Skywork Open Reasoner 1 Technical Report
Paper • 2505.22312 • Published • 56
Datasets:
id large_stringlengths 12 12 | problem large_stringlengths 87 791 | answer large_stringlengths 1 38 | source large_stringclasses 4
values | qwen3.5_4b_rl_solved int64 32 155 | qwen3.5_4b_rl_attempts int64 296 336 | qwen3.5_4b_base_pass_at_4 float64 0 0.5 |
|---|---|---|---|---|---|---|
f0add24795a8 | At a fish market, there are 10 stalls, all selling the same 10 types of fish. All the fish are caught in either the North Sea or the Mediterranean Sea, and each stall has only one sea of origin for each type of fish. A number, $k$, of customers buy one fish from each stall such that they have one fish of each type. Fur... | 2^{10}-10 | olympiads_ref | 32 | 304 | 0 |
ca632e79b4da | For a real number $a$ , let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$ . Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$ . The region $\mathcal{R}$ is completely contained in a disk of ra... | 132 | aime | 36 | 328 | 0 |
886607c6b0b9 | 21. N4 (FRA) For any positive integer $x_{0}$, three sequences $\left\{x_{n}\right\},\left\{y_{n}\right\}$, and $\left\{z_{n}\right\}$ are defined as follows: (i) $y_{0}=4$ and $z_{0}=1$; (ii) if $x_{n}$ is even for $n \geq 0, x_{n+1}=\frac{x_{n}}{2}, y_{n+1}=2 y_{n}$, and $z_{n+1}=z_{n}$; (iii) if $x_{n}$ is odd for $... | 31 | olympiads_ref | 35 | 312 | 0 |
dd0e36c518c6 | Let $n \geq 3$ be an integer and consider an $n \times n$ board, divided into $n^{2}$ unit squares. We have for each $m \geq 1$ arbitrarily many $1 \times m$ rectangles (type I) and arbitrarily many $m \times 1$ rectangles (type II) available. We cover the board with $N$ of these rectangles, which do not overlap and do... | 2n-1 | olympiads_ref | 37 | 320 | 0 |
fb23907ef77a | A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $... | 258 | aime | 37 | 320 | 0 |
a9a6dc6a1451 | Find the number of ordered pairs of integers $(a,b)$ such that the sequence \[3,4,5,a,b,30,40,50\] is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. | 228 | aime | 38 | 320 | 0 |
7ca450f68e04 | 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 | 37 | 304 | 0 |
f9e1f1a85f1d | For $n\ge1$ call a finite sequence $(a_1,a_2,\ldots,a_n)$ of positive integers progressive if $a_i<a_{i+1}$ and $a_i$ divides $a_{i+1}$ for $1\le i\le n-1$ . Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360.$ | 47 | aime | 40 | 320 | 0 |
31b60f1a0590 | Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and
\[|x - y| \leq |f(x) - f(y)| \leq 3|x - y|\]
for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$. | 185 | amc_aime | 45 | 336 | 0 |
4fb2e55d52de | 16. (UKR) Determine the smallest integer \( n \geq 4 \) for which one can choose four different numbers \( a, b, c \), and \( d \) from any \( n \) distinct integers such that \( a+b-c-d \) is divisible by 20. | 9 | olympiads_ref | 43 | 320 | 0 |
b0e6892779bc | 25. N3 (GBR) Let \( a_{1}=11^{11}, a_{2}=12^{12}, a_{3}=13^{13} \), and
\[ a_{n}=\left|a_{n-1}-a_{n-2}\right|+\left|a_{n-2}-a_{n-3}\right|, \quad n \geq 4 \]
Determine \( a_{14^{14}} \). | 1 | olympiads_ref | 48 | 328 | 0 |
e08fda51bc78 | Let $S$ be the set of positive integers $k$ such that the two parabolas \[y=x^2-k~~\text{and}~~x=2(y-20)^2-k\] intersect in four distinct points, and these four points lie on a circle with radius at most $21$ . Find the sum of the least element of $S$ and the greatest element of $S$ . | 285 | aime | 50 | 320 | 0 |
2905a4234928 | Find all positive integers $n>1$ with the following property:
