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": "The equations $x^{2}+5 x+6=0$ and $x^{2}+5 x-6=0$ each have integer solutions whe... | math | {
"ground_truth": "$\\frac{a b}{2}$",
"style": "rule"
} | {
"index": 300,
"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 $k$, with $k \\neq 0$, for which the parabola\n\n$$\ny=k ... | math | {
"ground_truth": "$-1,-9$",
"style": "rule"
} | {
"index": 301,
"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 function $f(x)$ satisfies the equation $f(x)=f(x-1)+f(x+1)$ for all values of... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 302,
"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, b, c$, in that order, form a three term arithmetic sequence (see ... | math | {
"ground_truth": "$(27,20,13), (18,20,22)$",
"style": "rule"
} | {
"index": 303,
"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 average of three consecutive multiples of 3 is $a$.\n\nThe average of four co... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 304,
"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": "Billy and Crystal each have a bag of 9 balls. The balls in each bag are numbered ... | math | {
"ground_truth": "$\\frac{7}{27}$",
"style": "rule"
} | {
"index": 305,
"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 $2^{x+2} 5^{6-x}=10^{x^{2}}$ has two real solutions. Determine these... | math | {
"ground_truth": "$2,-\\log _{10} 250$",
"style": "rule"
} | {
"index": 306,
"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 solutions to the system of equations\n\n$$\n\\begin{aligned}\n... | math | {
"ground_truth": "$1,2,3$",
"style": "rule"
} | {
"index": 307,
"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 34 and 80 have exactly two positive common divisors, namely... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 308,
"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 function $f$ is defined so that\n\n- $f(1)=1$,\n- if $n$ is an even positive in... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 309,
"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 perimeter of equilateral $\\triangle P Q R$ is 12. The perimeter of regular h... | math | {
"ground_truth": "$\\frac{2}{3}$",
"style": "rule"
} | {
"index": 310,
"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 how many integers $k$ with $0<k<18$ is $\\frac{5 \\sin \\left(10 k^{\\circ}\\... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 311,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Eight people, including triplets Barry, Carrie and Mary, are going for a trip in ... | math | {
"ground_truth": "$\\frac{4}{7}$",
"style": "rule"
} | {
"index": 312,
"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": "Diagonal $W Y$ of square $W X Y Z$ has slope 2. Determine the sum of the slopes o... | math | {
"ground_truth": "$-\\frac{8}{3}$",
"style": "rule"
} | {
"index": 313,
"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 $x$ such that $\\log _{2 x}(48 \\sqrt[3]{3})=\\log _{3 x}... | math | {
"ground_truth": "$\\sqrt{6}$",
"style": "rule"
} | {
"index": 314,
"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 infinite array with two rows, the numbers in the top row are denoted $\\ldo... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 315,
"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 populations of Alphaville and Betaville were equal at the end of 1995. The po... | math | {
"ground_truth": "4.2",
"style": "rule"
} | {
"index": 316,
"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 coordinates of the points of intersection of the graphs of $y=\\log... | math | {
"ground_truth": "$(4, \\log _{10} 2)$",
"style": "rule"
} | {
"index": 317,
"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": "Charlie was born in the twentieth century. On his birthday in the present year (2... | math | {
"ground_truth": "1938",
"style": "rule"
} | {
"index": 318,
"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 $A, B$, and $C$ be randomly chosen (not necessarily distinct) integers betwee... | math | {
"ground_truth": "$\\frac{9}{25}$",
"style": "rule"
} | {
"index": 319,
"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": "Bobby, Peter, Greg, Cindy, Jan, and Marcia line up for ice cream. In an acceptabl... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 320,
"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 $A B C, a=12, b=17$, and $c=13$. Compute $b \\cos C-c \\cos B$.",
... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 321,
"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 sequence of words $\\left\\{a_{n}\\right\\}$ is defined as follows: $a_{1}=X,... | math | {
"ground_truth": "667",
"style": "rule"
} | {
"index": 322,
"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": "Compute the smallest positive integer $n$ such that $214 \\cdot n$ and $2014 \\cd... | math | {
"ground_truth": "19133",
"style": "rule"
} | {
"index": 323,
"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 the least integer greater than 20 that is a palindrome in both base 20... | math | {
"ground_truth": "105",
"style": "rule"
} | {
