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": "Find the largest possible integer $k$, such that the following statement is true:... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 100,
"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 $f$ from the set of positive integers into the set of pos... | math | {
"ground_truth": "$f(z)=z$",
"style": "rule"
} | {
"index": 101,
"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 integer $n \\geq 2$, let $N(n)$ be the maximal number of triples $\\left(... | math | {
"ground_truth": "$N(n)=\\left\\lfloor\\frac{2 n}{3}\\right\\rfloor+1$",
"style": "rule"
} | {
"index": 102,
"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 $999 \\times 999$ board a limp rook can move in the following way: From any ... | math | {
"ground_truth": "996000",
"style": "rule"
} | {
"index": 103,
"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 a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet... | math | {
"ground_truth": "$90^{\\circ}$,$60^{\\circ}$",
"style": "rule"
} | {
"index": 104,
"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 $n$ such that there exists a sequence of positive inte... | math | {
"ground_truth": "1,2,3,4",
"style": "rule"
} | {
"index": 105,
"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 we consider rectangles whose sides are parallel to the coordinate ax... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 106,
"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 coordinate plane consider the set $S$ of all points with integer coordinat... | math | {
"ground_truth": "180180",
"style": "rule"
} | {
"index": 107,
"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 $n$ and $k$ be fixed positive integers of the same parity, $k \\geq n$. We ar... | math | {
"ground_truth": "$2^{k-n}$",
"style": "rule"
} | {
"index": 108,
"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 $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ that satisfy the con... | math | {
"ground_truth": "$f(x)=x-1$",
"style": "rule"
} | {
"index": 109,
"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 $n \\geq 1$ be an integer. What is the maximum number of disjoint pairs of el... | math | {
"ground_truth": "$\\lfloor\\frac{2 n-1}{5}\\rfloor$",
"style": "rule"
} | {
"index": 110,
"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 a $999 \\times 999$ square table some cells are white and the remaining ones a... | math | {
"ground_truth": "$\\frac{4 \\cdot 999^{4}}{27}$",
"style": "rule"
} | {
"index": 111,
"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 with $N \\geq 2012$ coins and 2012 boxes arranged... | math | {
"ground_truth": "4022",
"style": "rule"
} | {
"index": 112,
"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 $(x, y, z)$ of positive integers such that $x \\leq y \\leq z$ a... | math | {
"ground_truth": "$(2,251,252)$",
"style": "rule"
} | {
"index": 113,
"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 $f: \\mathbb{Q} \\rightarrow \\mathbb{Q}$ such that the equati... | math | {
"ground_truth": "$f(x)=x,f(x)=-x$",
"style": "rule"
} | {
"index": 114,
"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 plane has a special point $O$ called the origin. Let $P$ be a set of 2021 point... | math | {
"ground_truth": "$2021 \\cdot 505 \\cdot 337$",
"style": "rule"
} | {
"index": 115,
"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 $k$ for which there exist a colouring of the p... | math | {
"ground_truth": "$k=3$",
"style": "rule"
} | {
"index": 116,
"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. Consider a $4 m \\times 4 m$ array of square unit ... | math | {
"ground_truth": "$6m$",
"style": "rule"
} | {
"index": 117,
"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 $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \\ldots$ is defined by ... | math | {
"ground_truth": "1,2",
"style": "rule"
} | {
"index": 118,
"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 $n$ contestants of an EGMO are named $C_{1}, \\ldots, C_{n}$. After the compe... | math | {
"ground_truth": "$2^{n}-n-1$",
"style": "rule"
} | {
"index": 119,
"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 real numbers such that $a b+b c+$ $c a=1$ and\n\n... | math | {
"ground_truth": "$(0,1,1),(0,-1,-1),(1,0,1),(-1,0,-1),(1,1,0)$,$(-1,-1,0),\\left(\\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}\\right)$,$\\left(-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}}\\right)$",
"style": "rule"
} | {
"index": 120,
"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. Dominoes are placed on a $2 n \\times 2 n$ board i... | math | {
"ground_truth": "$\\frac{n(n+1)}{2}$",
"style": "rule"
} | {
"index": 121,
"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 $n \\geq 2$, determine the largest positive integer $N$ ... | math | {
"ground_truth": "$N=n$",
"style": "rule"
} | {
"index": 122,
"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 $m$ for which the $m \\times m$ square can be dissected in... | math | {
"ground_truth": "11,13",
"style": "rule"
} | {
"index": 123,
"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. Lexi has a dictionary $\\mathcal{D}$ consisting of... | math | {
"ground_truth": "$2^{k-1}$",
"style": "rule"
} | {