for each two positive divisors $k, \ell<n$ of $n$, at least one of the numbers $2 k-\ell$ and $2 \ell-k$ is a (not necessarily positive) divisor of $n$ as well. | 6, 9, 15 | olympiads_ref | 54 | 328 | 0 |
4fc3bbae3cc0 | A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encount... | 342 | aime | 59 | 328 | 0 |
5431fca4a751 | A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and th... | 813 | aime | 59 | 328 | 0 |
202ef2d1a4e0 | (Luxembourg). Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or ... | \frac{n(n+1)}{2} | olympiads_ref | 60 | 320 | 0.25 |
971aa09390af | In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's inbox. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss deliver... | 704 | amc_aime | 65 | 336 | 0 |
804bd1675279 | Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$ . Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$ . Let $Y$ be the intersection of lines $XG$ and $BC$ . Given that the measures of $\angle ABC, \angle BCA,$ ... | 592 | aime | 65 | 328 | 0 |
d2b7729c02ee | Jacob starts with some complex number $x_{0}$ other than 0 or 1. He repeatedly flips a fair coin. If the $n^{\text {th }}$ flip lands heads, he lets $x_{n}=1-x_{n-1}$, and if it lands tails he lets $x_{n}=\frac{1}{x_{n-1}}$. Over all possible choices of $x_{0}$, what are all possible values of the probability that $x_{... | 1, \frac{2^{2011}+1}{3 \cdot 2^{2011}} | omnimath | 64 | 320 | 0 |
dd82b979bbaf | 2. (BEL 4) From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the ... | \frac{8}{15} | olympiads_ref | 65 | 312 | 0 |
220068d5331c | Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$ . | 797 | aime | 71 | 328 | 0.25 |
26cdb5d5825d | In isosceles triangle $ABC$ , $A$ is located at the origin and $B$ is located at (20,0). Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\circ}$ . If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$ -axis, the area of the region common t... | 875 | aime | 70 | 320 | 0.25 |
1513a53159f1 | Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bi... | 255 | aime | 74 | 336 | 0 |
e9f766799d54 | Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . | (n-2)2^n + 1 | omnimath | 70 | 312 | 0 |
ded8c6f20ed5 | 1. (SWE 2) Two ships sail on the sea with constant speeds and fixed directions. It is known that at 9:00 the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance? | 12 | olympiads_ref | 70 | 304 | 0 |
798d96a819e5 | Find the largest possible integer $k$, such that the following statement is true:
Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obta... | 1 | omnimath | 76 | 320 | 0 |
963e2620e1f7 | There are 2018 frogs in a pool and there is 1 frog on the shore. In each time-step thereafter, one random frog moves position. If it was in the pool, it jumps to the shore, and vice versa. Find the expected number of time-steps before all frogs are in the pool for the first time. | 2^{2018}-1 | omnimath | 77 | 320 | 0 |
1f245f1a7ece | Determine the number of $2021$-tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$. | 3^{2021} - 2^{2021} | omnimath | 81 | 336 | 0 |
2629d7d57d63 | The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\overline{AB}$ is $\frac{43}{99}$ and the area of octagon $ABCDEFGH$ is $\frac{m}{n}$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$ AIME 1999 Problem 4.png | 185 | aime | 81 | 320 | 0 |
1c030244b92c | Determine all pairs $(a, b)$ of positive integers for which
$$
a^{2}+b \mid a^{2} b+a \quad \text { and } \quad b^{2}-a \mid a b^{2}+b .