"index": 324,
"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": "$\\quad$ Compute the greatest integer $k \\leq 1000$ such that $\\left(\\begin{ar... | math | {
"ground_truth": "979",
"style": "rule"
} | {
"index": 325,
"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-valued function $f$ is called tenuous if $f(x)+f(y)>x^{2}$ for all pos... | math | {
"ground_truth": "136",
"style": "rule"
} | {
"index": 326,
"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 $T=(0,0), N=(2,0), Y=(6,6), W=(2,6)$, and $R=(0,2)$. Compute the area of pent... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 327,
"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 $T=20$. The lengths of the sides of a rectangle are the zeroes of the polynom... | math | {
"ground_truth": "$20 \\sqrt{7}$",
"style": "rule"
} | {
"index": 328,
"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 $T=20 \\sqrt{7}$. Let $w>0$ be a real number such that $T$ is the area of the... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 329,
"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": "Compute the least positive integer $n$ such that $\\operatorname{gcd}\\left(n^{3}... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 330,
"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 $T=8$. At a party, everyone shakes hands with everyone else exactly once, exc... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 331,
"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 $T=7$. Given the sequence $u_{n}$ such that $u_{3}=5, u_{6}=89$, and $u_{n+2}... | math | {
"ground_truth": "233",
"style": "rule"
} | {
"index": 332,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In each town in ARMLandia, the residents have formed groups, which meet each week... | math | {
"ground_truth": "136",
"style": "rule"
} | {
"index": 333,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In each town in ARMLandia, the residents have formed groups, which meet each week... | math | {
"ground_truth": "$d=\\frac{1+\\sqrt{1+8 n}}{2}$",
"style": "rule"
} | {
"index": 334,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In each town in ARMLandia, the residents have formed groups, which meet each week... | math | {
"ground_truth": "1680",
"style": "rule"
} | {
"index": 335,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In each town in ARMLandia, the residents have formed groups, which meet each week... | math | {
"ground_truth": "$\\frac{d !}{2 k(d-k) !}$",
"style": "rule"
} | {
"index": 336,
"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 student computed the repeating decimal expansion of $\\frac{1}{N}$ for some int... | math | {
"ground_truth": "606",
"style": "rule"
} | {
"index": 337,
"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 four-digit number whose square root is three times the sum of the di... | math | {
"ground_truth": "2916",
"style": "rule"
} | {
"index": 338,
"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": "Compute the sum of the reciprocals of the positive integer divisors of 24.",
... | math | {
"ground_truth": "$\\frac{5}{2}$",
"style": "rule"
} | {
"index": 339,
"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 exists a digit $Y$ such that, for any digit $X$, the seven-digit number $\\... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 340,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A point is selected at random from the interior of a right triangle with legs of ... | math | {
"ground_truth": "$(\\frac{1}{4}, \\frac{1}{27})$",
"style": "rule"
} | {
"index": 341,
"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 square $A R M L$ is contained in the $x y$-plane with $A=(0,0)$ and $M=(1,1)$... | math | {
"ground_truth": "$\\frac{2}{7} \\sqrt{53}$",
"style": "rule"
} | {
"index": 342,
"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 $S_{k}$ denote the infinite arithmetic sequenc... | math | {
"ground_truth": "326",
"style": "rule"
} | {
"index": 343,
"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": "Compute the sum of all values of $k$ for which there exist positive real numbers ... | math | {
"ground_truth": "$\\frac{43}{48}$",
"style": "rule"
} | {
"index": 344,
"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 $W=(0,0), A=(7,0), S=(7,1)$, and $H=(0,1)$. Compute the number of ways to til... | math | {
"ground_truth": "3432",
"style": "rule"
} | {
"index": 345,
"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": "Compute $\\sin ^{2} 4^{\\circ}+\\sin ^{2} 8^{\\circ}+\\sin ^{2} 12^{\\circ}+\\cdo... | math | {
"ground_truth": "$\\frac{45}{2}$",
"style": "rule"
} | {
"index": 346,
"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": "Compute the area of the region defined by $x^{2}+y^{2} \\leq|x|+|y|$.",
"role... | math | {
"ground_truth": "$2+\\pi$",
"style": "rule"
} | {
"index": 347,
"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 arithmetic sequences $a_{1}, a_{2}, a_{3}, \\ldots, a_{20}$ and $b_{1}, b_{2}... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 348,
