{v})}$ over all $\\mathbf{v} \\in V_n$. Evaluate $M(2020)$.", - "informal_solution" : "Show that the answer is $\\frac{1}{4040}$.", - "tags" : ["algebra"] - } -] \ No newline at end of file diff --git a/informal/batch_50.json b/informal/batch_50.json deleted file mode 100644 index db0a9cc6..00000000 --- a/informal/batch_50.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_1966_a1", - "informal_statement" : "Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \\dots$, where $a_n = \\frac{n}{2}$ if $n$ is even and $\\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1966_a3", - "informal_statement" : "If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \\ge 1$, prove that $\\lim_{n \\to \\infty} nx_n = 1$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1966_a4", - "informal_statement" : "Prove that the $n$th item in the ascending list of non-perfect-square positive integers equals $n + \\{\\sqrt{n}\\}$, where $\\{m\\}$ denotes the closest integer to $m$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1966_a5", - "informal_statement" : "Let $C$ be the set of continuous functions $f : \\mathbb{R} \\to \\mathbb{R}$. Let $T : C \\to C$ satisfty the following two properties:\n\\begin{enumerate}\n\\item Linearity: $T(af + bg) = aT(f) + bT(g)$ for all $a, b \\in \\mathbb{R}$ and all $f, g \\in C$.\n\\item Locality: If $f \\in C$ and $g \\in C$ are identical on some interval $I$, then $T(f)$ and $T(g)$ are also identical on $I$.\n\\end{enumerate}\nProve that there exists some function $f \\in C$ such that $T(g(x)) = f(x)g(x)$ for all $g \\in C$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1966_a6", - "informal_statement" : "Prove that $$\\sqrt {1 + 2 \\sqrt {1 + 3 \\sqrt {1 + 4 \\sqrt {1 + 5 \\sqrt {\\dots}}}}} = 3.$$", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1966_b2", - "informal_statement" : "Prove that, for any ten consecutive integers, at least one is relatively prime to all of the others.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1966_b3", - "informal_statement" : "Let $p_1, p_2, \\dots$ be a sequence of positive real numbers. Prove that if $\\sum_{n=1}^{\\infty} \\frac{1}{p_n}$ converges, then $$\\sum_{n=1}^{\\infty} \\frac {n^2 p_n}{(\\sum_{i=1}^{n} p_i)^2}$$ also converges.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1966_b4", - "informal_statement" : "Let $a_1, a_2, ...$ be an increasing sequence of $mn + 1$ positive integers. Prove that there exists either a subset of $m + 1$ $a_i$ such that no element of the subset divides any other, or a subset of $n + 1$ $a_i$ such that each element of the subset (except the greatest) divides the next greatest element.", - "informal_solution" : "None.", - "tags" : ["number_theory", "combinatorics"] - }, - { - "problem_name" : "putnam_1966_b6", - "informal_statement" : "Prove that any solution $y(x)$ to the differential equation $y'' + e^{x}y = 0$ remains bounded as $x$ goes to $+\\infty$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_51.json b/informal/batch_51.json deleted file mode 100644 index d88ec3ed..00000000 --- a/informal/batch_51.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_1965_a2", - "informal_statement" : "Prove that $$\\sum_{r=0}^{\\lfloor\\frac{n-1}{2}\\rfloor} \\left(\\frac{n - 2r}{n} {n \\choose r}\\right)^2 = \\frac{1}{n} {{2n - 2} \\choose {n - 1}}$$ for every positive integer $n$.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1965_a3", - "informal_statement" : "Prove that, for any sequence of real numbers $a_1, a_2, \\dots$, $$\\lim_{n \\to \\infty} \\frac{\\sum_{k = 1}^{n} e^{ia_k}}{n} = \\alpha$$ if and only if $$\\lim_{n \\to \\infty} \\frac{\\sum_{k = 1}^{n} e^{ia_{k^2}}}{n^2} = \\alpha.$$", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1965_a4", - "informal_statement" : "At a party, no boy dances with every girl, but each girl dances with at least one boy. Prove that there exist girls $g$ and $h$ and boys $b$ and $c$ such that $g$ dances with $b$ and $h$ dances with $c$, but $h$ does not dance with $b$ and $g$ does not dance with $c$.", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1965_a5", - "informal_statement" : "How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$?", - "informal_solution" : "There are $2^{n-1}$ such orderings.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1965_a6", - "informal_statement" : "Prove that the line $ux + vy = 1$ (where $u \\ge 0$ and $v \\ge 0$) will lie tangent to the curve $x^m + y^m = 1$ (where $m > 1$) if and only if $u^n + v^n = 1$ for some $n$ such that $m^{-1} + n^{-1} = 1$.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1965_b2", - "informal_statement" : "A round-robin tournament has $n > 1$ players $P_1, P_2, \\dots, P_n$, who each play one game with each other player. Each game results in a win for one player and a loss for the other. If $w_r$ and $l_r$ denote the number of games won and lost, respectively, by $P_r$, prove that $$\\sum_{r=1}^{n} w_r^2 = \\sum_{r=1}^{n} l_r^2.$$", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1965_b3", - "informal_statement" : "Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter.", - "informal_solution" : "None.", - "tags" : ["algebra", "geometry"] - }, - { - "problem_name" : "putnam_1965_b4", - "informal_statement" : "Let $$f(x, n) = \\frac{{n \\choose 0} + {n \\choose 2}x + {n \\choose 4}x^2 + \\cdots}{{n \\choose 1} + {n \\choose 3}x + {n \\choose 5}x^2 + \\cdots}$$ for all real numbers $x$ and positive integers $n$. Express $f(x, n+1)$ as a rational function involving $f(x, n)$ and $x$, and find $\\lim_{n \\to \\infty} f(x, n)$ for all $x$ for which this limit converges.", - "informal_solution" : "We have $$f(x, n+1) = \\frac{f(x, n) + x}{f(x, n) + 1};$$ $\\lim_{n \\to \\infty} f(x, n) = \\sqrt{x}$ for all $x \\ge 0$ and diverges otherwise.", - "tags" : ["algebra", "analysis"] - }, - { - "problem_name" : "putnam_1965_b5", - "informal_statement" : "Prove that, if $4E \\le V^2$, there exists a graph with $E$ edges and $V$ vertices with no triangles (cycles of length $3$).", - "informal_solution" : "None.", - "tags" : ["combinatorics"] - }, - { - "problem_name" : "putnam_1965_b6", - "informal_statement" : "Let $A$, $B$, $C$, and $D$ be four distinct points for which every circle through $A$ and $B$ intersects every circle through $C$ and $D$. Prove that $A$, $B$, $C$ and $D$ are either collinear (all lying on the same line) or cocyclic (all lying on the same circle).", - "informal_solution" : "None.", - "tags" : ["geometry"] - } -] \ No newline at end of file diff --git a/informal/batch_52.json b/informal/batch_52.json deleted file mode 100644 index a54d40ae..00000000 --- a/informal/batch_52.json +++ /dev/null @@ -1,58 +0,0 @@ -[ - { - "problem_name" : "putnam_1964_a1", - "informal_statement" : "Let $A_1, A_2, A_3, A_4, A_5, A_6$ be distinct points in the plane. Let $D$ be the longest distance between any pair, and let $d$ the shortest distance. Show that $\\frac{D}{d} \\geq \\sqrt 3$.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1964_a2", - "informal_statement" : "Let $\\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \\to (0, \\infty)$ such that \n\\begin{align*}\n\\int_0^1 f(x) dx &= 1, \\\\\n\\int_0^1 x f(x) dx &= \\alpha, \\\\\n\\int_0^1 x^2 f(x) dx &= \\alpha^2. \\\\\n\\end{align*}", - "informal_solution" : "Prove that there are no such functions.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1964_a3", - "informal_statement" : "The distinct points $x_n$ are dense in the interval $(0, 1)$. For all $n \\geq 1$, $x_1, x_2, \\dots , x_{n-1}$ divide $(0, 1)$ into $n$ sub-intervals, one of which must contain $x_n$. This part is divided by $x_n$ into two sub-intervals, lengths $a_n$ and $b_n$. Prove that $\\sum_{n=1}^{\\infty} a_nb_n(a_n + b_n) = \\frac{1}{3}$.", - "tags" : ["analysis", "algebra"] - }, - { - "problem_name" : "putnam_1964_a4", - "informal_statement" : "The sequence of integers $u_n$ is bounded and satisfies \n\\[\nu_n = \\frac{u_{n-1} + u_{n-2} + u_{n-3}u_{n-4}}{u_{n-1}u_{n-2} + u_{n-3} + u_{n-4}}.\n\\] \nShow that it is periodic for sufficiently large $n$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1964_a5", - "informal_statement" : "Prove that there exists a constant $k$ such that for any sequence $a_i$ of positive numbers, \n\\[\n\\sum_{n=1}^{\\infty} \\frac{n}{a_1 + a_2 + \\dots + a_n} \\leq k \\sum_{n=1}^{\\infty}\\frac{1}{a_n}.\n\\]", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1964_a6", - "informal_statement" : "Let $S$ be a finite set of collinear points. Let $k$ be the maximum distance between any two points of $S$. Given a pair of points of $S$ a distance $d < k$ apart, we can find another pair of points of $S$ also a distance $d$ apart. Prove that if two pairs of points of $S$ are distances $a$ and $b$ apart, then $\frac{a}{b}$ is rational.