problem stringlengths 0 14.3k | solution stringlengths 0 6.9k ⌀ | tests stringlengths 33 74M ⌀ | domain stringclasses 8
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values | difficulty stringlengths 24 320 ⌀ | metadata stringlengths 2 145k | guessability stringclasses 9
values | guessability_samples stringlengths 40 3.2k ⌀ | verifiability bool 1
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Problem 6. (3 points)
Let $f(x)=\sin (\pi x)$. How many roots does the function $\underbrace{f(f(f(\ldots f(x) \ldots)))}_{20 \text { times }}$ have on the interval $[0 ; 1]$? | 524289=2^{19}+1 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "20", "1", "20", "0", "20", "0"] | null |
We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if
(i) $ A_1 \cap A_2 \cap \cdots \cap A_n \equal{} A$,
(ii) $ A_i \cap A_j \neq \emptyset$.
Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A \equal{} \{1, 2, \ldots, m\}$, there exists a set $ A_... | 56 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "15/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["17", "42", "12", "15", "15", "15", "15", "15"] | null |
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ . | 15^\circ | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "6/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["30", "60", "60", "30", "30", "60", "30", "30"] | null |
Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$ | n | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "14/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["4", "4", "1", "4", "1", "1", "1", "2"] | null |
Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for her to see him. | 16.2 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "4/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["12", "12", "12", "12", "12", "12", "12", "30"] | null |
A6. Wouter is walking from his house to his sports club. He could have taken his racing bike; with that, the journey would be seven times faster. But he left it at home. After $1 \mathrm{~km}$, he has reached a point where it makes no difference in time whether he continues walking or goes back home to get his racing b... | \frac{4}{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["\\frac{4}{3", "C", "\\frac{5}{4", "\\frac{8}{7", "C", "\\frac{6}{5", "\\frac{5}{4", "A"] | null |
7.049. $\lg (5-x)-\frac{1}{3} \lg \left(35-x^{3}\right)=0$. | 2;3 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2", "3", "3", "1", "1", "2", "3", "2"] | null |
Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$. | 64 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "12/16", "DeepSeek-R1-Distill-Qwen-32B": "7/16", "DeepSeek-R1-Distill-Qwen-7B": "11/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["100", "100", "256", "256", "256", "256", "256", "256"] | null |
21. Given that the internal angles of hexagon $A B C D E F$ are all equal, the area of the triangle formed by the lines containing sides $A B, C D, E F$ is $192 \sqrt{3}$, and the area of the triangle formed by the lines containing sides $B C, D E, F A$ is $324 \sqrt{3}$. If the perimeter of hexagon $A B C D E F$ is $m... | 55 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["52", "14", "58", "52", "52", "52", "52", "58"] | null |
Some positive integers are initially written on a board, where each $2$ of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+... | -3 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "11/16", "DeepSeek-R1-Distill-Qwen-32B": "7/16", "DeepSeek-R1-Distill-Qwen-7B": "11/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["5", "1", "8", "4", "1", "6", "8", "1"] | null |
3. (3 points) The sum of five consecutive natural numbers is 2010, the largest one is | 404 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "1/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["604", "605", "604", "603", "604", "604", "603", "604"] | null |
Example 3: In a table tennis match between A and B, the score is tied at $14: 14$. How many score sequences (denoted as $M$ in total) are there in the game process, where A is leading except for one exact tie during the game? | \frac{1}{14}C_{26}^{13} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["150", "81", "132", "15", "1023", "28", "200", "63"] | null |
20.95 As shown in the figure, $A B \perp B C, B C \perp C D, B C$ is tangent to the circle $O$ with diameter $A D$. In which of the following cases is the area of $A B C D$ an integer?
(A) $A B=3, C D=1$.
(B) $A B=5, C D=2$.
(C) $A B=7, C D=3$.
(D) $A B=9, C D=4$.
(E) $A B=11, C D=5$.