"index": 124,
"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 an increasing sequence of numbers with an odd number of terms, the difference ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 125,
"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 two increasing sequences of five consecutive integers, each of which ha... | math | {
"ground_truth": "10,11,12,13,14,-2,-1,0,1,2",
"style": "rule"
} | {
"index": 126,
"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": "If $f(t)=\\sin \\left(\\pi t-\\frac{\\pi}{2}\\right)$, what is the smallest posit... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 127,
"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 integer values of $x$ such that $\\left(x^{2}-3\\right)\\left(x^{2}... | math | {
"ground_truth": "-1,0,1",
"style": "rule"
} | {
"index": 128,
"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": "At present, the sum of the ages of a husband and wife, $P$, is six times the sum ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 129,
"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": "What is the value of $x$ such that $\\log _{2}\\left(\\log _{2}(2 x-2)\\right)=2$... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 130,
"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 $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine... | math | {
"ground_truth": "210",
"style": "rule"
} | {
"index": 131,
"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, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x... | math | {
"ground_truth": "(-\\infty,-5)",
"style": "rule"
} | {
"index": 132,
"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": "If $2 \\leq x \\leq 5$ and $10 \\leq y \\leq 20$, what is the maximum value of $1... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 133,
"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 functions $f$ and $g$ satisfy\n\n$$\n\\begin{aligned}\n& f(x)+g(x)=3 x+5 \\\\... | math | {
"ground_truth": "-84",
"style": "rule"
} | {
"index": 134,
"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": "Three different numbers are chosen at random from the set $\\{1,2,3,4,5\\}$.\n\nT... | math | {
"ground_truth": "$\\frac{2}{5}$",
"style": "rule"
} | {
"index": 135,
"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": "What is the largest two-digit number that becomes $75 \\%$ greater when its digit... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 136,
"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": "Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. ... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 137,
"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": "Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabol... | math | {
"ground_truth": "$\\frac{32}{9}$,$\\frac{1}{2}$",
"style": "rule"
} | {
"index": 138,
"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 75 year old person has a $50 \\%$ chance of living at least another 10 years.\n... | math | {
"ground_truth": "62.5%",
"style": "rule"
} | {
"index": 139,
"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 values of $x$ for which $2^{\\log _{10}\\left(x^{2}\\right)}=3\\lef... | math | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": 140,
"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 Sieve of Sundaram uses the following infinite table of positive integers:\n\n... | math | {
"ground_truth": "4090",
"style": "rule"
} | {
"index": 141,
"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 Sieve of Sundaram uses the following infinite table of positive integers:\n\n... | math | {
"ground_truth": "$2RC+R+C$",
"style": "rule"
} | {
"index": 142,
"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 $\\lfloor x\\rfloor$ denote the greatest integer less than or equal to $x$. F... | math | {
"ground_truth": "4085",
"style": "rule"
} | {
"index": 143,
"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 $\\lfloor x\\rfloor$ denote the greatest integer less than or equal to $x$. F... | math | {
"ground_truth": "55",
"style": "rule"
} | {
"index": 144,
"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": "Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly se... | math | {
"ground_truth": "$\\frac{14}{15}$",
"style": "rule"
} | {
"index": 145,
"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 goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it ... | math | {
"ground_truth": "(6,5)",
"style": "rule"
} | {
"index": 146,
"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 possible values of $r$ such that the three term geometric sequence ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 147,
"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": "If $f(x)=\\sin ^{2} x-2 \\sin x+2$, what are the minimum and maximum values of $f... | math | {
"ground_truth": "5,1",
"style": "rule"
} | {
"index": 148,
"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": "What is the sum of the digits of the integer equal to $\\left(10^{3}+1\\right)^{2... | math | {
"ground_truth": "1002001",
"style": "rule"
} | {
"index": 149,
"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 bakery sells small and large cookies. Before a price increase, the price of eac... | math | {
"ground_truth": "$8 \\%$",
"style": "rule"