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1964_b1", - "informal_statement" : "Let $a_n$ be a sequence of positive integers such that $\\sum_{n=1}^{\\infty} 1/a_n$ converges. For all $n$, let $b_n$ be the number of $a_n$ which are at most $n$. Prove that $\\lim_{n \\to \\infty} b_n/n = 0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1964_b2", - "informal_statement" : "Let $S$ be a finite set. A set $P$ of subsets of $S$ has the property that any two members of $P$ have at least one element in common and that $P$ cannot be extended (whilst keeping this property). Prove that $P$ contains exactly half of the subsets of $S$.", - "tags" : ["set_theory", "combinatorics"] - }, - { - "problem_name" : "putnam_1964_b3", - "informal_statement" : "Suppose $f : \\mathbb{R} \\to \\mathbb{R}$ is continuous and for every $\\alpha > 0$, $\\lim_{n \\to \\infty} f(n\\alpha) = 0$. Prove that $\\lim_{x \\to \\infty} f(x) = 0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1964_b5", - "informal_statement" : "Let $a_n$ be a strictly monotonic increasing sequence of positive integers. Let $b_n$ be the least common multiple of $a_1, a_2, \\dots , a_n$. Prove that $\\sum_{n=1}^{\\infty} 1/b_n$ converges.", - "tags" : ["analysis", "number_theory"] - }, - { - "problem_name" : "putnam_1964_b6", - "informal_statement" : "Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \\cap B = \\emptyset$ and $A \\cup B = D$.", - "tags" : ["geometry"] - } -] \ No newline at end of file diff --git a/informal/batch_55.json b/informal/batch_55.json deleted file mode 100644 index 54b8a530..00000000 --- a/informal/batch_55.json +++ /dev/null @@ -1,50 +0,0 @@ -[ - { - "problem_name" : "putnam_1962_a1", - "informal_statement" : "Given five points in a plane, no three of which lie on a straight line, show that some four of these points form the vertices of a convex quadrilateral.", - "informal_solution" : "None.", - "tags" : ["geometry"] - }, - { - "problem_name" : "putnam_1962_a2", - "informal_statement" : "Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.", - "informal_solution" : "Show that \\[ f(x) = \\frac{a}{(1 - cx)^2} \\begin{cases} \\text{for } 0 \\le x < \\frac{1}{c}, & \\text{if } c > 0\\\\ \\text{for } 0 \\le x < \\infty, & \\text{if } c \\le 0, \\end{cases} \\] where $a > 0$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1962_a4", - "informal_statement" : "Assume that $\\lvert f(x) \\rvert \\le 1$ and $\\lvert f''(x) \\rvert \\le 1$ for all $x$ on an interval of length at least 2. Show that $\\lvert f'(x) \\rvert \\le 2$ on the interval.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1962_a5", - "informal_statement" : "Evaluate in closed form \\[ \\sum_{k=1}^n {n \\choose k} k^2. \\]", - "informal_solution" : "Show that the expression equals $n(n+1)2^{n-2}$.", - "tags" : ["algebra", "combinatorics"] - }, - { - "problem_name" : "putnam_1962_a6", - "informal_statement" : "Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \\[ r \\in S, -r \\in S, r = 0. \\] Prove that $S$ is the set of all positive rational numbers.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1962_b1", - "informal_statement" : "Let $x^{(n)} = x(x-1)\\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \\[ (x+y)^{(n)} = \\sum_{k=0}^n {n \\choose k} x^{(k)} y^{(n-k)}. \\]", - "informal_solution" : "None.", - "tags" : ["algebra", "combinatorics"] - }, - { - "problem_name" : "putnam_1962_b5", - "informal_statement" : "Prove that for every integer $n$ greater than 1: \\[ \\frac{3n+1}{2n+2} < \\left( \\frac{1}{n} \\right)^n + \\left(\\frac{2}{n} \\right)^n + \\cdots + \\left(\\frac{n}{n} \\right)^n < 2. \\]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1962_b6", - "informal_statement" : "Let \\[ f(x) = \\sum_{k=0}^n a_k \\sin kx + b_k \\cos kx, \\] where $a_k$ and $b_k$ are constants. Show that, if $\\lvert f(x) \\rvert \\le 1$ for $0 \\le x \\le 2 \\pi$ and $\\lvert f(x_i) \\rvert = 1$ for $0 \\le x_1 < x_2 < \\cdots < x_{2n} < 2 \\pi$, then $f(x) = \\cos (nx + a)$ for some constant $a$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] \ No newline at end of file diff --git a/informal/batch_6.json b/informal/batch_6.json deleted file mode 100644 index 4b7ca5e4..00000000 --- a/informal/batch_6.json +++ /dev/null @@ -1,56 +0,0 @@ -[ - { - "problem_name" : "putnam_2014_a2", - "informal_statement" : "Let $A$ be the $n \\times n$ matrix whose entry in the $i$-th row and $j$-th column is $\\frac{1}{\\min(i,j)}$ for $1 \\leq i,j \\leq n$. Compute $\\det(A)$.", - "informal_solution" : "Show that the determinant is $\\frac{(-1)^{n-1}}{(n-1)!n!}$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2015_a2", - "informal_statement" : "Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \\geq 2$. Find an odd prime factor of $a_{2015}$.", - "informal_solution" : "Show that one possible answer is $181$.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2015_a3", - "informal_statement" : "Compute $\\log_2 \\left( \\prod_{a=1}^{2015}\\prod_{b=1}^{2015}(1+e^{2\\pi iab/2015}) \\right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$).", - "informal_solution" : "Show that the answer is $13725$.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_2015_a5", - "informal_statement" : "Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $00$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2016_b6", - "informal_statement" : "Evaluate $\\sum_{k=1}^\\infty \\frac{(-1)^{k-1}}{k} \\sum_{n=0}^\\infty \\frac{1}{k2^n+1}$.", - "informal_solution" : "Show that the desired sum equals $1$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2019_a4", - "informal_statement" : "Let $f$ be a continuous real-valued function on $\\mathbb{R}^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals $0$. Must $f(x,y,z)$ be identically 0?", - "informal_solution" : "Show that the answer is no.", - "tags" : ["analysis"] - } -] diff --git a/informal/batch_7.json b/informal/batch_7.json deleted file mode 100644 index f3b9d1bb..00000000 --- a/informal/batch_7.json +++ /dev/null @@ -1,50 +0,0 @@ -[ - { - "problem_name" : "putnam_2013_a3", - "informal_statement" : "Suppose that the real numbers \\( a_0, a_1, \\ldots, a_n \\) and \\( x \\), with \\( 0 < x < 1 \\), satisfy $ \\frac{a_0}{1-x} + \\frac{a_1}{(1-x)^2} + \\cdots + \\frac{a_n}{(1-x)^{n+1}} = 0. $ Prove that there exists a real number \\( y \\) with \\( 0 < y < 1 \\) such that $ a_0 + a_1y + \\cdots + a_ny^n = 0. $.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2014_a3", - "informal_statement" : "Let \\( a_0 = \\frac{5}{2} \\) and \\( a_k = a_{k-1}^2 - 2 \\) for \\( k \\geq 1 \\). Compute \\( \\prod_{k=0}^{\\infty} \\left(1 - \\frac{1}{a_k}\\right) \\) in closed form.", - "informal_solution" : "Show that the solution is 3/7.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2014_a6", - "informal_statement" : "Let \\( n \\) be a positive integer. What is the largest \\( k \\) for which there exist \\( n \\times n \\) matrices \\( M_1, \\ldots, M_k \\) and \\( N_1, \\ldots, N_k \\) with real entries such that for all \\( i \\) and \\( j \\), the matrix product \\( M_i N_j \\) has a zero entry somewhere on its diagonal if and only if \\( i \\neq j \\)?", - "informal_solution" : "Show that the solution has the form k \\<= n ^ n.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2014_b2", - "informal_statement" : "Suppose that \\( f \\) is a function on the interval \\([1,3]\\) such that \\(-1 \\leq f(x) \\leq 1\\) for all \\( x \\) and \\( \\int_{1}^{3} f(x) \\, dx = 0 \\). How large can \\(\\int_{1}^{3} \\frac{f(x)}{x} \\, dx \\) be?", - "informal_solution" : "Show that the solution is log (4 / 3).", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2014_b4", - "informal_statement" : "Show that for each positive integer \\( n \\), all the roots of the polynomial $ \\sum_{k=0}^{n} 2^k(n-k)x^k $ are real numbers.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2018_a2", - "informal_statement" : "Let \\( S_1, S_2, \\ldots, S_{2^n-1} \\) be the nonempty subsets of \\( \\{1, 2, \\ldots, n\\} \\) in some order, and let \\( M \\) be the \\( (2^n - 1) \\times (2^n - 1) \\) matrix whose \\((i, j)\\) entry is $m_{ij} = \\begin{cases} 0 & \\text{if } S_i \\cap S_j = \\emptyset; \\\\ 1 & \\text{otherwise}. \\end{cases} $ Calculate the determinant of \\( M \\).", - "informal_solution" : "Show that the solution is 1 if n = 1, and otherwise -1.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2019_a6", - "informal_statement" : "Let \\( g \\) be a real-valued function that is continuous on the closed interval \\([0,1]\\) and twice differentiable on the open interval \\((0,1)\\). Suppose that for some real number $\\( r > 1 \\),\\lim_{{x \\to 0^+}} \\frac{g(x)}{x^r} = 0.$ Prove that either $\\lim_{{x \\to 0^+}} g'(x) = 0$ or $\\limsup_{{x \\to 0^+}} x^r |g''(x)| = \\infty.$", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2019_b6", - "informal_statement" : "Let \\( \\mathbb{Z}^n \\) be the integer lattice in \\( \\mathbb{R}^n \\). Two points in \\( \\mathbb{Z}^n \\) are called neighbors if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers \\( n \\geq 1 \\) does there exist a set of points \\( S \\subset \\mathbb{Z}^n \\) satisfying the following two conditions? \\begin{enumerate} \\item If \\( p \\) is in \\( S \\), then none of the neighbors of \\( p \\) is in \\( S \\). \\item If \\( p \\in \\mathbb{Z}^n \\) is not in \\( S \\), then exactly one of the neighbors of \\( p \\) is in \\( S \\). \\end{enumerate}", - "informal_solution" : "Show that the above statement is True.", - "tags" : ["algebra"] - } -] diff --git a/informal/batch_8.json b/informal/batch_8.json deleted file mode 100644 index 523854c6..00000000 --- a/informal/batch_8.json +++ /dev/null @@ -1,62 +0,0 @@ -[ - { - "problem_name" : "putnam_2012_a2", - "informal_statement" : "Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$.", - "informal_solution" : "None.", - "tags" : ["abstract_algebra"] - }, - { - "problem_name" : "putnam_2013_a6", - "informal_statement" : "Define a function $w:\\mathbb{Z} \\times \\mathbb{Z} \\to \\mathbb{Z}$ as follows. For $|a|,|b| \\leq 2$, let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0$.