(39th American High School Mathema... | D | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "10/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "2/8"} | ["B", "A", "D", "C", "E", "D", "E", "C"] | null |
Task 6. Find all values that the expression
$$
3 \arcsin x - 2 \arccos y
$$
can take under the condition $x^{2} + y^{2} = 1$. | [-\frac{5\pi}{2};\frac{\pi}{2}] | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "15/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["[-\\pi, \\pi]", "1", "0", "0", "0", "0, \\pi", "0", "4\\pi"] | null |
Leon has cards with digits from 1 to 7. How many ways are there to combine these cards into two three-digit numbers (one card will not be used) so that each of them is divisible by 9? | 36 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "3/16", "DeepSeek-R1-Distill-Qwen-32B": "2/16", "DeepSeek-R1-Distill-Qwen-7B": "4/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["140", "210", "14", "14", "35", "0", "4", "0"] | null |
From city $A$, two cars start simultaneously on the same route to $B$, the first at speed $v_{1}$, the second at speed $v_{2}\left(<v_{1}\right)$. $t_{1}$ hours later, a third car starts after them and $t_{2}$ hours after overtaking the first car, it arrives in $B$. After another $t_{3}$ time, the second car also arriv... | 2 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["240", "240", "240", "1200", "1600", "240", "1600} km", "320"] | null |
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. ... | 051 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "6/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["3", "3", "4", "3", "3", "6", "4", "1"] | null |
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers. | \frac{41}{2} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "11/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["3", "8", "8", "3", "4", "7", "8", "3"] | null |
Given the function $f(x) = x + 1 + |3 - x|$, where $x \geq -1$.
1. Find the solution set for the inequality $f(x) \leq 6$.
2. If the minimum value of $f(x)$ is $n$, and the positive numbers $a$ and $b$ satisfy $2nab = a + 2b$, find the minimum value of $2a + b$. | \frac{9}{8} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2", "2", "(-\\infty, 1] \\cup [4, +\\infty)", "[-1, 2]", "2", "1", "1", "1"] | null |
Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.
[i]A. Khrabov[/i] | p = 3 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "6/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["41", "4", "11", "5", "3", "2", "41", "11"] | null |
Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a coffee grinder four times faster than they can disassemble it. When they came to the workshop in the morning, several coffee grinders were already assembled there.
At 7:00, Hubert started assembling and Robert started disassemb... | 15 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "13/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "4/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["10", "8", "16", "15", "120", "12", "24", "60"] | null |
10.068. The length of the base of the triangle is 36 cm. A line parallel to the base divides the area of the triangle in half. Find the length of the segment of this line enclosed between the sides of the triangle. | 18\sqrt{2} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["12", "18", "18", "18} cm", "18", "18", "18", "18"] | null |
Among all pairs of real numbers $(x, y)$ such that $\cos \sin x = \cos \sin y$ with $-\frac{15\pi}{2} \le x, y \le \frac{15\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{4} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1/2", "0.5", "\\frac{1}{2", "1/2", "\\frac{1}{2", "\\frac{1}{2", "1", "\\frac{1}{2"] | null |
20. In a mob of kangaroos, the two lightest kangaroos together weigh $25 \%$ of the total weight of the mob. The three heaviest kangaroos together weigh $60 \%$ of the total weight. How many kangaroos are in the mob?
A 6
B 7
C 8
D 15
E 20 | 6 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "15/16", "DeepSeek-R1-Distill-Qwen-7B": "14/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["E", "15", "C", "E", "C", "15", "8", "15"] | null |
Given that $x = \frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction. | \frac{5}{6} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "11/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "11/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["\\frac{2}{5", "\\frac{2}{3", "-\\frac{2}{5", "\\frac{2}{5", "\\frac{2}{3", "\\frac{2}{5", "-\\frac{3}{5", "\\frac{2}{5"] | null |
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$ . | 72 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "9/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["16", "444", "120", "24", "16", "10", "0", "240"] | null |
A circular piece of metal with a radius of 10 units has a maximum square cut out from it. Then, the largest possible circle is cut from this square. Calculate the total amount of metal wasted.
A) $50\pi - 200$
B) $200 - 50\pi$
C) $100\pi$
D) $50\pi$
E) None of these | 50\pi - 200 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["A", "A", "A", "A", "E", "A", "A", "A"] | null |
A sequence of numbers has 6 as its first term, and every term after the first is defined as follows: If a term, $t$, is even, the next term in the sequence is $\frac{1}{2} t$. If a term, $s$, is odd, the next term is $3 s+1$. Thus, the first four terms in the sequence are $6,3,10,5$. The 100th term is
(A) 1
(B) 2
(C) 3... | 4 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "15/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["81", "4", "C", "E", "81", "C", "E", "E"] | null |
Given an integer $ c \geq 1 $. To each subset $ A $ of the set $ \{1,2, \ldots ,n\} $, we assign a number $ w(A) $ from the set $ \{1,2, \ldots ,c\} $ such that the following condition is satisfied:
Let $ a(n) $ be the number of such assignments. Calculate $ \lim_{n\to \infty}\sqrt[n]{a(n)} $.
