RefGrader: Automated Grading of Mathematical Competition Proofs using Agentic Workflows
Paper • 2510.09021 • Published
problem_id stringlengths 32 32 | problem_statement stringlengths 88 1.12k | num_expert_solutions int64 4 44 | expert_solutions stringlengths 14.4k 278k | student_solutions stringlengths 5.1k 35.8k | problem_metadata stringlengths 14.6k 278k | student_metadata stringlengths 124 1.32k | created_at stringdate 2025-10-04 20:12:17 2025-10-04 20:20:25 |
|---|---|---|---|---|---|---|---|
problem_6733857f52d7e82fe11d62ef | Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
$$Version 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
$$
\sqrt... | 14 | [{"text": "We will prove $\\frac{n}{2}$ is a solution for all $n\\geq 2$ (the case $n=1$ is trivial). Let $F(x)$ denote $\\left( \\frac{x^{2n}+1}{2} \\right)^\\frac{1}{n}$ . Calculating the first three derivatives of $F$ , we get the following identities: \\begin{align*}\n F(1)&=1,\\\\\n F'(1)&=1,\\\\\n F''(1... | {"S1": "Let the given inequality be $f(x) \\le g(x)$, where $f(x) = \\sqrt[N]{\\frac{x^{2 N}+1}{2}}$ and $g(x) = a_{n}(x-1)^{2}+x$. Here $N=2^n$.\nWe are looking for the smallest real number $a_n$ such that $f(x) \\le g(x)$ for all $x \\in \\mathbb{R}$.\n\nLet's analyze the behavior of $f(x)$ and $g(x)$ near $x=1$.\n$f... | {"url": "https://artofproblemsolving.com/community/c6h2625913p22698446", "mongo_id": "6733857f52d7e82fe11d62ef", "tags": ["algebra"], "has_final_answer": true, "posts": [{"text": "We will prove $\\frac{n}{2}$ is a solution for all $n\\geq 2$ (the case $n=1$ is trivial). Let $F(x)$ denote $\\left( \\frac{x^{2n}+1}{2} \... | {"S1": {"correctness": "Some Correct Information", "thought_tokens": NaN, "final_comments": "1- The solution correctly realizes that it should focus on the Taylor series expansion of $f$ and $g$. \n2- It mistakenly focuses on the Taylor expansion of $h = g - f$ and correctly realizes that $a_n \\geq \\frac{N}{2}.$\n3- ... | 2025-10-04T20:12:17.193716 |
problem_6733857b52d7e82fe11d6105 | "Let $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coef(...TRUNCATED) | 24 | "[{\"text\": \"solution: We claim that the minimal $n$ is $n=4$ . Firstly note that the set $\\\\mat(...TRUNCATED) | "{\"S1\": \"Let $\\\\mathcal{A} = \\\\mathbb{Z}[x, y, z]$ be the ring of polynomials in three variab(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625905p22698387\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Correct\", \"thought_tokens\": NaN, \"final_comments\": \"The solution(...TRUNCATED) | 2025-10-04T20:12:23.232037 |
problem_6733857552d7e82fe11d59a9 | "Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest p(...TRUNCATED) | 29 | "[{\"text\": \"one liner: $$ S = \\\\frac ab + \\\\frac bc + \\\\frac cd + \\\\frac da \\\\ge 2\\\\l(...TRUNCATED) | "{\"S1\": \"Solution:\\nLet the given condition be\\n$$(a+c)(b+d)=ac+bd$$\\nExpanding the left side,(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625839p22697887\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Some Correct Information\", \"thought_tokens\": NaN, \"final_comments\(...TRUNCATED) | 2025-10-04T20:12:29.146683 |
problem_6733857b52d7e82fe11d610a | "A magician intends to perform the following trick. She announces a positive integer $n$, along with(...TRUNCATED) | 11 | "[{\"text\": \"Solution: The answer is no, the magician fails. Consider the following linear system (...TRUNCATED) | "{\"S1\": \"Solution:\\nLet $\\\\mathcal{P}_n$ denote the set of polynomials of degree at most $n$ w(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625872p22698165\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Incorrect\", \"thought_tokens\": NaN, \"final_comments\": \"Although t(...TRUNCATED) | 2025-10-04T20:12:34.807904 |
problem_6733859052d7e82fe11d76dd | "Find all functions $f : \\mathbb{Z}\\rightarrow \\mathbb{Z}$ satisfying\n$$f^{a^{2} + b^{2}}(a+b) =(...TRUNCATED) | 17 | "[{\"text\": \"Any function which maps anything but $0$ into $0$ is a solution. Now let's find the r(...TRUNCATED) | "{\"S1\": \"Solution:\\nLet $P(a, b)$ be the assertion $f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)$.\\nHe(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625860p22698062\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Some Correct Information\", \"thought_tokens\": NaN, \"final_comments\(...TRUNCATED) | 2025-10-04T20:12:40.519328 |