\n\\begin{center}\n\\begin{tabular}{|cc|r|r|r|r|r|}\n\\hline\n\\multicolumn{2}{|c|}{\\multirow{2}{*}{$w(a,b)$}} & \\multicolumn{5}{|c|}{$b$} \\\\\n& & -2 & -1 & 0 & 1 & 2 \\\\\n\\hline\n& -2 & -1 & -2 & 2 & -2 & -1 \\\\\n& -1 & -2 & 4 & -4 & 4 & -2 \\\\\n$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\\\\n& 1 & -2 & 4 & -4 & 4 & -2 \\\\\n& 2 & -1 & -2 & 2 & -2 & -1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nFor every finite subset $S$ of $\\mathbb{Z} \\times \\mathbb{Z}$, define $A(S)=\\sum_{(\\mathbf{s},\\mathbf{s}') \\in S \\times S} w(\\mathbf{s}-\\mathbf{s}')$. Prove that if $S$ is any finite nonempty subset of $\\mathbb{Z} \\times \\mathbb{Z}$, then $A(S)>0$. (For example, if $S=\\{(0,1),(0,2),(2,0),(3,1)\\}$, then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4$.)", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2013_b1", - "informal_statement" : "For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\\sum_{n=1}^{2013} c(n)c(n+2)$.", - "informal_solution" : "Show that the desired sum is $-1$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2013_b4", - "informal_statement" : "For any continuous real-valued function $f$ defined on the interval $[0,1]$, let $\\mu(f)=\\int_0^1 f(x)\\,dx,\\text{Var}(f)=\\int_0^1 (f(x)-\\mu(f))^2\\,dx,M(f)=\\max_{0 \\leq x \\leq 1} |f(x)|$. Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1]$, then $\\text{Var}(fg) \\leq 2\\text{Var}(f)M(g)^2+2\\text{Var}(g)M(f)^2$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2013_b5", - "informal_statement" : "Let $X=\\{1,2,\\dots,n\\}$, and let $k \\in X$. Show that there are exactly $k \\cdot n^{n-1}$ functions $f:X \\to X$ such that for every $x \\in X$ there is a $j \\geq 0$ such that $f^{(j)}(x) \\leq k$. [Here $f^{(j)}$ denotes the $j$\\textsuperscript{th} iterate of $f$, so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f(f^{(j)}(x))$.]", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_a2", - "informal_statement" : "Let $Q_0(x)=1$, $Q_1(x)=x$, and $Q_n(x)=\\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}$ for all $n \\geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2017_a3", - "informal_statement" : "Let $a$ and $b$ be real numbers with $a 0$, \n \\item[(b)] $g(0) = 0$, \n \\item[(c)] $|f'(x)| \\leq |g(x)|$ for all $x$, \n \\item[(d)] $|g'(x)| \\leq |f(x)|$ for all $x$, and \n \\item[(e)] $f(r) = 0$. \\end{enumerate}", - "informal_solution" : "Show that the solution is $r = \\pi/2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2023_a5", - "informal_statement" : "For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that \\[ \\sum_{k=0}^{3^{1010}-1} (-2)^{f(k)} (z+k)^{2023} = 0. \\]", - "informal_solution" : "Show that the solution is the set of complex numbers $\\{- \\frac{3^{1010} - 1}{2} \\pm \\frac{\\sqrt{9^{1010} - 1}}{4} * i \\}$", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2022_a1", - "informal_statement" : "Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \\ln(1+x^2)$ in exactly one point.", - "informal_solution" : "Show that the solution is the set of ordered pairs $(a,b)$ which satisfy at least one of (1) $a = b = 0$, (2) $|a| \\geq 1$, and (3) $0 < |a| < 1$ and $b < \\log(1 - r_{-})^2 - |a|r_{-}$ or $b > \\log(1 + r_{+})^2 + |a|r_{+}$ where $r_{\\pm} = \\frac{1 \\pm \\sqrt{1 - a^2}}{a}$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2022_a2", - "informal_statement" : "Let $n$ be an integer with $n \\geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$?", - "informal_solution" : "Show that the solution is $2n - 2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2022_a3", - "informal_statement" : "Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \\dots$ such that $a_n \\in \\{1, 2, \\dots, p-1\\}$ and $a_n a_{n+2} \\equiv 1 + a_{n+1} \\pmod{p}$ for all $n \\geq 1$. Prove that $f(p)$ is congruent to 0 or 2 $\\pmod{5}$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2019_b3", - "informal_statement" : "Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u \\in \\mathbb{R}^n$ be a unit column vector (that is, $u^T u = 1$). Let $P = I - 2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.", - "informal_solution" : "None.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2016_a1", - "informal_statement" : "Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \\[ p^{(j)}(k) = \\left. \\frac{d^j}{dx^j} p(x) \\right|_{x=k} \\] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.", - "informal_solution" : "Show that the solution is $18$.", - "tags" : ["algebra", "number_theory"] - }, - { - "problem_name" : "putnam_2013_b3", - "informal_statement" : "Let $\\mathcal{P}$ be a nonempty collection of subsets of $\\{1,\\dots, n\\}$ such that: \\begin{enumerate} \\item[(i)] if $S, S' \\in \\mathcal{P}$, then $S \\cup S' \\in \\mathcal{P}$ and $S \\cap S' \\in \\mathcal{P}$, and \\item[(ii)] if $S \\in \\mathcal{P}$ and $S \\neq \\emptyset$, then there is a subset $T \\subset S$ such that $T \\in \\mathcal{P}$ and $T$ contains exactly one fewer element than $S$. \\end{enumerate} Suppose that $f: \\mathcal{P} \\to \\mathbb{R}$ is a function such that $f(\\emptyset) = 0$ and \\[f(S \\cup S') = f(S) + f(S') - f(S \\cap S') \\mbox{ for all $S,S' \\in \\mathcal{P}$.} \\] Must there exist real numbers $f_1,\\dots,f_n$ such that\\[f(S) = \\sum_{i \\in S} f_i\\] \\n for every $S \\in \\mathcal{P}$?", - "informal_solution" : "None.", - "tags" : ["set_theory"] - }, - { - "problem_name" : "putnam_2012_a5", - "informal_statement" : "Let $\\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\\FF_p^n$, let $M$ be an $n \\times n$ matrix with entries of $\\FF_p$, and define $G: \\FF_p^n \\to \\FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\\dots,p^n$ are distinct.", - "informal_solution" : "Show that the solution is the pairs $(p,n)$ with $n = 1$ as well as the single pair $(2,2)$.", - "tags" : ["linear_algebra"] - }, - { - "problem_name" : "putnam_2012_b5", - "informal_statement" : "Prove that, for any two bounded functions $g_1, g_2: \\RR \\to [1, \\infty)$, there exist functions $h_1, h_2: \\RR \to \\RR$ such that, for every $x \\in \\RR$, \\[ \\sup_{s \\in \\RR} (g_1(s)^x g_2(s)) = \\max_{t \\in \\RR} (x h_1(t) + h_2(t)).\\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2012_b6", - "informal_statement" : "Let $p$ be an odd prime number such that $p \\equiv 2 \\pmod{3}$. Define a permutation $\\pi$ of the residue classes modulo $p$ by $\\pi(x) \\equiv x^3 \\pmod{p}$. Show that $\\pi$ is an even permutation if and only if $p \\equiv 3 \\pmod{4}$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2011_a2", - "informal_statement" : "Let $a_1,a_2,\\dots$ and $b_1,b_2,\\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \\[ S = \\sum_{n=1}^\\infty \\frac{1}{a_1...a_n} \\] converges, and evaluate $S$.", - "informal_solution" : "Show that the solution is $S = 3/2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2010_b1", - "informal_statement" : "Is there an infinite sequence of real numbers $a_1, a_2, a_3, \\dots$ such that \\[ a_1^m + a_2^m + a_3^m + \\cdots = m \\] for every positive integer $m$?", - "informal_solution" : "Show that the solution is no such infinite sequence exists.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2010_b5", - "informal_statement" : "Is there a strictly increasing function $f: \\mathbb{R} \\to \\mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?", - "informal_solution" : "Show that the solution is no such function exists.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2009_a3", - "informal_statement" : "Let $d_n$ be the determinant of the $n \\times n$ matrix whose entries, from left to right and then from top to bottom, are $\\cos 1, \\cos 2, \\dots, \\cos n^2$. (For example,\\[ d_3 = \\left|\\begin{matrix} \\cos 1 & \\cos 2 & \\cos 3 \\\\ \\cos 4 & \\cos 5 & \\cos 6 \\\\ \\cos 7 & \\cos 8 & \\cos 9 \\end{matrix} \\right|. \\]The argument of $\\cos$ is always in radians, not degrees.) Evaluate $\\lim_{n\\to\\infty} d_n$.", - "informal_solution" : "Show that the limit is 0.", - "tags" : ["linear_algebra", "analysis"] - }, { - "problem_name" : "putnam_2008_b1", - "informal_statement" : "What is the maximum number of rational points that can lie on a circle in $\\mathbb{R}^2$ whose center is not a rational point? (A \\emph{rational point} is a point both of whose coordinates are rational numbers.)", - "informal_solution" : "Show that the maximum number is $2$.", - "tags" : ["geometry", "number_theory"] - }, - { - "problem_name" : "putnam_2006_b6", - "informal_statement" : "Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \\[ a_{n+1} = a_n + \\frac{1}{\\sqrt[k]{a_n}} \\] for $n > 0$. Evaluate \\[\\lim_{n \\to \\infty} \\frac{a_n^{k+1}}{n^k}.\\]", - "informal_solution" : "Show that the solution is $(\\frac{k+1}{k})^k$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2005_a1", - "informal_statement" : "Show that every positive integer is a sum of one or more numbers of the form $2^r 3^s$, where $r$ and $s$ are nonnegative integers and no summand divides another.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_2005_a5", - "informal_statement" : "Evaluate $\\int_0^1 \\frac{\\ln(x+1)}{x^2+1}\\,dx$.", - "informal_solution" : "Show that the solution is $\\pi / 8 * \\log 2$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2003_b6", - "informal_statement" : "Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \\[ \\int_0^1 \\int_0^1 | f(x) + f(y) |\\,dx\\,dy \\geq \\int_0^1 |f(x)|\\,dx. \\]", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_2002_a3", - "informal_statement" : "Let $n \\geq 2$ be an integer and $T_n$ be the number of non-empty subsets $S$ of $\\{1, 2, 3, \\dots, n\\}$ with the property that the average of the elements of $S$ is an integer. Prove that $T_n - n$ is always even.", - "informal_solution" : "None.