Note: $ \min(x,y) $ is t... | c | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "5/16", "DeepSeek-R1-Distill-Qwen-32B": "4/16", "DeepSeek-R1-Distill-Qwen-7B": "4/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1", "1", "0", "1", "0", "0", "1", "0"] | null |
Given that the vertex of a parabola is at the origin and the center of the circle $(x-2)^2 + y^2 = 4$ is exactly the focus of the parabola.
1. Find the equation of the parabola.
2. A line with a slope of 2 passes through the focus of the parabola and intersects the parabola at points A and B. Find the area of triangle... | 4\sqrt{5} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2x^2", "y^2 = 8x", "y = x^2", "y^2 = 8x", "y^2 = 8x", "y^2 = 16x", "8", "8"] | null |
# Problem 6.
There are $n$ houses in a row, painted in $k$ different colors, and for any color, there are 100 consecutive houses, among which the number of houses of this color is strictly greater than the number of houses of any other color. What is the largest $k$ for which this is possible if: a) $n=404$? b) $n=406... | 202 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["13", "50", "202", "16", "50", "503", "51", "10"] | null |
Evochkimov M.A.
Vasya thought of a two-digit number and told Petya the product of the digits in the number, and Sasha the sum of these digits. The boys had the following conversation:
Petya: "I can guess the number in three attempts, but two might not be enough".
Sasha: "If so, then I can guess it in four attempts, ... | 10 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "3/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["42", "18", "81", "18", "42", "24", "81", "24"] | null |
Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Evaluate $m + n$. | 9 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "12/16", "DeepSeek-R1-Distill-Qwen-32B": "5/16", "DeepSeek-R1-Distill-Qwen-7B": "6/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["11", "1", "15400", "1", "15", "1", "1024", "120"] | null |
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
How many different paths can the particle take from $(0,0)$ to $(5,5)$?
| 83 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "12/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["16", "10", "24", "0", "120", "240", "10", "120"] | null |
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9,10\}$ and arranges them in ascending order to form a 3-digit number, while Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,10\}$ and also arranges them in ascending order to form a 3-digit number. Determine the probabi... | 0.395 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-still3 | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["\\frac{1}{2", "0.5", "1/2", "1", "1/2", "\\frac{1}{2", "\\frac{1}{2", "\\frac{1}{2"] | null |
Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves? | 960 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "8/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["120", "1440", "720", "720", "240", "720", "720", "120"] | null |
What is the smallest positive value of $m$ such that the equation $10x^2 - mx + 660 = 0$ has integral solutions? | 170 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "8/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["33", "11", "33", "11", "33", "33", "33", "33"] | null |
A large supermarket purchased a popular disinfectant laundry detergent. Due to the rise in raw material prices, the cost price per bottle of detergent this year increased by $4$ compared to last year. The quantity of detergent purchased for $1440$ yuan this year is the same as the quantity purchased for $1200$ yuan las... | 8100 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "12/16", "DeepSeek-R1-Distill-Qwen-32B": "13/16", "DeepSeek-R1-Distill-Qwen-7B": "13/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["12", "32", "40", "10", "12", "12", "8", "60"] | null |
2. Real numbers $x, y, z$ satisfy the relations:
$$
4 x^{2}-2 x-30 y z=25 y^{2}+5 y+12 x z=9 z^{2}-3 z-20 x y .
$$
Find the maximum of the sum $a+b+c$, where $a=2 x+5 y, b=3 z+5 y, c=3 z-2 x$. | 2 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "14/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "3/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["7", "100", "100", "12", "10", "100", "16", "25"] | null |
Let $S_-$ be the semicircular arc defined by
\[
(x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1.
\]
Let $S_+$ be the semicircular arc defined by
\[
(x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1.