$$ | (a, a+1) | olympiads_ref | 80 | 312 | 0 |
564c82ab1faa | Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that $$ \left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leqslant C $$ | \frac{1}{2} | olympiads_ref | 87 | 336 | 0.25 |
cf8f0e8e723b | In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ su... | 6 | omnimath | 85 | 320 | 0 |
506cf2e4074b | Find the smallest possible value of
$$
x y+y z+z x+\frac{1}{x}+\frac{2}{y}+\frac{5}{z},
$$
for positive real numbers $x, y$ and $z$. | 3 \sqrt[3]{36} | olympiads_ref | 88 | 328 | 0 |
b04cd7aee213 | Let $A_1A_2A_3\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$ . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely.... | 19 | aime | 92 | 336 | 0.25 |
751916970825 | Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$ . Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$ . Find the number of such sequences for which $a_n=0$ for some $n$ . | 494 | aime | 88 | 320 | 0 |
75cb7a3b985f | In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$. | 2016 | olympiads_ref | 89 | 320 | 0 |
ae289c766957 | Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\] | (2, 251, 252) | omnimath | 96 | 320 | 0.25 |
a81f9a37da01 | The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$ . The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$ ... | 131 | aime | 101 | 336 | 0 |
07424b769238 | A $10\times10\times10$ grid of points consists of all points in space of the form $(i,j,k)$ , where $i$ , $j$ , and $k$ are integers between $1$ and $10$ , inclusive. Find the number of different lines that contain exactly $8$ of these points. | 168 | aime | 100 | 320 | 0 |
8264c4e354e8 | Find all the triples of positive integers $(a, b, c)$ for which the number
$$
\frac{(a+b)^{4}}{c}+\frac{(b+c)^{4}}{a}+\frac{(c+a)^{4}}{b}
$$
is an integer and $a+b+c$ is a prime. | (1,1,1),(1,2,2),(2,3,6) | olympiads_ref | 105 | 328 | 0 |
2e2c9e9094db | It is said that an integer $n \geqslant 1$ is Japanese if it admits three divisors $d_{1}, d_{2}$, and $d_{3}$ such that $1 \leqslant d_{1}<d_{2}<d_{3}$ and $d_{1}+d_{2}+d_{3}=2022$.
What is the smallest Japanese integer? | 1344 | olympiads_ref | 116 | 328 | 0.25 |
5c2f7c41125d | In $\triangle ABC, AB = 3, BC = 4,$ and $CA = 5$ . Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B, \overline{BC}$ at $B$ and $D,$ and $\overline{AC}$ at $F$ and $G$ . Given that $EF=DF$ and $\frac{DG}{EG} = \frac{3}{4},$ length $DE=\frac{a\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers... | 41 | aime | 120 | 320 | 0.25 |
1ecce6ccb197 | Let $\overline{AB}$ be a chord of a circle $\omega$ , and let $P$ be a point on the chord $\overline{AB}$ . Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$ . Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$ . Circles $\omega_1$ and $\omega_2$ intersec... | 65 | aime | 120 | 320 | 0.25 |
43cba3af3882 | For integers $a$ and $b$ consider the complex number \[\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number. | 103 | aime | 123 | 328 | 0 |
c77d45d9b28c | Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$, the sum of the first $m$ terms or the sum of the last $m$ terms of the sequence is an integer. Determine the minimum number of integers in a complete sequence of $n$ numbers. | 2 | olympiads_ref | 125 | 328 | 0.25 |
88813147fbac | Two players, Blake and Ruby, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with Blake. On Blake's turn, Blake selects one white unit square and colors it blue. On Ruby's turn, Ruby selects two white unit squares and colors them red. The players... | 4 | olympiads_ref | 126 | 328 | 0.5 |
c1d6800f7071 | 13. (BUL) ${ }^{\mathrm{IMO} 6}$ Determine the least possible value of $f(1998)$, where $f$ is a function from the set $\mathbb{N}$ of positive integers into itself such that for all $m, n \in \mathbb{N}$,
$$ f\left(n^{2} f(m)\right)=m[f(n)]^{2} $$ | 120 | olympiads_ref | 126 | 328 | 0 |
cd9ddf80d2bf | There are 100 people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair le... | \frac{2^{100}}{\binom{100}{50}}-1 | omnimath | 127 | 328 | 0 |
a61228a81a75 | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R ... | 4 | omnimath | 124 | 320 | 0.5 |
10d3d19b6e15 | 51. (SWE 4) A subset \( S \) of the set of integers \( 0, \ldots, 99 \) is said to have property A if it is impossible to fill a crossword puzzle with 2 rows and 2 columns with numbers in \( S \) (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets \( S \) with property A. | 25 | olympiads_ref | 127 | 312 | 0.25 |