"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": "Compute the ordered triple $(x, y, z)$ representing the farthest lattice point fr... | math | {
"ground_truth": "$(-266,-3,-28)$",
"style": "rule"
} | {
"index": 349,
"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 sequence $a_{1}, a_{2}, a_{3}, \\ldots$ is a geometric sequence with $a_{20}=... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 350,
"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 $T=1$. Circles $L$ and $O$ are internally tangent and have radii $T$ and $4 T... | math | {
"ground_truth": "$2 \\sqrt{2}$",
"style": "rule"
} | {
"index": 351,
"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 $T=2 \\sqrt{2}$. In a right triangle, one leg has length $T^{2}$ and the othe... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 352,
"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": "$\\quad$ Let $T=40$. If $x+9 y=17$ and $T x+(T+1) y=T+2$, compute $20 x+14 y$.",
... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 353,
"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 $T=8$. Let $f(x)=a x^{2}+b x+c$. The product of the roots of $f$ is $T$. If $... | math | {
"ground_truth": "$\\frac{8}{5}$",
"style": "rule"
} | {
"index": 354,
"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 $T=\\frac{8}{5}$. Let $z_{1}=15+5 i$ and $z_{2}=1+K i$. Compute the smallest ... | math | {
"ground_truth": "25",
"style": "rule"
} | {
"index": 355,
"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 $T=25$. Suppose that $T$ people are standing in a line, including three peopl... | math | {
"ground_truth": "$\\frac{47}{300}$",
"style": "rule"
} | {
"index": 356,
"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 $A$ be the number you will receive from position 7 and let $B$ be the number ... | math | {
"ground_truth": "$\\frac{94}{4225}$",
"style": "rule"
} | {
"index": 357,
"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 $T=13$. If $r$ is the radius of a right circular cone and the cone's height i... | math | {
"ground_truth": "$\\frac{12}{169}$",
"style": "rule"
} | {
"index": 358,
"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 $T=650$. If $\\log T=2-\\log 2+\\log k$, compute the value of $k$.",
"rol... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 359,
"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 $T=100$. Nellie has a flight from Rome to Athens that is scheduled to last fo... | math | {
"ground_truth": "650",
"style": "rule"
} | {
"index": 360,
"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 $T=9$. Compute $\\sqrt{\\sqrt{\\sqrt[T]{10^{T^{2}-T}}}}$.",
"role": "user... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 361,
"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 $T=3$. Regular hexagon $S U P E R B$ has side length $\\sqrt{T}$. Compute the... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 362,
"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 $T=70$. Chef Selma is preparing a burrito menu. A burrito consists of: (1) a ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 363,
"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": "Compute the smallest positive integer $N$ such that $20 N$ is a multiple of 14 an... | math | {
"ground_truth": "70",
"style": "rule"
} | {
"index": 364,
"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 positive integer fibbish if each digit, after the leftmost two, is at leas... | math | {
"ground_truth": "10112369",
"style": "rule"
} | {
"index": 365,
"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 ARMLbar is a $7 \\times 7$ grid of unit squares with the center unit square re... | math | {
"ground_truth": "96",
"style": "rule"
} | {
"index": 366,
"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": "Regular hexagon $A B C D E F$ and regular hexagon $G H I J K L$ both have side le... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 367,
"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": "Compute the largest base-10 integer $\\underline{A} \\underline{B} \\underline{C}... | math | {
"ground_truth": "5762",
"style": "rule"
} | {
"index": 368,
"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 $X$ be the number of digits in the decimal expansion of $100^{1000^{10,000}}$... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 369,
"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": "Compute the smallest possible value of $n$ such that two diagonals of a regular $... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 370,
"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": "Compute the number of quadratic functions $f(x)=a x^{2}+b x+c$ with integer roots... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 371,
"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 bubble in the shape of a hemisphere of radius 1 is on a tabletop. Inside the bu... | math | {
"ground_truth": "$\\frac{1}{54}$",
"style": "rule"
} | {
"index": 372,
"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": "Compute the smallest positive integer base $b$ for which $16_{b}$ is prime and $9... | math | {
"ground_truth": "53",
"style": "rule"
} | {
"index": 373,