\]
Let $R$ be the locus of points $P$ such that $P$ is the intersection of two l... | \frac{1}{6} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2\\pi", "2\\pi", "4\\pi", "2\\pi", "2\\pi", "8\\pi", "16\\pi", "4\\pi"] | null |
5. In the country of Lemonia, coins in circulation have denominations of $2^{n}, 2^{n-1} \cdot 3, 2^{n-2} \cdot 3^{2}$, $2^{n-3} \cdot 3^{3}, \ldots, 2 \cdot 3^{n-1}, 3^{n}$ piastres, where $n$ is a natural number. A resident of the country went to the bank without any cash. What is the largest amount that the bank wil... | 3^{n+1}-2^{n+2} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "9/16", "DeepSeek-R1-Distill-Qwen-7B": "11/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1", "81", "81", "1", "10", "6", "81", "1"] | null |
Expand the following product: $\frac{2}{5}\left(\frac{5}{x} + 10x^2\right)$. | \frac{2}{x} + 4x^2 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["2", "\\frac{2}{5}\\left(\\frac{5}{x} + 10x^2\\right)", "\\frac{2}{5}(5 + 50x^2)", "\\frac{10}{x} + 4", "\\frac{20}{x} + 4", "\\frac{2}{x} + 4", "\\frac{20}{x} + 4", "\\frac{20}{x} + 4"] | null |
35. Let $f(x)=x^{2}+a x+b \cos x$, find all pairs of real numbers $(a, b)$, such that the equation $f(x)=0$ and $f(f(x))=0$ have the same and non-empty set of real solutions. | (,b)\mid0\leqslant<4,b=0 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "15/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["(-1, 0)", "(-1, 0)", "0", "0", "0", "0", "0", "0"] | null |
1. The numbers from 1 to 2150 are written on a board. Every minute, each number undergoes the following operation: if the number is divisible by 100, it is divided by 100; if it is not divisible by 100, 1 is subtracted from it. Find the largest number on the board after 87 minutes. | 2012 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "3/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["19", "2150", "2150", "2150", "2150", "2150", "2150", "2150"] | null |
Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that:
(a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$
(b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$
($P \circ Q$ indicates $P(Q(x))$.) | f(P) = c \cdot \deg(P) | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "15/16", "DeepSeek-R1-Distill-Qwen-32B": "12/16", "DeepSeek-R1-Distill-Qwen-7B": "12/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "0", "0", "0", "0", "0", "0"] | null |
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$. | \frac{\pi}{12} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "7/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["Shift by \\frac{\\pi}{12} to the right", "\\frac{\\pi}{6", "\\frac{\\pi}{6", "\\frac{\\pi}{6", "\\frac{\\pi}{6", "\\frac{\\pi}{6", "\\frac{\\pi}{6", "\\frac{\\pi}{6"] | null |
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds. | a^b + 1 \geq b(a + 1) | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "8/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "9/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["a \\geq b-1", "When \\(a=2\\) and \\(b=2\\), equality holds", "a=2, b=3", "a=2, b=3", "a=2, b=3", "a \\geq b-1", "a=2, b=3", "When \\(a = 1\\) and \\(b = 2\\), equality holds"] | null |
II. (25 points) As shown in Figure 3, circles $\odot O_{1}$ and $\odot O_{2}$ are externally tangent at point $O$. Line $AB$ is tangent to $\odot O_{1}$ and $\odot O_{2}$ at points $B$ and $A$, respectively, and intersects the $x$-axis and $y$-axis at points $M(2 \sqrt{3}, 0)$ and $C(0,2)$.
(1) Find the radius of $\odo... | P(0,2) \text{ or } P(-4 \sqrt{3}, 6) | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_cn_contest | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["No", "Yes", "No", "Yes", "No solution", "2", "\\text{No", "\\text{Yes"] | null |
2. Determine the fraction that is equal to the fraction $\frac{73}{95}$, and the sum of the numerator and denominator is the smallest possible square of a natural number. | \frac{3066}{3990} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "4/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "2/8"} | ["\\frac{17}{25", "\\frac{73}{95", "\\frac{19}{31", "\\frac{73}{95", "\\frac{19}{25", "\\frac{17}{25", "\\frac{1}{2", "\\frac{17}{25"] | null |
10. The function $f(x)=\mathrm{e}^{x}-a \ln (a x-a)+a(a>0)$, if $f(x)>0$ always holds, then the range of the real number $a$ is
$\qquad$. | (0,\mathrm{e}^{2}) | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["a \\in (0,1)", "a > 1", "e^{2", "a > 0", "1", "0 < a < 1", "a > 1", "[1,+\\infty)"] | null |
267. $x+\sqrt{x^{2}-x}=2$.
