Datasets:

kushasareen
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contest-math-hardest

id large_stringlengths 12 12	problem large_stringlengths 237 790	answer large_stringlengths 1 37	source large_stringclasses 4 values	qwen3.5_4b_rl_solved int64 0 9	qwen3.5_4b_rl_attempts int64 184 208
20306c835460	Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=$ 2021 is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player inc...	6561	olympiads_ref	0	192
2bc87ff2e9d2	On a $2023 \times 2023$ board, some squares contain a beetle, with at most one beetle per square. After one minute, each beetle moves one square to the left, right, up, or down. After each subsequent minute, each beetle moves one square further, but always makes a 90-degree turn. If a beetle moved left or right in one ...	2022^{2}+1	olympiads_ref	0	192
3fbb88bb7c0e	11. (POL 1) Find the number of partitions of the set $\{1,2, \ldots, n\}$ into three subsets $A_{1}, A_{2}, A_{3}$, some of which may be empty, such that the following conditions are satisfied: (i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have dif...	2^{n-1}	olympiads_ref	0	192
43813ac4d937	Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $\|x-y\|$ on the blackboard. The game terminates as soon as, at the end of some round...	7	omnimath	0	208
44500d62dfd9	Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is list...	\frac{n(n+1)(2n+1)}{6}	omnimath	0	200
6318adfb3397	A baseball league has 6 teams. To decide the schedule for the league, for each pair of teams, a coin is flipped. If it lands head, they will play a game this season, in which one team wins and one team loses. If it lands tails, they don't play a game this season. Define the imbalance of this schedule to be the minimum ...	\frac{5055}{2^{15}}	omnimath	0	192
6b01ef84ac90	Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.	\frac{4}{\sqrt{3}} \sin^2 80^\circ	omnimath	0	192
6ca08b5b4423	2. (BRA 3) A polyhedron has 12 faces and is such that: (i) all faces are isosceles triangles, (ii) all edges have length either $x$ or $y$, (iii) at each vertex either 3 or 6 edges meet, and (iv) all dihedral angles are equal. Find the ratio $x / y$.	\frac{5}{3}	olympiads_ref	0	200
8e57b426a70c	A finite sequence of integers $a_{1}, a_{2}, \ldots, a_{n}$ is called regular if there exists a real number $x$ satisfying $$ \lfloor k x\rfloor=a_{k} \quad \text { for } 1 \leq k \leq n $$ Given a regular sequence $a_{1}, a_{2}, \ldots, a_{n}$, for $1 \leq k \leq n$ we say that the term $a_{k}$ is forced if the follo...	985	olympiads_ref	0	192
a38b542bc5f8	A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.	384	amc_aime	0	192
d41876078d03	The L shape made by adjoining three congruent squares can be subdivided into four smaller L shapes. Each of these can in turn be subdivided, and so forth. If we perform 2005 successive subdivisions, how many of the $4^{2005}$ L's left at the end will be in the same orientation as the original one?	4^{2004}+2^{2004}	omnimath	0	200
efb6cc85f49b	For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $...	3024	omnimath	0	208
f60fc20b1705	6. C6 (FRA) Let $p$ and $q$ be relatively prime positive integers. A subset $S$ of $\{0,1,2, \ldots\}$ is called ideal if $0 \in S$ and for each element $n \in S$, the integers $n+p$ and $n+q$ belong to $S$. Determine the number of ideal subsets of $\{0,1,2, \ldots\}$.	\frac{1}{p+q}\binom{p+q}{p}	olympiads_ref	0	200
45d5440e20ff	For positive integers $a$ , $b$ , and $c$ with $a < b < c$ , consider collections of postage stamps in denominations $a$ , $b$ , and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $...	188	aime	1	200
63589d4282d7	A ten-level 2-tree is drawn in the plane: a vertex $A_{1}$ is marked, it is connected by segments with two vertices $B_{1}$ and $B_{2}$, each of $B_{1}$ and $B_{2}$ is connected by segments with two of the four vertices $C_{1}, C_{2}, C_{3}, C_{4}$ (each $C_{i}$ is connected with one $B_{j}$ exactly); and so on, up to ...	2^{2^{7}}	omnimath	1	192
b7e36d51a2ea	Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every...	\frac{k(4k^2 + k - 1)}{2}	omnimath	1	192
c0b2295ece42	Ash and Gary independently come up with their own lineups of 15 fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team's lineup; if there is a dr...	1-\frac{2^{15}}{3^{15}}	omnimath	1	192
cf1e42c8228b	A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. (Turkey)	\binom{2 n}{n}^{2}	olympiads_ref	1	200
8f8dfc0a37ab	The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relati...	85	aime	2	192
9fe9444e8312	Inside the square $A B C D$, the equilateral triangle $\triangle A B E$ is constructed. Let $M$ be an interior point of the triangle $\triangle A B E$ such that $M B=\sqrt{2}, M C=\sqrt{6}, M D=\sqrt{5}$ and $M E=\sqrt{3}$. Find the area of the square $A B C D$.	3+\sqrt{6}	olympiads_ref	5	192
a41fb388fb05	Let $S=\{1,2, \ldots, n\}$, where $n \geq 1$. Each of the $2^{n}$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue. Determine the number of colorings that...	1+3^{n}	olympiads_ref	5	200
f28455155b08	12. (GDR 3) Let $R$ be a set of exactly 6 elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$; (ii) For any three sets $X, Y, Z$ in $F, X \cup Y \cup Z \neq R$, (iii) $\bigcup_{X \in F} X=...	11	olympiads_ref	5	200
33e5b4ffb5a5	For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.	1, 3, 5	omnimath	6	200
f83a5669fbe4	Let $m$ and $n$ be odd integers greater than $1.$ An $m\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$ , those in the second row are numbered left to right with the integers $n + 1$ through $2n$ , and so on. Square $200$ is in ...	248	aime	6	192
e5e743fda132	Rishabh has 2024 pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops.	\frac{4^{2024}}{\binom{4048}{2024}}-2	omnimath	7	192
675317d46284	In a country, there are 2018 cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any arbitrary city to any other arbitrary city via one or more roads. Consider for each pair of cities the shortest route between these two cities. What is the ...	1511	olympiads_ref	8	184
812446f0c637	In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrou...	2271380	omnimath	9	208