"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 $n$, let $C(n)$ equal the number of pairs of consecutive 1... | math | {
"ground_truth": "448",
"style": "rule"
} | {
"index": 374,
"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 set $S$ contains thirteen distinct positive integers whose sum is 120 . Compute... | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 375,
"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 $T=11$. Compute the least positive integer $b$ such that, when expressed in b... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 376,
"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 $T=5$. Suppose that $a_{1}=1$, and that for all positive integers $n, a_{n+1}... | math | {
"ground_truth": "491",
"style": "rule"
} | {
"index": 377,
"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": "Compute the smallest $n$ such that in the regular $n$-gon $A_{1} A_{2} A_{3} \\cd... | math | {
"ground_truth": "37",
"style": "rule"
} | {
"index": 378,
"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 $T=37$. A cube has edges of length $T$. Square holes of side length 1 are dri... | math | {
"ground_truth": "8640",
"style": "rule"
} | {
"index": 379,
"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 $T=8640$. Compute $\\left\\lfloor\\log _{4}\\left(1+2+4+\\cdots+2^{T}\\right)... | math | {
"ground_truth": "4320",
"style": "rule"
} | {
"index": 380,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "$\\frac{1}{3},1,\\frac{1}{9}$",
"style": "rule"
} | {
"index": 381,
"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 ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "81",
"style": "rule"
} | {
"index": 382,
"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 ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "27,54,108,135,189,216,270,297,351,378",
"style": "rule"
} | {
"index": 383,
"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 ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 384,
"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 ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "$1/3$",
"style": "rule"
} | {
"index": 385,
"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 ARMLopolis, every house number is a positive integer, and City Hall's address ... | math | {
"ground_truth": "$\\frac{1}{3}, 1, 9$",
"style": "rule"
} | {
"index": 386,
"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 $A R M L$ be a trapezoid with bases $\\overline{A R}$ and $\\overline{M L}$, ... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 387,
"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 regular hexagon has side length 1. Compute the average of the areas of the 20 t... | math | {
"ground_truth": "$\\frac{9 \\sqrt{3}}{20}$",
"style": "rule"
} | {
"index": 388,
"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": "Paul was planning to buy 20 items from the ARML shop. He wanted some mugs, which ... | math | {
"ground_truth": "120",
"style": "rule"
} | {
"index": 389,
"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 $x$ be the smallest positive integer such that $1584 \\cdot x$ is a perfect c... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 390,
"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": "Emma goes to the store to buy apples and peaches. She buys five of each, hands th... | math | {
"ground_truth": "1525",
"style": "rule"
} | {
"index": 391,
"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": "Circle $O$ has radius 6. Point $P$ lies outside circle $O$, and the shortest dist... | math | {
"ground_truth": "272",
"style": "rule"
} | {
"index": 392,
"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 palindrome is a positive integer, not ending in 0 , that reads the same forward... | math | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": 393,
"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": "Positive integers $x, y, z$ satisfy $x y+z=160$. Compute the smallest possible va... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 394,
"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": "Compute $\\cos ^{3} \\frac{2 \\pi}{7}+\\cos ^{3} \\frac{4 \\pi}{7}+\\cos ^{3} \\f... | math | {
"ground_truth": "$-\\frac{1}{2}$",
"style": "rule"
} | {
"index": 395,
"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 right triangle $A B C$ with right angle $C$, line $\\ell$ is drawn through $C$... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 396,
"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": "Compute the sum of all real numbers $x$ such that\n\n$$\n\\left\\lfloor\\frac{x}{... | math | {
"ground_truth": "-21",
"style": "rule"
} | {
"index": 397,
"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 $S=\\{1,2, \\ldots, 20\\}$, and let $f$ be a function from $S$ to $S$; that i... | math | {
"ground_truth": "140",
"style": "rule"
} | {
"index": 398,
"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": "Compute the smallest positive integer $n$ such that $n^{2}+n^{0}+n^{1}+n^{3}$ is ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 399,
"split": "default"
} |
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