Задача из собрания примеров на латинском языке из той же рукописи. | \frac{4}{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2", "2", "267", "2", "1", "2", "2", "3"] | null |
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 peopl... | 46 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "10/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["440", "54", "110", "46", "410", "506", "56", "61"] | null |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | 15 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["8", "10", "5", "10", "10", "5", "5", "10"] | null |
Given that $0 < \alpha < \pi$ and $\cos{\alpha} = -\frac{3}{5}$, find the value of $\tan{\alpha}$ and the value of $\cos{2\alpha} - \cos{(\frac{\pi}{2} + \alpha)}$. | \frac{13}{25} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["\\text{Not enough information", "1", "1", "1", "8", "-\\frac{4}{3", "-\\frac{4}{3", "1"] | null |
Exercise 15. In how many ways can $10^{6}$ be written as a product $A \times B \times C$ of three natural numbers? (N.B. For example, the writings $1 \times 1000 \times 1000$ and $1000 \times 1 \times 1000$ are considered different.) | 784 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["3628800", "504", "1000", "400", "400000000", "120", "3", "0"] | null |
Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26. | 31 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "10/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["3", "26", "26", "25", "13", "5", "13", "13"] | null |
54. How many five-digit numbers are there in which
a) the digit 5 appears exactly once?
b) the digit 5 appears no more than once?
c) the digit 5 appears at least once? | 37512 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "7/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["80", "80", "30000", "120", "240", "10000", "30000", "14400"] | null |
5. (10 points) “Xiong Da” $\times$ “Xiong Er” $=$ “Xiong Xiong Di”. If the same Chinese character represents the same digit from 0 to 9, different characters represent different digits, and “Da” > “Er”, then the sum of all three-digit numbers represented by “Xiong Xiong Di” that satisfy the condition is $\qquad$ | 686 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "15/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1000", "9000", "360", "1530", "12900", "201", "100", "999"] | null |
Remove all perfect squares from the sequence of positive integers $1, 2, 3, \ldots$ to obtain a new sequence, and find the 2003rd term of this new sequence. | 2048 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["15121", "2003", "142857", "142857", "142857", "142857", "142", "2016"] | null |
A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decr... | 2640 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "9/16", "DeepSeek-R1-Distill-Qwen-32B": "6/16", "DeepSeek-R1-Distill-Qwen-7B": "3/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["280", "D", "52} yuan", "46", "D", "50", "C", "D"] | null |
In order to test students' mastery of high school mathematics knowledge, two opaque boxes, Box A and Box B, are prepared. Box A contains 2 conceptual description questions and 2 calculation questions; Box B contains 2 conceptual description questions and 3 calculation questions (all questions are different). Two studen... | \frac{3}{7} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "13/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0.5", "1/2", "0.5", "0.5", "1", "1", "1", "1"] | null |
Find the area of a triangle if two of its sides are equal to 1 and $\sqrt{13}$, and the median drawn to the third side is equal to 2.
# | \sqrt{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "4/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["4", "4", "4", "2", "6", "6", "3", "4"] | null |
For nonnegative integers $p$, $q$, $r$, let \[
f(p, q, r) = (p!)^p (q!)^q (r!)^r.
\]Compute the smallest positive integer $n$ such that for any triples $(a,b,c)$ and $(x,y,z)$ of nonnegative integers satisfying $a+b+c = 2020$ and $x+y+z = n$, $f(x,y,z)$ is divisible by $f(a,b,c)$.
[i]Proposed by Brandon Wang[/i] | 6052 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2020", "5040", "2020", "2020", "5040", "2020", "4040", "1610"] | null |
164*. Find four whole (positive) numbers such that the square of each of them, added to the sum of the other three, is also a perfect square.
Find four whole (positive) numbers such that the square of each of them, added to the sum of the other three, is also a perfect square. | 96,57,u=40;11,u=6;k(3k\2),u=1;u=1 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "8/16", "DeepSeek-R1-Distill-Qwen-7B": "5/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["8", "81", "169", "4, 4, 4, 4", "4, 4, 4, 4", "6, 7, 8, 9", "8", "8"] | null |
The figure shown consists of a right triangle and two squares. If the figure's total area equals 850 square inches, what is the value of $x$ in inches? [asy]
unitsize(5mm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((0,5)--(0,-2)--(-2,-2)--(-2,0)--(5,0)--(5,5)--cycle--(-2,0));
draw(scale(0.2)*((-1,0)--(-1,1)--(1... | 5 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["10", "10", "10", "10", "10", "10", "10", "15"] | null |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $\sin A + \sin B = [\cos A - \cos (π - B)] \sin C$.