Contest Math Hardest

27 competition math problems with short, checkable final answers, chosen because a 4B model trained on them with reinforcement learning almost never solves them. It is meant as a small, very hard test set for small models.

How the problems were chosen

The problems come from a pool of contest problems (AIME, AMC, Omni-MATH and olympiad reference solutions) taken from Skywork-OR1-RL-Data, filtered to drop problems with figures, multiple choice, multi-part questions and prose answers.

Qwen3.5-4B (no fine-tuning) was sampled 4 times per problem at 16k tokens. Problems it solved 0 to 2 times out of 4 were kept.
Qwen3.5-4B was then trained with reinforcement learning (CISPO) on those problems for about 210 steps, which produced 184 to 208 sampled solutions per problem. Problems solved in fewer than 5% of those samples were candidates.
Every candidate was checked by hand. Problems with a wrong or unverifiable reference answer, garbled text, more than one valid answer, or more than one question were removed. One reference with a missing closing brace was repaired.

Fields

field	meaning
`id`	stable identifier
`problem`	the problem statement
`answer`	the reference final answer, in LaTeX
`source`	the sub-collection it came from (`aime`, `amc_aime`, `omnimath`, `olympiads_ref`)
`qwen3.5_4b_rl_solved`, `qwen3.5_4b_rl_attempts`	how often the RL-trained Qwen3.5-4B solved it during training
`qwen3.5_4b_base_pass_at_4`	the untrained model's solve rate over 4 samples (0 for every problem here)

13 problems were never solved by the trained model. For two of them (20306c835460, d41876078d03), the training log showed a few solves that were scoring errors on very large numbers; their count is set to 0.

Caveats

The model saw these problems during training, so the solve counts describe a model that was trained on them.
Several answers are expressions in n, k, p, q or contain very large numbers. Use a symbolic checker (for example math-verify) and compare large integers exactly, not as floats.
The problems are from public competitions and may appear in model pretraining data.

Source and credit

Problems are drawn from Skywork-OR1-RL-Data, which does not state a license; the problems themselves are from public mathematics competitions. If you use this set, please also cite the Skywork Open Reasoner 1 report (He et al., 2025, arXiv:2505.22312).

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Paper for kushasareen/contest-math-hardest

Skywork Open Reasoner 1 Technical Report

Paper • 2505.22312 • Published May 28, 2025 • 56