data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | sub_source stringclasses 8
values |
|---|---|---|---|---|---|
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $k$ for which there exists a nonlinear function $f:\\mathbb{Z} \\rightarrow\\mathbb{Z}$ such that the equation $$ f(a)+f(b)+f(c)=\\frac{f(a-b)+f(... | math | {
"ground_truth": "k = \\{0, 1, 3, 9\\}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the system of equations: \\[ x+y+z=a \\] \\[ x^2+y^2+z^2=b^2 \\] \\[ xy=z^2 \\] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisf... | math | {
"ground_truth": "x, y = \\frac{a^2 + b^2 \\pm \\sqrt{(3a^2 - b^2)(3b^2 - a^2)}}{4a}, z = \\frac{a^2 - b^2}{2a}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A class has 100 students, and after a math exam, the scores are normally distributed with a mean of 100 and a standard deviation of 10, denoted as ξ～N(100,10^2). Given that P(90 ... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many ways can a student schedule $4$ mathematics courses—algebra, geometry, number theory, and calculus—in a $7$-period day if no two mathematics courses can be taken in cons... | math | {
"ground_truth": "120",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In right triangle $PQR$ with legs $PQ$ and $PR$, points $S$ and $T$ lie on legs $PQ$ and $PR$, respectively, such that $PS:SQ = PT:TR = 1:3$. If $QT = 20$ units, and $SR = 36$ un... | math | {
"ground_truth": "\\sqrt{1085.44}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What fraction of $\\frac{3}{4}$ is $\\frac{1}{6}$?",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2}{9}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A chessboard has 13 rows and 17 columns. Each small square contains a number, starting from the top-left corner. The first row is filled sequentially with $1, 2, \\cdots, 17$; th... | math | {
"ground_truth": "555",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the equation of the plane passing through $(2,-2,1)$ and $(5,1,2),$ and which is perpendicular to the plane $2x - y + 4z = 7.$ Enter your answer in the form\n\\[Ax + By + Cz... | math | {
"ground_truth": "11x - 10y - 9z - 33 = 0",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be the vertex of the graph of the equation $y=x^2 - 4x + 7$. Let $B$ be the vertex of the graph of the equation $y=x^2 + 6x + 20$. What is the distance between $A$ and $B... | math | {
"ground_truth": "\\sqrt{89}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\theta\\in[0, \\dfrac{\\pi}{2}]$ , the following inequality $\\sqrt{2}(2a+3)\\cos(\\theta-\\dfrac{\\pi}{4})+\\dfrac{6}{\\sin\\theta+\\cos\\theta}-2\\sin2\\theta<3a+6$ is... | math | {
"ground_truth": "a > 3",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate the expression $(2)^{3^{2}} + 1^{3^{3}}$.\nA) 511\nB) 512\nC) 513\nD) 514",
"role": "user"
}
] | math | {
"ground_truth": "C) 513",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$\\sin^{4}(2025x) + \\cos^{2019}(2016x) \\cdot \\cos^{2018}(2025x) = 1$.",
"role": "user"
}
] | math | {
"ground_truth": "x = \\frac{\\pi}{4050} + \\frac{n\\pi}{2025}, \\, n \\in \\mathbb{Z}, \\quad \\text{or} \\quad x = \\frac{k\\pi}{9}, \\, k \\in \\mathbb{Z}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The number of positive even integers less than $1000$ divisible by neither $3$ nor $11$ is:\nA) 100\nB) 108\nC) 110\nD) 112\nE) 120",
"role": "user"
}
] | math | {
"ground_truth": "B) 108",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given points A(1, -1) and B(2, 2), and point P lies on the line y = x. Find the coordinates of point P for which the value of $|PA|^2 + |PB|^2$ is minimized.",
"role": "user"... | math | {
"ground_truth": "P(1, 1)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\\(2 \\cos^{6}(2t) - \\cos^{4}(2t) + 1.5 \\sin^{2}(4t) - 3 \\sin^{2}(2t) = 0\\)",
"role": "user"
}
] | math | {
"ground_truth": "t = \\frac{\\pi}{8}(2k + 1), \\quad k \\in \\mathbb{Z}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular sports field has a diagonal of 20 meters and an area of 120 square meters. What is the perimeter of the sports field in meters?\nA) 48\nB) 50\nC) 51\nD) 53\nE) 55",... | math | {
"ground_truth": "C) 51",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the line $l: ax-2y+2=0 \\ (a \\in \\mathbb{R})$ \n(1) If it is parallel to the line $m: x+(a-3)y+1=0 \\ (a \\in \\mathbb{R})$, find $a$; \n(2) If line $l$ always bisects ... | math | {