1. Determine whether triangle $ABC$ is a right triangle and explain your reasoning.
2. If $a + b + c = 1 + \sqrt{2}$, find the maximum a... | \frac{1}{4} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "5/16", "DeepSeek-R1-Distill-Qwen-32B": "2/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["Yes", "Right Triangle", "1", "Rt", "\\text{not a right triangle", "right triangle", "D", "\\text{Yes"] | null |
If $x\geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}}=$
$\textbf{(A) } x\sqrt{x}\qquad \textbf{(B) } x\sqrt[4]{x}\qquad \textbf{(C) } \sqrt[8]{x}\qquad \textbf{(D) } \sqrt[8]{x^3}\qquad \textbf{(E) } \sqrt[8]{x^7}$ | \textbf{(E)}\sqrt[8]{x^7} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_amc_aime | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["E", "E", "C", "E", "A", "C", "C", "C"] | null |
1. Every natural number $n$ greater than 2 can be expressed as the sum of several distinct natural numbers. Let the maximum number of such distinct natural numbers be $A(n)$, find $A(n)$ (expressed in terms of $n$). | A(n)=\left[\frac{\sqrt{8 n+1}-1}{2}\right] | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_cn_contest | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["A(n) = n-1", "2", "A(n) = n-1", "A(n) = \\left\\lfloor \\frac{n}{2} \\right\\rfloor + 1", "A(n) = \\left\\lfloor \\frac{n}{2} \\right\\rfloor + 1", "A(n) = 2", "A(n) = 2^{n-2", "A(n) = 2^{n-2"] | null |
6. (7 points) There is a piece of paper in the shape of an isosceles right triangle. Fold this triangle along the altitude to its hypotenuse; fold it again along the altitude to its hypotenuse. At this point, you get an isosceles right triangle with the length of the legs being 2 cm (as shown in the shaded part of the ... | 16 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["4", "2", "12", "16", "8", "32", "12", "8"] | null |
8.2. On a certain island, only knights, who always tell the truth, and liars, who always lie, live. One day, 1001 inhabitants of this island stood in a circle, and each of them said: "All ten people following me in a clockwise direction are liars." How many knights could there be among those standing in the circle? | 91 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "1/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["499", "0", "334", "500", "499", "0", "500", "500"] | null |
Consider the ellipse $C\_1$: $\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\_2$ has the coordinates $(\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\_2$ at points $A$ and $B$, and the midpoint $H$ of the line segment ... | -\frac{1}{2} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "8/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["C", "C", "Ellipse C_2: \\frac{x^2}{4} + \\frac{y^2}{3} = 1", "C", "B", "C", "B", "B"] | null |
Given two distinct numbers \(a\) and \(b\) such that \(\frac{a}{b} + a = \frac{b}{a} + b\), find \(\frac{1}{a} + \frac{1}{b}\). | -1 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["\\frac{1}{2", "2", "\\frac{1}{2", "0", "-1", "\\frac{1}{2", "1", "\\frac{1}{2"] | null |
【Question 12】Cut a $4 \times 4 \times 4$ cube into 64 $1 \times 1 \times 1$ small cubes, then dye 16 of the $1 \times 1 \times 1$ small cubes red, requiring that among any 4 small cubes parallel to any edge, exactly 1 small cube is dyed red. The number of different coloring methods is $\qquad$ (coloring methods that ar... | 576 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "7/16", "DeepSeek-R1-Distill-Qwen-32B": "3/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["24", "120", "24", "120", "24", "1200", "40320", "3840"] | null |
Exercise 1. Find the number of odd integers between 1 and 2019 inclusive.
