data_source stringclasses 8 values | prompt listlengths 2 2 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Given a positive integer $n$, determine the largest real number $\\mu$ satisfying... | math | {
"ground_truth": "$\\frac{1}{2 n+2}$",
"style": "rule"
} | {
"index": 1,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find (in closed form) the difference between the number of positive integers at m... | math | {
"ground_truth": "$2^{1009}$",
"style": "rule"
} | {
"index": 2,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all positive integers $n$ satisfying the following condition: for every... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 3,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-emp... | math | {
"ground_truth": "$2n-2$",
"style": "rule"
} | {
"index": 4,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying ... | math | {
"ground_truth": "$(1,8,19), (2,7,13), (4,5,7)$",
"style": "rule"
} | {
"index": 5,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial wi... | math | {
"ground_truth": "$2n$",
"style": "rule"
} | {
"index": 6,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 7,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. S... | math | {
"ground_truth": "$\\binom{2n}{n}$",
"style": "rule"
} | {
"index": 8,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Given positive integers $m$ and $n \\geq m$, determine the largest number of domi... | math | {
"ground_truth": "$m n-\\lfloor m / 2\\rfloor$",
"style": "rule"
} | {
"index": 9,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, ... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 10,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n\... | math | {
"ground_truth": "$f(x)=2 x$",
"style": "rule"
} | {
"index": 11,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n>1$ be an integer. In the space, consider the set\n$$\nS=\\{(x, y, z) \\mid... | math | {
"ground_truth": "$3 n$",
"style": "rule"
} | {
"index": 12,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all positive integers $n$, for which the numbers in the set $S=\\{1,2, \\ldo... | math | {
"ground_truth": "$69$,$84$",
"style": "rule"
} | {
"index": 13,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine the smallest positive real number $k$ with the following property.\n\nL... | math | {
"ground_truth": "$k=1$",
"style": "rule"
} | {
"index": 14,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{... | math | {
"ground_truth": "$(2,4)$",
"style": "rule"
} | {
"index": 15,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all surjective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that... | math | {
"ground_truth": "$f(n)=n$",
"style": "rule"
} | {
"index": 16,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all pairs $(f, g)$ of functions from the set of positive integers to it... | math | {
"ground_truth": "$f(n)=n$, $g(n)=1$",
"style": "rule"
} | {
"index": 17,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine the greatest positive integer $k$ that satisfies the following property... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 18,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m... | math | {
"ground_truth": "$\\frac{3 m}{2}-1$",
"style": "rule"
} | {
"index": 19,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "On a square table of 2011 by 2011 cells we place a finite number of napkins that ... | math | {
"ground_truth": "3986729",
"style": "rule"
} | {
"index": 20,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Dete... | math | {
"ground_truth": "1,3,5",
"style": "rule"
} | {
"index": 21,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $x_{1}, \\ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1... | math | {
"ground_truth": "$\\frac{25}{2}$",
"style": "rule"
} | {
"index": 22,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Denote by $\\mathbb{Q}^{+}$the set of all positive rational numbers. Determine al... | math | {
"ground_truth": "$f(x)=\\frac{1}{x}$",
"style": "rule"
} | {
"index": 23,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "On some planet, there are $2^{N}$ countries $(N \\geq 4)$. Each country has a fla... | math | {
"ground_truth": "$M=2^{N-2}+1$",
"style": "rule"
} | {
"index": 24,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n(... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 25,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find the least positive integer $n$ for which there exists a set $\\left\\{s_{1},... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 26,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all pairs $(m, n)$ of nonnegative integers for which\n\n$$\nm^{2}+2 \\cdot 3... | math | {
"ground_truth": "$(6,3),(9,3),(9,5),(54,5)$",
"style": "rule"
} | {
"index": 27,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \\l... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 28,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine the smallest number $M$ such that the inequality\n\n$$\n\\left|a b\\lef... | math | {
"ground_truth": "$M=\\frac{9}{32} \\sqrt{2}$",
"style": "rule"
} | {
"index": 29,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A diagonal of a regular 2006-gon is called odd if its endpoints divide the bounda... | math | {
"ground_truth": "$1003$",
"style": "rule"
} | {
"index": 30,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ ... | math | {
"ground_truth": "$\\angle B E A_{1}=90$,$\\angle A E B_{1}=90$",
"style": "rule"
} | {
"index": 31,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all pairs $(x, y)$ of integers satisfying the equation\n\n$$\n1+2^{x}+2... | math | {
"ground_truth": "$(0,2),(0,-2),(4,23),(4,-23)$",
"style": "rule"
} | {
"index": 32,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Given a positive integer $n$, find the smallest value of $\\left\\lfloor\\frac{a_... | math | {
"ground_truth": "$\\left\\lfloor\\log _{2} n\\right\\rfloor+1$",
"style": "rule"
} | {
"index": 33,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n \\geqslant 3$ be an integer. An integer $m \\geqslant n+1$ is called $n$-c... | math | {