} | {
"index": 150,
"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": "Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average ag... | math | {
"ground_truth": "7,14,18",
"style": "rule"
} | {
"index": 151,
"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 parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola ... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 152,
"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": "If $3\\left(8^{x}\\right)+5\\left(8^{x}\\right)=2^{61}$, what is the value of the... | math | {
"ground_truth": "$\\frac{58}{3}$",
"style": "rule"
} | {
"index": 153,
"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 some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists... | math | {
"ground_truth": "1,-1,7,-7",
"style": "rule"
} | {
"index": 154,
"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": "Chinara starts with the point $(3,5)$, and applies the following three-step proce... | math | {
"ground_truth": "(-7,-1)",
"style": "rule"
} | {
"index": 155,
"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 $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \\ldots, t_{n-2}, t... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 156,
"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$ and $r$ are real numbers. A geometric sequence with first term $... | math | {
"ground_truth": "$a=2$, $a=-6-4 \\sqrt{2}$",
"style": "rule"
} | {
"index": 157,
"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 bag contains 3 green balls, 4 red balls, and no other balls. Victor removes bal... | math | {
"ground_truth": "$\\frac{4}{7}$",
"style": "rule"
} | {
"index": 158,
"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 $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\\log _{\\fr... | math | {
"ground_truth": "$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$",
"style": "rule"
} | {
"index": 159,
"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=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which ... | math | {
"ground_truth": "$(3,2),(-3,2),(3,-2),(-3,-2)$",
"style": "rule"
} | {
"index": 160,
"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 values of $x$ for which $0<\\frac{x^{2}-11}{x+1}<7$.",
"role": ... | math | {
"ground_truth": "$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$",
"style": "rule"
} | {
"index": 161,
"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 numbers $a_{1}, a_{2}, a_{3}, \\ldots$ form an arithmetic sequence with $a_{1... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"index": 162,
"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 some positive integers $k$, the parabola with equation $y=\\frac{x^{2}}{k}-5$... | math | {
"ground_truth": "1,2,5,8,9",
"style": "rule"
} | {
"index": 163,
"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 the following system of equations in which all logarithms have base 10:\... | math | {
"ground_truth": "$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$",
"style": "rule"
} | {
"index": 164,
"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": "Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the pr... | math | {
"ground_truth": "$\\frac{11}{36}$",
"style": "rule"
} | {
"index": 165,
"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": "If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possi... | math | {
"ground_truth": "$(3,0),(-3,1)$",
"style": "rule"
} | {
"index": 166,
"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": "Digital images consist of a very large number of equally spaced dots called pixel... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 167,
"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": "If $T=x^{2}+\\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}... | math | {
"ground_truth": "-3,0",
"style": "rule"
} | {
"index": 168,
"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 Skolem sequence of order $n$ is a sequence $\\left(s_{1}, s_{2}, \\ldots, s_{2 ... | math | {
"ground_truth": "(4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3)",
"style": "rule"
} | {
"index": 169,
"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 Skolem sequence of order $n$ is a sequence $\\left(s_{1}, s_{2}, \\ldots, s_{2 ... | math | {
"ground_truth": "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)",
"style": "rule"
} | {
"index": 170,
"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 three-digit positive integer $m$ is odd and has three distinct digits. If the... | math | {
"ground_truth": "623",
"style": "rule"
} | {
"index": 171,
"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": "Eleanor has 100 marbles, each of which is black or gold. The ratio of the number ... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 172,
"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 $n$ is a positive integer and that the value of $\\frac{n^{2}+n+15}{... | math | {
"ground_truth": "1, 3, 5, 15",
"style": "rule"
} | {
"index": 173,
"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": "Ada starts with $x=10$ and $y=2$, and applies the following process:\n\nStep 1: A... | math | {
"ground_truth": "340",
"style": "rule"
} | {
"index": 174,