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_2001_a5", - "informal_statement" : "Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - } -] \ No newline at end of file diff --git a/informal/putnam_1963.json b/informal/putnam_1963.json deleted file mode 100644 index 1ccbef3e..00000000 --- a/informal/putnam_1963.json +++ /dev/null @@ -1,44 +0,0 @@ -[ - { - "problem_name" : "putnam_1963_a2", - "informal_statement" : "Let $\\{f(n)\\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$.", - "informal_solution" : "None.", - "tags" : ["number_theory", "algebra"] - }, - { - "problem_name" : "putnam_1963_a4", - "informal_statement" : "Let $\\{a_n\\}$ be a sequence of positive real numbers. Show that $\\limsup_{n \\to \\infty} n\\left(\\frac{1+a_{n+1}}{a_n}-1\\right) \\geq 1$. Show that the number $1$ on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol $\\limsup$ is sometimes written $\\overline{\\lim}$.)", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_b1", - "informal_statement" : "For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?", - "informal_solution" : "Show that $a=2$.", - "tags" : ["algebra"] - }, - { - "problem_name" : "putnam_1963_b2", - "informal_statement" : "Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$?", - "informal_solution" : "Show that $S$ is dense in $P$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_b3", - "informal_statement" : "Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$.", - "informal_solution" : "Show that the solution is the sets of functions $f(u)=A\\sinh ku$, $f(u)=Au$, and $f(u)=A\\sin ku$ with $A,k \\in \\mathbb{R}$.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_b5", - "informal_statement" : "Let $\\{a_n\\}$ be a sequence of real numbers satisfying the inequalities $0 \\leq a_k \\leq 100a_n$ for $n \\leq k \\leq 2n$ and $n=1,2,\\dots$, and such that the series $\\sum_{n=0}^\\infty a_n$ converges. Prove that $\\lim_{n \\to \\infty}na_n=0$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - }, - { - "problem_name" : "putnam_1963_b6", - "informal_statement" : "Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\\dots$. Prove that $A_2=A_3=\\cdots$. (A one-point set should be considered to be a special case of a closed segment.)", - "informal_solution" : "None.", - "tags" : ["geometry", "linear_algebra"] - } -] diff --git a/informal/putnam_1967.json b/informal/putnam_1967.json deleted file mode 100644 index cbb39c4f..00000000 --- a/informal/putnam_1967.json +++ /dev/null @@ -1,20 +0,0 @@ -[ - { - "problem_name" : "putnam_1967_a6", - "informal_statement" : "Given real numbers $\\{a_i\\}$ and $\\{b_i\\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \\neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $4$-tuple of plus and minus signs $(\\text{signum }x_1,\\text{signum }x_2,\\text{signum }x_3,\\text{signum }x_4)$. Determine, with a proof, the maximum number of distinct $4$-tuples possible.", - "informal_solution" : "Show that the maximum number of distinct $4$-tuples is eight.", - "tags" : ["algebra", "geometry"] - }, - { - "problem_name" : "putnam_1967_b4", - "informal_statement" : "A certain locker room contains $n$ lockers numbered $1,2,3,\\cdots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1,T_2,\\cdots,T_n$ whereby with the operation $T_k$, $1 \\leq k \\leq n$, the condition of being locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of $k$. After all the $n$ operations have been performed it is observed that all lockers whose numbers are perfect squares (and only those lockers) are now open or unlocked. Prove this mathematically.", - "informal_solution" : "None.", - "tags" : ["number_theory"] - }, - { - "problem_name" : "putnam_1967_b6", - "informal_statement" : "Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2+y^2 \\leq 1$ and is such that $|f(x,y)| \\leq 1$. Show that there exists a point $(x_0,y_0)$ in the interior of the unit circle such that $\\left(\\frac{\\partial f}{\\partial x} (x_0,y_0)\\right)^2+\\left(\\frac{\\partial f}{\\partial y} (x_0,y_0)\\right)^2 \\leq 16$.", - "informal_solution" : "None.", - "tags" : ["analysis"] - } -] From ec48d6f99737a1a6a276a1f569878a88c613312a Mon Sep 17 00:00:00 2001 From: George Tsoukalas Date: Wed, 17 Jul 2024 19:09:04 +0000 Subject: [PATCH 03/12] Added website leaderboard, modified Lean files. --- README.md | 26 +- coq/README.md | 3 +- docs/index.html | 85 ++- docs/leaderboard.html | 563 ++++++++++++++++++++ docs/results.json | 114 ++++ isabelle/README.md | 2 + isabelle/{ => scripts}/convert_to_json.py | 0 isabelle/{ => scripts}/rewrite_solutions.py | 0 lean4/scripts/rewrite_solutions.py | 6 +- lean4/src/putnam_1962_a3.lean | 3 +- lean4/src/putnam_1963_a6.lean | 7 +- lean4/src/putnam_1965_a1.lean | 2 +- lean4/src/putnam_1966_a2.lean | 7 +- lean4/src/putnam_1967_a5.lean | 3 +- lean4/src/putnam_1967_b1.lean | 11 +- lean4/src/putnam_1970_b6.lean | 9 +- lean4/src/putnam_1973_a1.lean | 3 +- 17 files changed, 777 insertions(+), 67 deletions(-) create mode 100644 docs/leaderboard.html create mode 100644 docs/results.json rename isabelle/{ => scripts}/convert_to_json.py (100%) rename isabelle/{ => scripts}/rewrite_solutions.py (100%) diff --git a/README.md b/README.md index e7435778..31b20159 100644 --- a/README.md +++ b/README.md @@ -4,11 +4,11 @@ PutnamBench is a benchmark for evaluation of theorem-proving algorithms on compe PutnamBench aims to support research in automated mathematical reasoning by providing a multilingual benchmark for evaluating theorem-proving algorithms. It is released under permissive licenses (Apache 2.0 for Lean 4 and Isabelle, MIT for Coq). The informal statements are also available with permission from the MAA. -We strongly encourage community feedback! Please let us know if you have any comments for improving PutnamBench. If you notice any mistakes, please raise an issue on the repository and we will address it. We kindly ask that you do not write formal proofs for any of the problems in an effort to reduce contamination. +PutnamBench includes factored solutions for problems which require exhibiting a numerical answer in addition to its proof of correctness. For these problems, one can attempt two tasks: proving the problem with the numerical answer written into the theorem statement, or additionally producing the answer along with the proof. -## Citation -The associated paper for PutnamBench is {TODO}. Please consider including the following citation if you find PutnamBench useful. -{TODO} +We are hosting a [**leaderboard**](https://github.com/trishullab/PutnamBench) and will readily receive evaluation results which are accompanied by a preprint or publication. + +**We strongly encourage community feedback!** Please let us know if you have any comments for improving PutnamBench. If you notice any mistakes, please raise an issue on the repository and we will address it. We kindly ask that you do not write formal proofs for any of the problems in an effort to reduce contamination. ## Statistics | Language | Count | @@ -17,7 +17,7 @@ The associated paper for PutnamBench is {TODO}. Please consider including the fo | Isabelle | 640 | | Coq | 417 | -We also report the number of problems in a certain category. Note that some problems fall under multiple categories. +We also report the number of problems in a certain category. Note that some problems fall under multiple categories. While the categories are intended to capture general features of the problems, we indicate that there is a high variance of problems inside an individual category. | Category | Total Quantity | | ---------------- | -------------- | @@ -31,8 +31,20 @@ We also report the number of problems in a certain category. Note that some prob | Probability | 9 | | Set Theory | 8 | - - ## Versioning - Version: `v0` - In preliminary release to allow for initial community feedback. We seek to release an official first version following several weeks of discussion with the community. + +## Citation +The associated paper for PutnamBench is [available at this link](https://arxiv.org/abs/2407.11214). Please consider including the following citation if you find PutnamBench useful. +``` +@misc{tsoukalas2024putnambenchevaluatingneuraltheoremprovers, + title={PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition}, + author={George Tsoukalas and Jasper Lee and John Jennings and Jimmy Xin and Michelle Ding and Michael Jennings and Amitayush Thakur and Swarat Chaudhuri}, + year={2024}, + eprint={2407.11214}, + archivePrefix={arXiv}, + primaryClass={cs.AI}, + url={https://arxiv.org/abs/2407.11214}, +} +``` \ No newline at end of file diff --git a/coq/README.md b/coq/README.md index 2489d7bc..70243bcf 100644 --- a/coq/README.md +++ b/coq/README.md @@ -1,5 +1,6 @@ -Here lies the Coq formalizations, currently in the format of batches of formalizations. +Note: We are continuing to make modifications to the Coq formalizations, in particular those which were recently added. We encourage feedback, but keep this in mind. The format of the formalizations will be standardized with the other languages upon completion. +We also note that while the dependencies listed in each formalization are sufficient to *state* the problem, one may need further mathematical theory developed in Coq to write a proof. # Installation You need to install `opam` and then run `setup.sh` to install the necessary dependencies. diff --git a/docs/index.html b/docs/index.html index e537b00a..356be7f7 100644 --- a/docs/index.html +++ b/docs/index.html @@ -34,6 +34,7 @@ + @@ -115,42 +116,42 @@