Only a numerical answer is expected here. | 1010 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "5/8"} | ["1010", "1009", "1010", "1010", "1009", "1009", "1010", "1010"] | null |
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$. For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac... | \frac{34}{19} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "10/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "2/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "1", "1", "0", "1", "1", "1", "1"] | null |
Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text... | 1 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "0", "4", "E", "0", "E", "0"] | null |
1. Given the function
$$
f(x)=\frac{1+2 x-x^{2}}{(1+x)\left(1+x^{2}\right)} \text {. }
$$
Let $\alpha, \beta, \gamma$ be the three interior angles of an arbitrary acute triangle. Then
$$
\begin{array}{l}
f(\tan \alpha)+f(\tan \beta)+f(\tan \gamma)+ \\
f(\cot \alpha)+f(\cot \beta)+f(\cot \gamma)=
\end{array}
$$ | 3 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "3/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "1/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "0", "0", "0", "0", "0", "0"] | null |
Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A... | \frac{(N-1)(3N-1)}{8} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "15/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["4", "0", "1", "4", "3", "0", "1", "0"] | null |
36.3. In triangle $A B C$, the bisectors $A A_{1}$ and $B B_{1}$ intersect at point $O$. Find the ratio $A A_{1}: O A_{1}$, if $A B=6, B C=5$, $C A=4$. | 3:1 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "3/16", "DeepSeek-R1-Distill-Qwen-32B": "1/16", "DeepSeek-R1-Distill-Qwen-7B": "3/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "3/8"} | ["2", "3", "3", "2", "1", "1", "1", "3"] | null |
Nine children, constituting three sets of three siblings, will occupy a van for a trip. The van has three rows, each containing three seats. Siblings should not sit next to each other in the same row, and no child may sit directly in front of or directly behind a sibling. Calculate the total number of possible seating ... | 648 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-still3 | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "1440", "0", "0", "1440", "0", "360", "1440"] | null |
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola. | y = \frac{1}{4} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "14/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "13/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["y=-x^2", "a circle", "a circle with radius equal to the distance between the vertices of the two parabolas", "A", "The locus of the focus of the moving parabola is a circle with radius 1 centered at (0, 1)", "The locus of the focus of the moving parabola is a circle with radius 2 centered at the origin", "y^2 = x", "... | null |
Given $a\in\{1,3,5\}$ and $b\in\{2,4,8\}$, find the probability that the function $y=\log_{\frac{b}{a}}{\frac{1}{x}}$ is an increasing function. | \frac{1}{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "3/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "\\frac{1}{4", "0", "1", "0", "\\frac{1}{2", "0"] | null |
A projectile is launched with an initial velocity of $u$ at an angle of $\alpha$ from the ground. The trajectory can be modeled by the parametric equations:
\[
x = ut \cos \alpha, \quad y = ut \sin \alpha - \frac{1}{2} kt^2,
\]
where $t$ denotes time and $k$ denotes a constant acceleration, forming a parabolic arch.
S... | \frac{\pi}{8} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "11/16", "DeepSeek-R1-Distill-Qwen-32B": "10/16", "DeepSeek-R1-Distill-Qwen-7B": "9/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1", "4", "81", "81", "81", "0", "81", "\\frac{4u^4}{k^2"] | null |
Given the polar equation of curve $C$ is $\rho=1$, with the pole as the origin of the Cartesian coordinate system, and the polar axis as the positive half-axis of $x$, establish the Cartesian coordinate system. The parametric equation of line $l$ is $\begin{cases} x=-1+4t \\ y=3t \end{cases}$ (where $t$ is the paramete... | \dfrac {8}{5} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2", "2", "2", "2", "2", "2", "2\\sqrt{5", "2"] | null |
2. 12 friends agree to go out for a meal once a week, each time evenly divided into 3 tables. Each table seats 4 people, until any two people have eaten at the same table at least once. How many weeks are needed at minimum? | 5 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["9", "9", "4", "12", "4", "4", "24", "9"] | null |
An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point | 26\% | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "7/16", "DeepSeek-R1-Distill-Qwen-32B": "13/16", "DeepSeek-R1-Distill-Qwen-7B": "8/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["14", "14", "30", "21", "12", "14", "21", "21"] | null |
The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. How many such integer points $(x, y)$ are there? | 20 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "1/8"} | ["13", "20", "0", "25", "28", "16", "12", "16"] | null |
Question 1 Let $a_{1}, a_{2}, \cdots, a_{2018}$ be the roots of the polynomial
$$
x^{2018}+x^{2017}+\cdots+x^{2}+x-1345=0
$$
Calculate $\sum_{n=1}^{2018} \frac{1}{1-a_{n}}$.
(2018, Stanford University Mathematics Competition) | 3027 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "4/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["0", "0", "0", "1345", "0", "0", "2017", "0"] | null |
4. A natural number $N$ has exactly 12 divisors (including 1 and $N$), and these divisors are numbered in increasing order: $d_{1}<$ $d_{2}<\cdots<d_{12}$. The divisor with the index $d{ }_{4}-1$ equals $\left(d_{1}+d_{2}+d_{1}\right) \times d_{8}$, find $N$.