"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 $k$, with $k \\neq 0$, for which the parabola with equatio... | math | {
"ground_truth": "-2,-1,1,2",
"style": "rule"
} | {
"index": 175,
"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 positive integers $a$ and $b$ have no common divisor larger than 1 . If the d... | math | {
"ground_truth": "$\\frac{19}{34}$",
"style": "rule"
} | {
"index": 176,
"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 geometric sequence has first term 10 and common ratio $\\frac{1}{2}$.\n\nAn ari... | math | {
"ground_truth": "$-\\frac{30}{17}$",
"style": "rule"
} | {
"index": 177,
"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 real number $x$, define $f(x)$ to be the number of prime number... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 178,
"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 $(x, y, z)$ of real numbers that satisfy the following syst... | math | {
"ground_truth": "(1,-4,-2),(3,2,3),(13,2,-2)",
"style": "rule"
} | {
"index": 179,
"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 the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and... | math | {
"ground_truth": "$4 \\pi^{2}-1$",
"style": "rule"
} | {
"index": 180,
"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 of angles $(x, y)$ with $0^{\\circ} \\leq x<180^{\\circ}$ and... | math | {
"ground_truth": "$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$",
"style": "rule"
} | {
"index": 181,
"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": "Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in wh... | math | {
"ground_truth": "$\\frac{1-p^{2}}{3}$",
"style": "rule"
} | {
"index": 182,
"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": "Three microphones $A, B$ and $C$ are placed on a line such that $A$ is $1 \\mathr... | math | {
"ground_truth": "$\\frac{41}{12}$",
"style": "rule"
} | {
"index": 183,
"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": "Kerry has a list of $n$ integers $a_{1}, a_{2}, \\ldots, a_{n}$ satisfying $a_{1}... | math | {
"ground_truth": "(1,7,103, 105), (3, 5, 101, 107)",
"style": "rule"
} | {
"index": 184,
"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 values of $x$ for which $\\frac{x^{2}+x+4}{2 x+1}=\\frac{4}{x}$.",
... | math | {
"ground_truth": "$-1$,$2$,$-2$",
"style": "rule"
} | {
"index": 185,
"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 number of positive divisors of 900, including 1 and 900, that are p... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 186,
"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": "Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\\angle A ... | math | {
"ground_truth": "$8$,$4$",
"style": "rule"
} | {
"index": 187,
"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 chemist has three bottles, each containing a mixture of acid and water:\n\n- bo... | math | {
"ground_truth": "17.5%",
"style": "rule"
} | {
"index": 188,
"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 $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimu... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 189,
"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 bag contains 40 balls, each of which is black or gold. Feridun reaches into the... | math | {
"ground_truth": "26",
"style": "rule"
} | {
"index": 190,
"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 geometric sequence with $n$ terms $t_{1}, t_{2}, \\ldots, t_{n-1}, t_{n}$ has... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 191,
"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": "If $\\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\\frac{1}{2}$, what is t... | math | {
"ground_truth": "4027",
"style": "rule"
} | {
"index": 192,
"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 real numbers $x$ for which\n\n$$\n\\left(\\log _{10} x\\right)^{\\l... | math | {
"ground_truth": "$10^{100}$,$10^{1 / 100}$",
"style": "rule"
} | {
"index": 193,
"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": "Without using a calculator, determine positive integers $m$ and $n$ for which\n\n... | math | {
"ground_truth": "$221,$8$",
"style": "rule"
} | {
"index": 194,
"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 $f(n)$ be the number of positive integers that have exactly $n$ digits and wh... | math | {
"ground_truth": "202",
"style": "rule"
} | {
"index": 195,
"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": "If $\\log _{10} x=3+\\log _{10} y$, what is the value of $\\frac{x}{y}$ ?",
"... | math | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": 196,
"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": "If $x+\\frac{1}{x}=\\frac{13}{6}$, determine all values of $x^{2}+\\frac{1}{x^{2}... | math | {
"ground_truth": "$\\frac{97}{36}$",
"style": "rule"
} | {
"index": 197,
"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 die, with the numbers $1,2,3,4,6$, and 8 on its six faces, is rolled. After thi... | math | {
"ground_truth": "$\\frac{2}{9}$",
"style": "rule"
} | {
"index": 198,
"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 table below gives the final standings for seven of the teams in the English C... | math | {
"ground_truth": "16,3,1,1",
"style": "rule"
} | {
"index": 199,
"split": "default"
} |
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