PutnamBench: A Multilingual Competition

- University of Texas at Austin + The University of Texas at Austin
- - Paper: TODO, points to github + Paper - - arXiv: TODO, points to github + arXiv - - Code + Dataset - @@ -205,7 +206,7 @@

Formal Theorem-Proving

Informal benchmarks such as MATH and GSM8K measure accuracy by exact match with the ground truth numerical answer. However, the reasoning path might be incorrect and cannot be checked automatically as the problem is specified in natural language. - By contrast, benchmarking formal theorem-proving through PutnamBench allows one to measure a model's capacity to produce precise logical arguments in a mathematical context. Furthermore, since no formal proofs have been written for any of our problems, direct contamination is not possible, whereas more measures must be taken to prevent contamination with informal reasoning benchmarks. + By contrast, benchmarking formal theorem-proving through PutnamBench allows one to measure a model's capacity to produce precise logical arguments in a mathematical context. Furthermore, since no formal proofs have been written for any of our problems, direct contamination is not possible. When training models for informal reasoning, one must take many more precautions to prevent test-set leakage.

@@ -229,7 +230,7 @@

The William-Lowell Putnam Mathematical Competition

The Putnam Competition is known for its challenging problems that require a deep understanding of undergraduate-level mathematics and creative problem-solving ability. The competition includes problems involving concepts from a wide variety of topics in the undergraduate curriculum, including analysis, abstract algebra, and linear algebra.