| 1989 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_cn_contest | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1680", "25", "105", "1200", "60", "8", "120", "144"] | null |
It is known that there exist integers $x_{1}, x_{2}, \cdots, x_{n}$, satisfying $x_{1}^{4}-x_{2}^{4}-\cdots+x_{n}^{4}=1599$, then the minimum value of the positive integer $n$ is
(A) 14
(B) 15
(C) 16
(D) 1599 | 15 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["D", "D", "D", "C", "D", "16", "D", "16"] | null |
[Median of a pyramid (tetrahedron) $)]$ [Median of a pyramid (tetrahedron) ]
Given a tetrahedron $A B C D$. All plane angles at vertex $D$ are right angles; $D A=1, D B=2, D C=3$. Find the median of the tetrahedron, drawn from vertex $D$. | \frac{\sqrt{14}}{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2\\sqrt{3", "2\\sqrt{2", "2\\sqrt{2", "2\\sqrt{2", "2\\sqrt{2", "\\sqrt{7", "2\\sqrt{2", "2\\sqrt{2"] | null |
Let $n \ge 3$ be an integer and $S$ be a set of $n$ elements. Determine the largest integer $k_n$ such that: for each selection of $k_n$ $3-$subsets of $S$, there exists a way to color elements of $S$ with two colors such that none of the chosen $3-$subset is monochromatic. | k_n = 6 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "2/8"} | ["6", "1", "3", "2", "4", "6", "2^{n-2", "2"] | null |
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(... | 59 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["61", "61", "61", "61", "61", "61", "60", "61"] | null |
12. Which of all the isosceles triangles inscribed in a given semicircle has the greatest base, if one of the equal sides lies on the diameter, and the other is a chord? | \frac{4}{3}R | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["The equilateral triangle", "Equilateral triangle", "Equilateral triangle", "Isosceles triangle", "120", "equilateral triangle", "The equilateral triangle", "isosceles triangle"] | null |
In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$.
| 177 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "0/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["17", "1009", "17", "1009", "17", "17", "1009", "17"] | null |
Given that the Riemann function defined on the interval $\left[0,1\right]$ is: $R\left(x\right)=\left\{\begin{array}{l}{\frac{1}{q}, \text{when } x=\frac{p}{q} \text{(p, q are positive integers, } \frac{p}{q} \text{ is a reduced proper fraction)}}\\{0, \text{when } x=0,1, \text{or irrational numbers in the interval } (... | \frac{5}{3} | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "16/16", "DeepSeek-R1-Distill-Qwen-7B": "16/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["-\\frac{1}{5", "0", "0", "0", "0", "0", "0", "0"] | null |
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. | 11111111100000000 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-deepscaler | {"DeepSeek-R1-Distill-Qwen-1.5B": "14/16", "DeepSeek-R1-Distill-Qwen-32B": "3/16", "DeepSeek-R1-Distill-Qwen-7B": "13/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1111111111", "111111111", "111111111", "1", "1111111111", "1111111111", "11111111111111111111", "111111111"] | null |
Let $P_{n}=(1+1)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{7}\right) \cdots$ $\left(1+\frac{1}{3 n-2}\right)$. Find the greatest integer part of $P_{2000}$. | 25 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_cn_contest | {"DeepSeek-R1-Distill-Qwen-1.5B": "14/16", "DeepSeek-R1-Distill-Qwen-32B": "14/16", "DeepSeek-R1-Distill-Qwen-7B": "15/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["1", "1", "1", "1", "1", "81", "1", "1"] | null |
18.5. On the sides $C B$ and $C D$ of the square $A B C D$, points $M$ and $K$ are taken such that the perimeter of triangle $C M K$ is equal to twice the side of the square. Find the measure of angle $M A K$. | 45 | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_olympiads | {"DeepSeek-R1-Distill-Qwen-1.5B": "2/16", "DeepSeek-R1-Distill-Qwen-32B": "0/16", "DeepSeek-R1-Distill-Qwen-7B": "0/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "3/8"} | ["90^\\circ", "45^\\circ", "90^\\circ", "45°", "90", "45^\\circ", "90°", "45"] | null |
A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possi... | (25,2),(12,4),(3,13) | null | math | Skywork/Skywork-OR1-RL-Data/train-math-numinamath1.5_aops_forum | {"DeepSeek-R1-Distill-Qwen-1.5B": "16/16", "DeepSeek-R1-Distill-Qwen-32B": "3/16", "DeepSeek-R1-Distill-Qwen-7B": "10/16"} | {} | {"Qwen/Qwen2.5-1.5B-Instruct": "0/8"} | ["2,1", "4, 3", "2", "8", "13, 3", "2", "2, 3", "26,26"] | null |
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