"ground_truth": "a=-2",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that Camilla had twice as many blueberry jelly beans as cherry jelly beans, and after eating 10 pieces of each kind, she now has three times as many blueberry jelly beans a... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $α$ and $β$ are acute angles, and $\\sin α= \\frac { \\sqrt {5}}{5}$, $\\sin β= \\frac { \\sqrt {10}}{10}$, calculate the value of $α+β$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{π}{4}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three workers are digging a hole. They take turns working, with each one working as long as it takes the other two to dig half the hole. Working in this way, they finished diggin... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ N$ denote the set of natural numbers. Let $ \\phi: N\\rightarrow N$ be a bijective function and assume that there exists a finite limit\r\n\\[ \\lim_{n\\rightarrow\\inf... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If \"0<x<1\" is a sufficient but not necessary condition for \"(x－a)\\[x－(a＋2)\\]≤0\", determine the range of the real number $a$.",
"role": "user"
}
] | math | {
"ground_truth": "[-1,0]",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the base prime representation of the number $441$.",
"role": "user"
}
] | math | {
"ground_truth": "0202",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sum of positive numbers \\( x_{1}, x_{2}, \\ldots, x_{n} \\) is equal to 1. Let \\( s \\) be the largest of the numbers\n\n\\[\n\\frac{x_{1}}{1+x_{1}}, \\frac{x_{2}}{1+x_{1}+... | math | {
"ground_truth": "1 - 2^{-\\frac{1}{n}}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The ratio of the number of games won to the number of games lost (no games ending in a tie nor cancelled) by the High Flyers is $17/8$. If it is known that 6 games were cancelled... | math | {
"ground_truth": "B) 30%",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the least integral value of m such that the equation \\( 3x(mx+6) - 2x^2 + 8 = 0 \\) has no real roots.",
"role": "user"
}
] | math | {
"ground_truth": "4",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are two spheres $\\left(O_{1}, r; O_{2}, r\\right)$, a point $A$, and a distance $d$. The radii of the spheres are equal. Construct a triangle $ABC$ where one vertex $(A)$ ... | math | {
"ground_truth": "0 < d < 2r",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Point $P$ $(5, -3)$ is reflected over the $y$-axis to point $Q$. Then $Q$ is reflected over the line $y = -x$ to point $R$. What is the area of triangle $PQR$?",
"role": "use... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$ .",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{5 \\cdot (10^{171k+1} + 35)}{171}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Paul bought 50 items each priced at $0.50, $4, or $5. If his total purchase price was $100.00, how many 50-cent items did he purchase?",
"role": "user"
}
] | math | {
"ground_truth": "40",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If \\(2x - 5 = 15\\), calculate the value of \\(x\\).",
"role": "user"
}
] | math | {
"ground_truth": "10",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The annual interest rates for fixed deposits in a bank are as follows: 11.7% for a two-year period, 12.24% for a three-year period, and 13.86% for a five-year period. If person A... | math | {
"ground_truth": "184.48 \\text{ (元)}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$ .",
"role": "user"
}
] | math | {
"ground_truth": "\\{1, 2, 3, 4\\}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "John is driving from his house to a conference center. He drives 25 miles in the first hour, but then calculates that he will be 90 minutes late if he continues at this speed. To... | math | {
"ground_truth": "\\textbf{(C) 123.4375 miles}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many ways are there to distribute 6 balls into 3 boxes if the balls are indistinguishable but the boxes are distinguishable?",
"role": "user"
}
] | math | {
"ground_truth": "28",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all triples of strictly positive integers \\((m, n, p)\\) with \\(p\\) being a prime number, such that \\(2^{m} p^{2} + 1 = n^{5}\\).",
"role": "user"
}
] | math | {
"ground_truth": "(1,3,11)",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a sequence $\\{a_n\\}$ such that:\n$$\na_{n+1}=\n\\begin{cases}\n2a_n & \\text{if } 0 \\leq a_n \\leq 1,\\\\\na_n - 1 & \\text{if } a_n > 1,\n\\end{cases}\n$$\nand $a_1 ... | math | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A circle is inscribed in a triangle, and the points of tangency with the sides of the triangle are connected to each other. A new circle is inscribed in the resulting triangle, a... | math | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sequence $4860, 1620, 540, \\ldots$ is created by repeatedly dividing by 3. How many integers are in this sequence?",
"role": "user"
}
] | math | {
"ground_truth": "6",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given sets $A=\\{x| -1 \\leqslant x < 3\\}$ and $B=\\{x| 2 < x \\leqslant 5\\}$, find:<br/>\n$(1) A \\cap B$;<br/>\n$(2) (\\complement _{R}A) \\cap B$.",
"role": "user"
}
] | math | {
"ground_truth": "\\{x| 3 \\leqslant x \\leqslant 5\\}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Convert the point $(2, -2, 2\\sqrt{2})$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\t... | math | {