- The competition is scored out of 120 points, with the median score typically being zero. The top five individual scorers are named Putnam Fellows, and in recent years regularly include former International Mathematical Olympiad (IMO) medalists. Hence, we find that PutnamBench is a well-suited benchmark for measuring progress towards the Artificial Intelligence Mathematical Olympiad and the IMO Grand Challenge." + The competition is scored out of 120 points, with the median score typically being zero. The top five individual scorers are named Putnam Fellows, and in recent years regularly include former International Mathematical Olympiad (IMO) medalists. Hence, we find that PutnamBench is a well-suited benchmark for measuring progress towards the Artificial Intelligence Mathematical Olympiad and the IMO Grand Challenge.

@@ -267,13 +268,13 @@

PutnamBench: Multilinguality & Factored Solutions

Evaluations & Leaderboard

- We perform evaluations with some recent models, as described in the TODO: PAPER. - We find that PutnamBench is hard, with an aggregration of all models tried unable to successfully prove even 1% of theorems. + We perform evaluations with some recent models, as described in the paper. + We find that PutnamBench is hard, with an aggregration of all methodologies tried unable to successfully prove even 1% of theorems. In particular, few-shot GPT-4o solves only a single problem in each language when allotted 10 attempts. PutnamBench highlights several deficiencies of large language models (LLMs): weak ability to (1) perform precise logical reasoning, (2) operate in low-resource languages.

- We are actively maintaining a TODO: Leaderboard to keep track of progress from upcoming approaches for neural-theorem proving. We will be incorporating results which are accompanied by a research publication (or arxiv preprint) detailing the general methodology. + We are actively maintaining a leaderboard to keep track of progress from upcoming approaches for neural-theorem proving. We will be incorporating results which are accompanied by a research publication (or arxiv preprint) detailing the general methodology.

@@ -281,6 +282,37 @@

Evaluations & Leaderboard

+
+
+ +
+
+

Sample Problem & Proof

+
+

Putnam 1988 B1. A composite (positive integer) is a produce \(ab\) with \(a\) and \(b\) not necessarily distinct integers in \(\{2,3,4,\dots\}\). Show that every composite is expressible as \(xy + xz + yz + 1\), with \(x,y,z\) positive integers.

+ 

+            theorem putnam_1988_b1 :
+            ∀ a ≥ 2, ∀ b ≥ 2, ∃ x y z : ℤ, 
+              x > 0 ∧ y > 0 ∧ z > 0 ∧ a * b = x * y + x * z + y * z + 1
+            := by
+              intro a ha b hb
+              use a - 1, b - 1, 1
+              constructor
+              linarith
+              constructor
+              linarith
+              constructor
+              linarith
+              ring
+
+

Here we include a formalization of Putnam 1988 B1 in Lean. The proof is generated by GPT-4o, and crucially requires the choice of \(x,y,z = a-1,b-1,1\). Once this choice is specified, the proof is completed with automated tactics for linear arithmetic and ring manipulations.

+

Putnam 1988 B1 is among the easiest problems to ever appear on the Putnam Competition. We believe progress on PutnamBench will come, in part, as a result of advancements in (1) proof planning & intermediate lemma synthesis and (2) ability to reason about and utilize existing formalized mathematical theory.

+
+
+
+
+
+
@@ -292,7 +324,8 @@

Licensing & Maintenance

PutnamBench is available under an Apache 2.0 License for Isabelle & Lean, and an MIT License for Coq. With a big thanks to the MAA, we also include the informal problem statements as sourced from the official archives.

- PutnamBench will be actively maintained for the foreseeable future. We plan to incorporate problem formalizations from Putnam 2024 in December. We are actively encouraging community feedback and discussion. We are interested in hearing how you might plan to use PutnamBench! Your feedback may help us iterate on the structure of PutnamBench, such as incorporating a validation/training split. Please reach out to us if you have (1) suggestions to improve the benchmark, (2) comments on maintaining benchmark longevity and usefulness, (3) noticed issues in certain formalizations which require modification. + PutnamBench will be actively maintained for the foreseeable future. We plan to incorporate problem formalizations from Putnam 2024 in December.

+

We are actively encouraging community feedback and discussion. We are interested in hearing how you might plan to use PutnamBench! Your feedback may help us iterate on the structure of PutnamBench, such as incorporating a validation/training split. Please reach out to us if you have (1) suggestions to improve the benchmark, (2) comments on maintaining benchmark longevity and usefulness, (3) noticed issues in certain formalizations which require modification

@@ -304,11 +337,14 @@

Licensing & Maintenance

BibTeX

- 
@article{TODO,
-  author    = {TODO},
-  title     = {TODO},
-  journal   = {TODO},
-  year      = {TODO},
+    
@misc{tsoukalas2024putnambenchevaluatingneuraltheoremprovers,
+      title={PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition}, 
+      author={George Tsoukalas and Jasper Lee and John Jennings and Jimmy Xin and Michelle Ding and Michael Jennings and Amitayush Thakur and Swarat Chaudhuri},
+      year={2024},
+      eprint={2407.11214},
+      archivePrefix={arXiv},
+      primaryClass={cs.AI},
+      url={https://arxiv.org/abs/2407.11214}, 
 }
@@ -329,16 +365,7 @@

BibTeX

- This website is licensed under a Creative - Commons Attribution-ShareAlike 4.0 International License. -

-

- This means you are free to borrow the source code of this website, - we just ask that you link back to this page in the footer. - Please remember to remove the analytics code included in the header of the website which - you do not want on your website. + we have borrowed this site template from the Nerfies webpage.

diff --git a/docs/leaderboard.html b/docs/leaderboard.html new file mode 100644 index 00000000..3ea429dd --- /dev/null +++ b/docs/leaderboard.html @@ -0,0 +1,563 @@ + + + + + + + + + PutnamBench Leaderboard + + + + + + + + + + + + +
+

🏆 PutnamBench Leaderboard 🏆

+

+ Benchmarking formal mathematical reasoning on the Putnam Mathematical Compeition.
 +

+
+ github + paper +
+
+ + + + + + +
+ +
+
+ +
+
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+ +
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+ +
+
+
+
+