"ground_truth": "$m_{\\max }=n^{2}-n-1$",
"style": "rule"
} | {
"index": 34,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine the largest $N$ for which there exists a table $T$ of integers with $N$... | math | {
"ground_truth": "$\\frac{100!}{2^{50}}$",
"style": "rule"
} | {
"index": 35,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all integers $n \\geqslant 1$ for which there exists a pair of positive... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 36,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all positive integers $n$ with the following property: the $k$ positive divi... | math | {
"ground_truth": "1,3",
"style": "rule"
} | {
"index": 37,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers writte... | math | {
"ground_truth": "$-2,0,2$",
"style": "rule"
} | {
"index": 38,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "An integer $n \\geqslant 3$ is given. We call an $n$-tuple of real numbers $\\lef... | math | {
"ground_truth": "$-(n-1) / 2$",
"style": "rule"
} | {
"index": 39,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n>1$ be an integer. An $n \\times n \\times n$ cube is composed of $n^{3}$ u... | math | {
"ground_truth": "$\\frac{n(n+1)(2 n+1)}{6}$",
"style": "rule"
} | {
"index": 40,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a given positive integer. In the Cartesian plane, each lattice point w... | math | {
"ground_truth": "$n^{2}+1$",
"style": "rule"
} | {
"index": 41,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "There are 2017 mutually external circles drawn on a blackboard, such that no two ... | math | {
"ground_truth": "6048",
"style": "rule"
} | {
"index": 42,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Call a rational number short if it has finitely many digits in its decimal expans... | math | {
"ground_truth": "807",
"style": "rule"
} | {
"index": 43,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number\n\n$$\n\... | math | {
"ground_truth": "$(3,2)$",
"style": "rule"
} | {
"index": 44,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$, or show that no such $n$ exists, with the... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 45,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find the smallest real constant $C$ such that for any positive real numbers $a_{1... | math | {
"ground_truth": "$\\frac{1}{2}$",
"style": "rule"
} | {
"index": 46,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The equation\n\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016)\n$$\n\n... | math | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": 47,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine the largest real number $a$ such that for all $n \\geqslant 1$ and for ... | math | {
"ground_truth": "$\\frac{4}{9}$",
"style": "rule"
} | {
"index": 48,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all positive integers $n$ for which all positive divisors of $n$ can be put ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 49,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Determine the smallest positive integer $k$ with t... | math | {
"ground_truth": "$2 n$",
"style": "rule"
} | {
"index": 50,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set\n\n$$\n\\... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 51,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Denote by $\\mathbb{N}$ the set of all positive integers. Find all functions $f: ... | math | {
"ground_truth": "$f(n)=n^{2}$",
"style": "rule"
} | {
"index": 52,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real num... | math | {
"ground_truth": "$\\frac{N}{2}$",
"style": "rule"
} | {
"index": 53,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 54,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b ... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 55,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions... | math | {
"ground_truth": "$f(x)=x+1$",
"style": "rule"
} | {
"index": 56,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be an integer with $n \\geqslant 2$. On a slope of a mountain, $n^{2}$ ch... | math | {
"ground_truth": "$n^{2}-n+1$",
"style": "rule"
} | {
"index": 57,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \\ldots$ are defined inductively by $... | math | {
"ground_truth": "$\\lceil n / 2\\rceil+1$",
"style": "rule"
} | {
"index": 58,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copi... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 59,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each sho... | math | {
"ground_truth": "$\\frac{1}{4} n(n+1)$",
"style": "rule"
} | {
"index": 60,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "On a flat plane in Camelot, King Arthur builds a labyrinth $\\mathfrak{L}$ consis... | math | {
"ground_truth": "$k=n+1$",
"style": "rule"
} | {
"index": 61,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "There are 60 empty boxes $B_{1}, \\ldots, B_{60}$ in a row on a table and an unli... | math | {
"ground_truth": "960",
"style": "rule"
} | {
"index": 62,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the uni... | math | {
"ground_truth": "$2^{k}$",
"style": "rule"
} | {
"index": 63,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all pairs $(m, n)$ of positive integers satisfying the equation\n\n$$\n\\lef... | math | {
"ground_truth": "$(1,1), (3,2)$",
"style": "rule"
} | {
"index": 64,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a... | math | {
"ground_truth": "$(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$",
"style": "rule"
} | {
"index": 65,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ with the proper... | math | {
"ground_truth": "$f(x)=-1$,$f(x)=x+1$",
"style": "rule"
} | {
"index": 66,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a fixed positive integer. Find the maximum possible value of\n\n$$\n\\... | math | {
"ground_truth": "$n(n-1)$",
"style": "rule"
} | {
"index": 67,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the equat... | math | {
"ground_truth": "$f(x)=x$,$f(x)=2-x$",
"style": "rule"
} | {