"ground_truth": "\\left( 4, \\frac{7 \\pi}{4}, \\frac{\\pi}{4} \\right)",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Simplify first, then find the value of $4x^{2}y-[6xy-3\\left(4xy-2\\right)-x^{2}y]+1$, where $|x+1|+\\left(y-2\\right)^{2}=0$.",
"role": "user"
}
] | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive two-digit integers are factors of $3^{18}-1$?\nA) 3\nB) 5\nC) 6\nD) 7 \nE) 8",
"role": "user"
}
] | math | {
"ground_truth": "B) 5",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$ .\n\nSL is an acronym for Special Lesson. this yea... | math | {
"ground_truth": "n = 3",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a wheel with a diameter of $10$ feet, calculate how many full revolutions the wheel will need to travel to cover half a mile.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{264}{\\pi}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the complex plane, $z$, $z^2$, and $z^3$, in some order, represent the vertices of an equilateral triangle. Calculate all possible side lengths of the triangle.",
"role": ... | math | {
"ground_truth": "\\frac{\\sqrt{481}}{4}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the real number $t$ , such that the following system of equations has a unique real solution $(x, y, z, v)$ :\n\n\\[ \\left\\{\\begin{array}{cc}x+y+z+v=0 (xy + yz +zv)+t(x... | math | {
"ground_truth": "t \\in \\left( \\frac{3 - \\sqrt{5}}{2}, \\frac{3 + \\sqrt{5}}{2} \\right)",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all natural numbers $ n$ for which $ 2^n\\minus{}n^2$ is divisible by $ 7$ .",
"role": "user"
}
] | math | {
"ground_truth": "n \\equiv 2, 4, 5, 6, 10, 15 \\pmod{21}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are balls that look identical, but their weights all di\u000ber by a little. We have a balance that can compare only two balls at a time. What is the minimum number of time... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to the square of an integer?",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The product of the two $101$-digit numbers $707,070,707,...,070,707$ and $909,090,909,...,090,909$ has tens digit $C$ and units digit $D$. Find the sum of $C$ and $D$.",
"rol... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function \\( f(x) = || x-1|-1| \\), if the equation \\( f(x) = m \\) (where \\( m \\in \\mathbf{R} \\)) has exactly 4 distinct real roots \\( x_{1}, x_{2}, x_{3}, x_{4}... | math | {
"ground_truth": "(-3,0)",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $y$ varies directly as $x$, and if $y=8$ when $x=4$, find the value of $y$ when $x=-8$.",
"role": "user"
}
] | math | {
"ground_truth": "-16",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Point $C$ is the midpoint of $\\overline{AB}$, point $D$ is the midpoint of $\\overline{AC}$, point $E$ is the midpoint of $\\overline{AD}$, point $F$ is the midpoint of $\\overl... | math | {
"ground_truth": "AB = 384",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each integer $n > 1$ , find a set of $n$ integers $\\{a_1, a_2,..., a_n\\}$ such that the set of numbers $\\{a_1+a_j | 1 \\le i \\le j \\le n\\}$ leave distinct remai... | math | {
"ground_truth": "\\{0, 1, 2, \\ldots, n-1\\}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the remainder function defined for all real numbers $x$ and $y$ (where $y \\neq 0$) by $\\text{rem}(x, y) = x - y \\left\\lfloor \\frac{x}{y} \\right\\rfloor$, calculate th... | math | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that x<sub>1</sub> = -1 is a root of the equation x<sup>2</sup> + mx - 5 = 0, find the value of m.",
"role": "user"
}
] | math | {
"ground_truth": "m = -4",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute\n\\[\n\\cot \\frac{\\pi}{18} - 2 \\sin \\frac{\\pi}{9}.\n\\]",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{4 - 2\\sqrt{3}}{\\sqrt{2} (\\sqrt{3} - 1)}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Simplify $((7p+4)-3p\\cdot3)(2)+(5-2\\div4)(4p-6)$ to a straightforward form $ap + b$, where $a$ and $b$ are integers.",
"role": "user"
}
] | math | {
"ground_truth": "14p - 19",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the set $M=\\{h(x) | \\text{the domain of } h(x) \\text{ is } \\mathbb{R}, \\text{ and for any } x, h(-x)=-h(x)\\}$. Let the function $f(x) = \\frac{-2^x + a}{2^{x+1} + b}$... | math | {
"ground_truth": "2k\\pi + \\frac{\\pi}{6} \\leq \\theta \\leq 2k\\pi + \\frac{5\\pi}{6}, k \\in \\mathbb{Z}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle paral... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x) = 3x - 4$ and $g(x) = x + 1$. What is the value of $f(1 + g(2))$?",
"role": "user"
}
] | math | {
"ground_truth": "8",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate: \\(\\sin 37^{\\circ} \\cos ^{2} 34^{\\circ} + 2 \\sin 34^{\\circ} \\cos 37^{\\circ} \\cos 34^{\\circ} - \\sin 37^{\\circ} \\sin ^{2} 34^{\\circ}\\).",
"role": "user... | math | {
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate the sum of all the positive divisors of $143$.",
"role": "user"
}
] | math | {