📝 Notes

+
+
    +
  1. + As no existing methods have been benchmarked on PutnamBench without numerical answers in the theorem statement, the leaderboard for that variant contains no entries. Please share you results with us and we will promptly update the leaderboard! +
    2. +
  2. + We thank the EvalPlus team for providing the leaderboard template. +
    3. +
+
+
+
+ + + + \ No newline at end of file diff --git a/docs/results.json b/docs/results.json new file mode 100644 index 00000000..f328e1d1 --- /dev/null +++ b/docs/results.json @@ -0,0 +1,114 @@ +{ + "DSP (GPT-4o)": { + "link": "https://arxiv.org/abs/2210.12283", + "open-data": "PARTIAL", + "pass@1": { + "isabelle-wsolution": 4, + "isabelle-nosolution": "NONE", + "lean-wsolution": "NONE", + "lean-nosolution": "NONE", + "coq-wsolution": "NONE", + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@10 w/ T=0.1" + }, + "GPT-4o": { + "link": "https://openai.com/index/hello-gpt-4o/", + "open-data": "NONE", + "pass@1": { + "isabelle-wsolution": 1, + "isabelle-nosolution": "NONE", + "lean-wsolution": 1, + "lean-nosolution": "NONE", + "coq-wsolution": 1, + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@10 w/ T=0.7" + }, + "COPRA (GPT-4o)": { + "link": "https://arxiv.org/abs/2310.04353", + "open-data": "PARTIAL", + "pass@1": { + "isabelle-wsolution": "NONE", + "isabelle-nosolution": "NONE", + "lean-wsolution": 1, + "lean-nosolution": "NONE", + "coq-wsolution": 1, + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ T=0.0" + }, + "CoqHammer": { + "link": "https://coqhammer.github.io/", + "open-data": "FULL", + "pass@1": { + "isabelle-wsolution": "NONE", + "isabelle-nosolution": "NONE", + "lean-wsolution": "NONE", + "lean-nosolution": "NONE", + "coq-wsolution": 0, + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ t=120s" + }, + "Sledgehammer": { + "link": "https://isabelle.in.tum.de/website-Isabelle2009-1/sledgehammer.html", + "open-data": "FULL", + "pass@1": { + "isabelle-wsolution": 3, + "isabelle-nosolution": "NONE", + "lean-wsolution": "NONE", + "lean-nosolution": "NONE", + "coq-wsolution": "NONE", + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ t=120s" + }, + "ReProver w/ retrieval": { + "link": "https://arxiv.org/abs/2306.15626", + "open-data": "FULL", + "pass@1": { + "isabelle-wsolution": "NONE", + "isabelle-nosolution": "NONE", + "lean-wsolution": 0, + "lean-nosolution": "NONE", + "coq-wsolution": "NONE", + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ t=600s" + }, + "ReProver w/o retrieval": { + "link": "https://arxiv.org/abs/2306.15626", + "open-data": "FULL", + "pass@1": { + "isabelle-wsolution": "NONE", + "isabelle-nosolution": "NONE", + "lean-wsolution": 0, + "lean-nosolution": "NONE", + "coq-wsolution": "NONE", + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ t=600s" + }, + "Tactician (LSH)": { + "link": "https://coq-tactician.github.io/", + "open-data": "FULL", + "pass@1": { + "isabelle-wsolution": "NONE", + "isabelle-nosolution": "NONE", + "lean-wsolution": "NONE", + "lean-nosolution": "NONE", + "coq-wsolution": 0, + "coq-nosolution": "NONE" + }, + "size": 0, + "note": "pass@1 w/ t=600s" + } +} \ No newline at end of file diff --git a/isabelle/README.md b/isabelle/README.md index f83e57a1..323d4090 100644 --- a/isabelle/README.md +++ b/isabelle/README.md @@ -1 +1,3 @@ The Isabelle formalizations largely rely on the HOL Library, and are present in a problem-per-file format by default. + +We also note that while the dependencies listed in each formalization are sufficient to *state* the problem, one may need further mathematical theory developed in Isabelle to write a proof. We do not believe this will be the case for many problems, but please do let us know if you have particular feedback on this or with any other concerns. diff --git a/isabelle/convert_to_json.py b/isabelle/scripts/convert_to_json.py similarity index 100% rename from isabelle/convert_to_json.py rename to isabelle/scripts/convert_to_json.py diff --git a/isabelle/rewrite_solutions.py b/isabelle/scripts/rewrite_solutions.py similarity index 100% rename from isabelle/rewrite_solutions.py rename to isabelle/scripts/rewrite_solutions.py diff --git a/lean4/scripts/rewrite_solutions.py b/lean4/scripts/rewrite_solutions.py index e2e1b724..0ff71c01 100644 --- a/lean4/scripts/rewrite_solutions.py +++ b/lean4/scripts/rewrite_solutions.py @@ -20,19 +20,17 @@ def rewrite_solutions(file): solution = lines[i + 1].split('--')[1].strip() # the type comes between the first and second colon in the ith line # the solution comes after the second colon in the ith line - # TODO: actually get the thing between the first : and the last := or last : type = lines[i].split(':')[1].strip() solution_dictionary[solution_name] = {'solution' : solution.replace("\n", ""), 'type' : type} - print(solution_dictionary) - with open(f"/home/gtsoukal/Projects/PUTNAM/lean4/solutions_replaced_new/{problem_name}_sol.lean", "w") as f: + with open(f"/lean4/solutions_replaced_new/{problem_name}_sol.lean", "w") as f: for line in lines: for solution_name, solution in solution_dictionary.items(): if solution_name in line and 'abbrev' not in line: if solution['solution'][0] == '(' and solution['solution'][-1] == ')': solution['solution'] = solution['solution'][1:-1] line = line.replace(solution_name, f"(({solution['solution']}) : {solution['type']})") - if 'abbrev' in line: # TODO: Modify this on release + if 'abbrev' in line: continue f.write(line) diff --git a/lean4/src/putnam_1962_a3.lean b/lean4/src/putnam_1962_a3.lean index 31de9517..bd70d8e0 100644 --- a/lean4/src/putnam_1962_a3.lean +++ b/lean4/src/putnam_1962_a3.lean @@ -3,9 +3,8 @@ open BigOperators open MeasureTheory --- TODO: (George), does not compile properly on my version, double check when migrating to v4.9.0 theorem putnam_1962_a3 -(A B C A' B' C' P Q R : EuclideanSpace ℝ (Fin 2)) +(A B C A' B' C' P Q R : Fin 2 → ℝ) (k : ℝ) (hk : k > 0) (hABC : ¬Collinear ℝ {A, B, C}) diff --git a/lean4/src/putnam_1963_a6.lean b/lean4/src/putnam_1963_a6.lean index 801e0cd4..b2445510 100644 --- a/lean4/src/putnam_1963_a6.lean +++ b/lean4/src/putnam_1963_a6.lean @@ -3,12 +3,11 @@ open BigOperators open Topology Filter --- TODO: (George) Euclidean.dist not properly compiling, check again when migrating to v4.9.0 theorem putnam_1963_a6 -(F1 F2 U V A B C D P Q : EuclideanSpace ℝ (Fin 2)) +(F1 F2 U V A B C D P Q : Fin 2 → ℝ) (r : ℝ) -(E : Set (EuclideanSpace ℝ (Fin 2)) := {H : EuclideanSpace ℝ (Fin 2) | Euclidean.dist F1 H + Euclidean.dist F2 H = r}) -(M : EuclideanSpace ℝ (Fin 2) := midpoint ℝ U V) +(E : Set (Fin 2 → ℝ) := {H : Fin 2 → ℝ | Euclidean.dist F1 H + Euclidean.dist F2 H = r}) +(M : Fin 2 → ℝ := midpoint ℝ U V) (hr : r > Euclidean.dist F1 F2) (hUV : U ∈ E ∧ V ∈ E ∧ U ≠ V) (hAB : A ∈ E ∧ B ∈ E ∧ A ≠ B) diff --git a/lean4/src/putnam_1965_a1.lean b/lean4/src/putnam_1965_a1.lean index 10ff45fb..2636b691 100644 --- a/lean4/src/putnam_1965_a1.lean +++ b/lean4/src/putnam_1965_a1.lean @@ -3,7 +3,7 @@ open BigOperators open EuclideanGeometry ---TODO: (George) Double check for errors here on Lean migration. +--TODO: (George) Double check for errors here on Lean migration to v4.9.0. Angles do not work with Fin 2 → ℝ, but dist does not work for EuclideanSpace ℝ (Fin 2). noncomputable abbrev putnam_1965_a1_solution : ℝ := sorry -- Real.pi / 15 theorem putnam_1965_a1 diff --git a/lean4/src/putnam_1966_a2.lean b/lean4/src/putnam_1966_a2.lean index 52df8020..51ed2325 100644 --- a/lean4/src/putnam_1966_a2.lean +++ b/lean4/src/putnam_1966_a2.lean @@ -1,17 +1,16 @@ import Mathlib open BigOperators --- TODO: (George) Euclidean.dist does not compile here, see if problem persists on Lean migration theorem putnam_1966_a2 (r : ℝ) -(A B C : EuclideanSpace ℝ (Fin 2)) +(A B C : Fin 2 → ℝ) (hABC : ¬Collinear ℝ {A, B, C}) (a : ℝ := Euclidean.dist B C) (b : ℝ := Euclidean.dist C A) (c : ℝ := Euclidean.dist A B) (p : ℝ := (Euclidean.dist B C + Euclidean.dist C A + Euclidean.dist A B)/2) -(hr : ∃ I : EuclideanSpace ℝ (Fin 2), -(∃! P : EuclideanSpace ℝ (Fin 2), Euclidean.dist I P = r ∧ Collinear ℝ {P, B, C}) ∧ +(hr : ∃ I : Fin 2 → ℝ, +(∃! P : Fin 2 → ℝ, Euclidean.dist I P = r ∧ Collinear ℝ {P, B, C}) ∧ (∃! Q : EuclideanSpace ℝ (Fin 2), Euclidean.dist I Q = r ∧ Collinear ℝ {Q, C, A}) ∧ (∃! R : EuclideanSpace ℝ (Fin 2), Euclidean.dist I R = r ∧ Collinear ℝ {R, A, B}) ∧ (∀ Z : EuclideanSpace ℝ (Fin 2), Euclidean.dist I Z ≤ r → Z ∈ convexHull ℝ {A, B, C})) diff --git a/lean4/src/putnam_1967_a5.lean b/lean4/src/putnam_1967_a5.lean index 73c2920f..f3bfa916 100644 --- a/lean4/src/putnam_1967_a5.lean +++ b/lean4/src/putnam_1967_a5.lean @@ -3,9 +3,8 @@ open BigOperators open Nat Topology Filter --- TODO: (George) Double-check on Euclidean.dist usage here theorem putnam_1967_a5 -(R : Set (EuclideanSpace ℝ (Fin 2))) +(R : Set (Fin 2 → ℝ)) (hR : Convex ℝ R ∧ (MeasureTheory.volume R).toReal > Real.pi / 4) : ∃ P ∈ R, ∃ Q ∈ R, Euclidean.dist P Q = 1 := sorry diff --git a/lean4/src/putnam_1967_b1.lean b/lean4/src/putnam_1967_b1.lean index cec55a26..160f9b59 100644 --- a/lean4/src/putnam_1967_b1.lean +++ b/lean4/src/putnam_1967_b1.lean @@ -3,15 +3,14 @@ open BigOperators open Nat Topology Filter --- TODO: (George) check Euclidean dist problems after migration theorem putnam_1967_b1 (r : ℝ) -(L : ZMod 6 → EuclideanSpace ℝ (Fin 2)) -(P : EuclideanSpace ℝ (Fin 2) := midpoint ℝ (L 1) (L 2)) -(Q : EuclideanSpace ℝ (Fin 2) := midpoint ℝ (L 3) (L 4)) -(R : EuclideanSpace ℝ (Fin 2) := midpoint ℝ (L 5) (L 0)) +(L : ZMod 6 → (Fin 2 → ℝ)) +(P : Fin 2 → ℝ := midpoint ℝ (L 1) (L 2)) +(Q : Fin 2 → ℝ := midpoint ℝ (L 3) (L 4)) +(R : Fin 2 → ℝ := midpoint ℝ (L 5) (L 0)) (hr : r > 0) -(hcyclic : ∃ O : EuclideanSpace ℝ (Fin 2), ∀ i : ZMod 6, Euclidean.dist O (L i) = r) +(hcyclic : ∃ O : Fin 2 → ℝ), ∀ i : ZMod 6, Euclidean.dist O (L i) = r) (horder : ∀ i j : ZMod 6, i + 1 = j ∨ i = j + 1 ∨ segment ℝ (L i) (L j) ∩ interior (convexHull ℝ {L k | k : ZMod 6}) ≠ ∅) (hlens : Euclidean.dist (L 0) (L 1) = r ∧ Euclidean.dist (L 2) (L 3) = r ∧ Euclidean.dist (L 4) (L 5) = r) (hdist : L 1 ≠ L 2 ∧ L 3 ≠ L 4 ∧ L 5 ≠ L 0) diff --git a/lean4/src/putnam_1970_b6.lean b/lean4/src/putnam_1970_b6.lean index 923a5781..5b923430 100644 --- a/lean4/src/putnam_1970_b6.lean +++ b/lean4/src/putnam_1970_b6.lean @@ -3,15 +3,14 @@ open BigOperators open Metric Set EuclideanGeometry Filter Topology --- TODO: (George) Check for Euclidean.dist errors on Lean version migration theorem putnam_1970_b6 -(L : ZMod 4 → EuclideanSpace ℝ (Fin 2)) -(S : Set (EuclideanSpace ℝ (Fin 2)) := {L i | i : ZMod 4}) +(L : ZMod 4 → (Fin 2 → ℝ)) +(S : Set (Fin 2 → ℝ) := {L i | i : ZMod 4}) (hSquad : S.ncard = 4 ∧ ∀ s ⊆ S, s.ncard = 3 → ¬ Collinear ℝ s) (hlens : Euclidean.dist (L 0) (L 1) > 0 ∧ Euclidean.dist (L 1) (L 2) > 0 ∧ Euclidean.dist (L 2) (L 3) > 0 ∧ Euclidean.dist (L 3) (L 0) > 0) (horder : ∀ i : ZMod 4, segment ℝ (L i) (L (i + 1)) ∩ interior (convexHull ℝ S) = ∅) -(hcircum : ∃ (O : EuclideanSpace ℝ (Fin 2)) (r : ℝ), O ∈ convexHull ℝ S ∧ r > 0 ∧ ∀ i : ZMod 4, -∃! I : EuclideanSpace ℝ (Fin 2), Collinear ℝ {I, L i, L (i + 1)} ∧ Euclidean.dist O I = r) +(hcircum : ∃ (O: Fin 2 → ℝ) (r : ℝ), O ∈ convexHull ℝ S ∧ r > 0 ∧ ∀ i : ZMod 4, +∃! I : Fin 2 → ℝ, Collinear ℝ {I, L i, L (i + 1)} ∧ Euclidean.dist O I = r) (harea : (MeasureTheory.volume (convexHull ℝ S)).toReal = Real.sqrt (Euclidean.dist (L 0) (L 1) * Euclidean.dist (L 1) (L 2) * Euclidean.dist (L 2) (L 3) * Euclidean.dist (L 3) (L 0))) : Cospherical S := diff --git a/lean4/src/putnam_1973_a1.lean b/lean4/src/putnam_1973_a1.lean index 677d0e85..cf95432e 100644 --- a/lean4/src/putnam_1973_a1.lean +++ b/lean4/src/putnam_1973_a1.lean @@ -3,9 +3,8 @@ open BigOperators open Nat Set MeasureTheory Topology Filter ---TODO: (George) Check for Euclidean.dist issues when migrating to v4.9.0 theorem putnam_1973_a1 -(A B C X Y Z : EuclideanSpace ℝ (Fin 2)) +(A B C X Y Z : Fin 2 → ℝ) (hnoncol : ¬Collinear ℝ {A, B, C}) (hX : X ∈ segment ℝ B C) (hY : Y ∈ segment ℝ C A) From b96c3d82e208ccb48f77817c577539722e649970 Mon Sep 17 00:00:00 2001 From: George Tsoukalas Date: Wed, 17 Jul 2024 19:19:55 +0000 Subject: [PATCH 04/12] Modify leaderboard, README --- README.md | 6 ++++++ docs/leaderboard.html | 2 +- 2 files changed, 7 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 31b20159..88d24f43 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,11 @@ # PutnamBench +