"index": 68,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For a finite set $A$ of positive integers, we call a partition of $A$ into two di... | math | {
"ground_truth": "3024",
"style": "rule"
} | {
"index": 69,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle ... | math | {
"ground_truth": "$\\sqrt{2}$",
"style": "rule"
} | {
"index": 70,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, ... | math | {
"ground_truth": "$(2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3)$",
"style": "rule"
} | {
"index": 71,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive inte... | math | {
"ground_truth": "$k \\geqslant 2$",
"style": "rule"
} | {
"index": 72,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For every positive integer $n$ with prime factorization $n=\\prod_{i=1}^{k} p_{i}... | math | {
"ground_truth": "$f(x)=a x+b$, where $b$ is an arbitrary integer, and $a$ is an arbitrary positive integer with $\\mho(a)=0$",
"style": "rule"
} | {
"index": 73,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For a sequence $x_{1}, x_{2}, \\ldots, x_{n}$ of real numbers, we define its pric... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 74,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying\n\n$... | math | {
"ground_truth": "$f(n) = 2 n+1007$",
"style": "rule"
} | {
"index": 75,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Consider all polynomials $P(x)$ with real coefficients that have the following pr... | math | {
"ground_truth": "$(-\\infty, 0) \\cup\\{1\\}$.",
"style": "rule"
} | {
"index": 76,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n \\geqslant 2$ be an integer. Consider an $n \\times n$ chessboard divided ... | math | {
"ground_truth": "$\\lfloor\\sqrt{n-1}\\rfloor$",
"style": "rule"
} | {
"index": 77,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "We are given an infinite deck of cards, each with a real number on it. For every ... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 78,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n \\geqslant 2$ be an integer, and let $A_{n}$ be the set\n\n$$\nA_{n}=\\lef... | math | {
"ground_truth": "$(n-2) 2^{n}+1$",
"style": "rule"
} | {
"index": 79,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $k \\geqslant 2$ be an integer. Find the smallest integer $n \\geqslant k+1$ ... | math | {
"ground_truth": "$n=k+4$",
"style": "rule"
} | {
"index": 80,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f... | math | {
"ground_truth": "$f(x)=\\frac{1}{x}$",
"style": "rule"
} | {
"index": 81,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all positive integers $n \\geqslant 2$ for which there exist $n$ real number... | math | {
"ground_truth": "2,3,4",
"style": "rule"
} | {
"index": 82,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "$A \\pm 1 \\text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \\ldots, a_{20... | math | {
"ground_truth": "506",
"style": "rule"
} | {
"index": 83,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In each square of a garden shaped like a $2022 \\times 2022$ board, there is init... | math | {
"ground_truth": "2271380",
"style": "rule"
} | {
"index": 84,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doin... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 85,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Alice fills the fields of an $n \\times n$ board with numbers from 1 to $n^{2}$, ... | math | {
"ground_truth": "$2 n^{2}-2 n+1$",
"style": "rule"
} | {
"index": 86,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{Z}_{\\geqslant 0}$ be the set of non-negative integers, and let $f:... | math | {
"ground_truth": "2500,7500",
"style": "rule"
} | {
"index": 87,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A number is called Norwegian if it has three distinct positive divisors whose sum... | math | {
"ground_truth": "1344",
"style": "rule"
} | {
"index": 88,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all positive integers $n>2$ such that\n\n$$\nn ! \\mid \\prod_{\\substack{p<... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 89,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find all triples of positive integers $(a, b, p)$ with $p$ prime and\n\n$$\na^{p}... | math | {
"ground_truth": "$(2,2,2),(3,4,3)$",
"style": "rule"
} | {
"index": 90,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine... | math | {
"ground_truth": "$f(x)=1$",
"style": "rule"
} | {
"index": 91,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers such that $a_{0}... | math | {
"ground_truth": "$\\frac{2016}{2017^{2}}$",
"style": "rule"
} | {
"index": 92,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Find the maximal value of\n\n$$\nS=\\sqrt[3]{\\frac{a}{b+7}}+\\sqrt[3]{\\frac{b}{... | math | {
"ground_truth": "$\\frac{8}{\\sqrt[3]{7}}$",
"style": "rule"
} | {
"index": 93,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Queenie and Horst play a game on a $20 \\times 20$ chessboard. In the beginning t... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 94,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $k$ be a positive integer. The organising committee of a tennis tournament is... | math | {
"ground_truth": "$k\\left(4 k^{2}+k-1\\right) / 2$",
"style": "rule"
} | {
"index": 95,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A circle $\\omega$ of radius 1 is given. A collection $T$ of triangles is called ... | math | {
"ground_truth": "$t(0,4]$",
"style": "rule"
} | {
"index": 96,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Find the smallest integer $k$ with the following p... | math | {
"ground_truth": "$k=2 n-1$",
"style": "rule"
} | {
"index": 97,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In the plane, 2013 red points and 2014 blue points are marked so that no three of... | math | {
"ground_truth": "2013",
"style": "rule"
} | {
"index": 98,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \\... | math | {
"ground_truth": "$f(n)=n$",
"style": "rule"
} | {
"index": 99,
"split": "default"
} |
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