8da6bbc4df4f | For any integer $k\geq 1$ , let $p(k)$ be the smallest prime which does not divide $k$ . Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$ , and $X(k)=1$ if $p(k)=2$ . Let $\{x_n\}$ be the sequence defined by $x_0=1$ , and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\geq 0$ . Find the ... | 149 | aime | 135 | 328 | 0.5 |
f91fc693f4d9 | 25. S3 (POL) For an integer \( x \geq 1 \), let \( p(x) \) be the least prime that does not divide \( x \), and define \( q(x) \) to be the product of all primes less than \( p(x) \). In particular, \( p(1)=2 \). For \( x \) such that \( p(x)=2 \), define \( q(x)=1 \). Consider the sequence \( x_{0}, x_{1}, x_{2}, \ldo... | 142 | olympiads_ref | 124 | 296 | 0 |
d9bb0f5b8fd9 | Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] | 5 | omnimath | 144 | 336 | 0.25 |
1b8845b35ce3 | Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a ... | 32 | olympiads_ref | 141 | 328 | 0.25 |
6e0d1b2a3e20 | Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$ . Find the remainder when $x$ is divided by $1000$ . | 896 | aime | 132 | 304 | 0 |
9fba11db3265 | Let $k \geq 1$ be an integer.
We consider $4 k$ chips, $2 k$ of which are red and $2 k$ of which are blue. A sequence of those $4 k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue ... | k | olympiads_ref | 143 | 320 | 0.25 |
ef4765c0e075 | Given that $x, y,$ and $z$ are real numbers that satisfy:
\begin{align*} x &= \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}, \\ y &= \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}, \\ z &= \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}, \end{align*}
and that $x+y+z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are posit... | 9 | amc_aime | 141 | 312 | 0.25 |
ad129d1dab9e | Let $\overline{MN}$ be a diameter of a circle with diameter $1$ . Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\dfrac 35$ . Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose end... | 14 | aime | 152 | 336 | 0.5 |
6d6b0d0decc9 | How many 48-tuples of positive integers \(\left(a_{1}, a_{2}, \ldots, a_{48}\right)\) between 0 and 100 inclusive have the property that for all \(1 \leq i<j \leq 48, a_{i} \notin\left\{a_{j}, a_{j}+1\right\}\) ? | 54^{48} | omnimath | 146 | 320 | 0.25 |
3b471940674d | Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top... | 2^{k-1} | omnimath | 150 | 328 | 0.25 |
ccd203a78e8e | In triangle $ABC$ , $AC=13$ , $BC=14$ , and $AB=15$ . Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$ . Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$ . Let $P$ be the point, other than $A$ , of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$... | 218 | aime | 151 | 328 | 0 |
139a6000fec3 | In $\triangle ABC$ , the sides have integer lengths and $AB=AC$ . Circle $\omega$ has its center at the incenter of $\triangle ABC$ . An excircle of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose th... | 20 | aime | 155 | 336 | 0 |
bedbecd06aed | Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that \[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\] Find the least possible value of $a+b.$ | 23 | aime | 148 | 320 | 0.25 |
a1d78bfcfc4f | Let \[P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let
$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$
where $m, n,$ and $p$ ar... | 15 | amc_aime | 145 | 312 | 0 |
Contest Math Intermediate
64 competition math problems with short, checkable final answers, chosen because a 4B model trained on them with reinforcement learning solves them between one time in ten and one time in two. It is the middle companion of contest-math-hardest, which holds the problems that model almost never solves.
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 350 steps, which produced 296 to 336 sampled solutions per problem. Problems solved in 10% to 50% of those samples are this set.
Every reference answer here was matched by the model's own answer at least 30 times, so the answers are consistent with what a solver produces; they were not checked by hand one by one.
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 |
Caveats
- The model saw these problems during training, so the solve counts describe a model that was trained on them.
- Several answers are expressions or contain 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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