+ + + +

+ PutnamBench is a benchmark for evaluation of theorem-proving algorithms on competition mathematics problems sourced from the William Lowell Putnam Mathematical Competition years 1962 - 2023. Our formalizations currently support three formal languages : Lean 4 $\land$ Isabelle $\land$ Coq. PutnamBench comprises of 1697 manually-crafted formalizations, aggregated over all languages. PutnamBench aims to support research in automated mathematical reasoning by providing a multilingual benchmark for evaluating theorem-proving algorithms. It is released under permissive licenses (Apache 2.0 for Lean 4 and Isabelle, MIT for Coq). The informal statements are also available with permission from the MAA. diff --git a/docs/leaderboard.html b/docs/leaderboard.html index 3ea429dd..e71e478f 100644 --- a/docs/leaderboard.html +++ b/docs/leaderboard.html @@ -553,7 +553,7 @@

📝 Notes

// displayChart("mbpp+"); }); - humanEvalRadio.click(); + wsolutionRadio.click(); // window.addEventListener("resize", () => { // this.chart.resize(); From ba98b1671619fb8bad73f1e3b534d5453de661eb Mon Sep 17 00:00:00 2001 From: George Tsoukalas Date: Wed, 17 Jul 2024 19:32:00 +0000 Subject: [PATCH 05/12] Fixed leaderboard (typo in html). --- docs/index.html | 5 +---- docs/leaderboard.html | 4 ++-- 2 files changed, 3 insertions(+), 6 deletions(-) diff --git a/docs/index.html b/docs/index.html index 356be7f7..ec7066be 100644 --- a/docs/index.html +++ b/docs/index.html @@ -153,10 +153,7 @@

PutnamBench: A Multilingual Competition - - - - Leaderboard + 🏆 Leaderboard diff --git a/docs/leaderboard.html b/docs/leaderboard.html index e71e478f..2bc096e4 100644 --- a/docs/leaderboard.html +++ b/docs/leaderboard.html @@ -156,7 +156,7 @@

-

📝 Notes

+

📝 Notes

  1. @@ -171,7 +171,7 @@

    📝 Notes