problem_6733857f52d7e82fe11d62e9 | "Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\\dots a_n$ of the\ns(...TRUNCATED) | 29 | "[{\"text\": \"solution: Let $x_n$ be the number of such permutations. If $a_n=n$ then the last ineq(...TRUNCATED) | "{\"S1\": \"Let $a_1, a_2, \\\\dots, a_n$ be a permutation of $1, 2, \\\\dots, n$.\\nThe given condi(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625893p22698309\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Correct\", \"thought_tokens\": NaN, \"final_comments\": \"Perfect solu(...TRUNCATED) | 2025-10-04T20:12:46.149727 |
problem_6733859452d7e82fe11d7d22 | "There is an integer $n \\gt 1$. There are $n^2$ stations on a slope of a mountain, all at differe(...TRUNCATED) | 29 | "[{\"text\": \"Is this the real problem? (Oh it is.)\", \"spans\": [], \"quotes\": [], \"like_count\(...TRUNCATED) | "{\"S1\": \"Let $N=n^2$ be the total number of stations. Let $s_1 < s_2 < \\\\dots < s_N$ be the sta(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2278648p17821585\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Some Correct Information\", \"thought_tokens\": NaN, \"final_comments\(...TRUNCATED) | 2025-10-04T20:12:52.069491 |
problem_6733857b52d7e82fe11d610d | "The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1(...TRUNCATED) | 12 | "[{\"text\": \"Solution: The answer is $\\\\left\\\\lceil\\\\frac n2\\\\right\\\\rceil+1$ . Obviousl(...TRUNCATED) | "{\"S1\": \"**1. Deep Comprehension & Deconstruction**\\n\\nThe problem asks for the smallest possib(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625914p22698447\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Some Correct Information\", \"thought_tokens\": NaN, \"final_comments\(...TRUNCATED) | 2025-10-04T20:12:57.279907 |
problem_6733859b52d7e82fe11d7ec0 | "Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1(...TRUNCATED) | 10 | "[{\"text\": \"In my opinion, this is one of the best olympiad combinatorics problems to ever be pub(...TRUNCATED) | "{\"S1\": \"The problem describes a game played by players A and B involving numbers on a blackboard(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2625868p22698110\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Incorrect\", \"thought_tokens\": NaN, \"final_comments\": \"The soluti(...TRUNCATED) | 2025-10-04T20:13:02.845887 |
problem_6733859452d7e82fe11d7d23 | "A deck of $n \\gt 1$ cards is given. A positive integer is written on each card. The deck has the(...TRUNCATED) | 29 | "[{\"text\": \"solution: They must be equal for all $n$ . Let the unique numbers on the cards be $a_(...TRUNCATED) | "{\"S1\": \"Let the given set of $n$ positive integers be $S = \\\\{x_1, x_2, \\\\ldots, x_n\\\\}$.\(...TRUNCATED) | "{\"url\": \"https://artofproblemsolving.com/community/c6h2278645p17821528\", \"mongo_id\": \"673385(...TRUNCATED) | "{\"S1\": {\"correctness\": \"Some Correct Information\", \"thought_tokens\": NaN, \"final_comments\(...TRUNCATED) | 2025-10-04T20:13:09.060491 |
End of preview. Expand in Data Studio
RefGrader: Data and Logs
Data and run artifacts from RefGrader, an automated system for grading mathematical competition proofs against reference solutions.
RefGrader derives problem-specific grading rubrics from reference solutions and applies them to score model-generated proofs, bringing automated grading substantially closer to expert human judgment than a single-pass baseline.
Dataset structure
Each row is an IMO Shortlist problem prepared for RefGrader evaluation:
problem_id- unique identifierproblem_statement- the competition problemnum_expert_solutions- countexpert_solutions- reference solutions (JSON)student_solutions- solutions to be graded (JSON)problem_metadata/student_metadata- source and grading metadata (JSON)created_at- timestamp
The complete raw run artifacts, including the full per-stage model caches, remain
available in imo-shortlist.json.
Related publications
- RefGrader: Automated Grading of Mathematical Competition Proofs (MATH-AI Workshop, NeurIPS 2025) - arXiv:2510.09021
- Brains vs. Bytes: Evaluating LLM Proficiency in Olympiad Mathematics (COLM 2025) - arXiv:2504.01995
Citation
If you use this data, please cite the RefGrader paper (arXiv:2510.09021).
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