"ground_truth": "168",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A certain association has 200 members, and it is necessary to draw 40 members as a sample using systematic sampling. All members are randomly numbered from 1 to 200 and evenly di... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $VABCD$ be a right square pyramid with $V$ as the apex and $ABCD$ as a square base. Let $P$ be a point inside square $ABCD$. Denote by $t$ the sum of the distances from $P$ t... | math | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A continuous function $ f(x)$ satisfies that $ \\frac {d}{dx}\\left(\\int_0^x f(x \\plus{} t)\\ dt \\right)\\equal{} 0$ . Find the necessary and sufficient condition such that... | math | {
"ground_truth": "f(2) = \\frac{1}{2}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For values of $x$ less than $1$ but greater than $-4$, find the maximum or minimum value of the expression $\\frac{x^2 - 2x + 2}{2x - 2}$.",
"role": "user"
}
] | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the random variable $X$ has variance $D(X) = 1$, find the value of $D(2X+3)$.",
"role": "user"
}
] | math | {
"ground_truth": "4",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "**p1.** Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresse... | math | {
"ground_truth": "\\frac{\\sqrt{35}}{25}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 12 meters along the fence... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\min(a, c)$ denote the smaller of the numbers a and c. Construct the graph of the function $y = \\min(x + 2, x^2 - 6x + 8)$, and use it to solve the inequality $\\min(x + 2... | math | {
"ground_truth": "x \\in [-2, 2] \\cup [4, +\\infty)",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If \\( C = \\frac{5}{9}(F-32) \\) and \\( C = K - 273 \\), determine the value of \\( F \\) when \\( K = 300 \\).",
"role": "user"
}
] | math | {
"ground_truth": "F = 80.6",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a cone with a fixed volume $V,$ and the total surface area (including the base) needs to be maximized. Find the ratio of height $H$ to the radius $R$ of the base for which ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let equilateral triangle $\\vartriangle ABC$ be inscribed in a circle $\\omega_1$ with radius $4$ . Consider another circle $\\omega_2$ with radius $2$ internally tangen... | math | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a circular cone with vertex \\( V \\), and let \\( ABC \\) be a triangle inscribed in the base of the cone, such that \\( AB \\) is a diameter and \\( AC = BC \\). Let \... | math | {
"ground_truth": "\\sqrt{\\frac{\\pi}{4 - \\pi}}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The points \\(A\\), \\(B\\), and \\(C\\) are the centers of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle \\(ABC\\) are 4, 5, and 6. Wha... | math | {
"ground_truth": "90\\sqrt{6}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The integers 256 and 125 are expressed in base 5 and added. What is the resulting sum, expressed in base 5?",
"role": "user"
}
] | math | {
"ground_truth": "3011_5",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $O$ be the center of the circle $\\omega$ circumscribed around the acute-angled triangle $\\vartriangle ABC$ , and $W$ be the midpoint of the arc $BC$ of the circle ... | math | {
"ground_truth": "\\angle BAC = 60^\\circ",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A high school has 54 students in Class 1 and 42 students in Class 2. Determine the number of students to be chosen from Class 1 and Class 2, respectively, when 16 students are se... | math | {
"ground_truth": "9, 7",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Each of the numbers \\(a\\), \\(b\\), and \\(c\\) is non-zero. It is known that the numbers \\((-2)^{8} a^{3} b^{3} c^{n-1}\\) and \\((-3)^{3} a^{2} b^{5} c^{n+1}\\) have the sam... | math | {
"ground_truth": "\\text{-a is negative, thus a is negative thus a < 0 }",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( r_1 \\) be the remainder when the polynomial \\( x^5 \\) is divided by \\( x - \\frac{1}{2} \\), and let \\( q_1(x) \\) be the respective quotient. Calculate the remainde... | math | {
"ground_truth": "C) \\( 2.9375 \\)",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Print 90,000 five-digit numbers\n$$\n10000, 10001, \\cdots, 99999\n$$\n\non cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806... | math | {
"ground_truth": "89100",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that s... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere... | math | {
"ground_truth": "\\frac{\\sqrt{3}\\pi}{2}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Kira has $3$ blocks with the letter $A$ , $3$ blocks with the letter $B$ , and $3$ blocks with the letter $C$ . She puts these $9$ blocks in a sequence. She wants to h... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any natural number $n$, let $d(n)$ denote the number of positive divisors of $n$ and $s(n)$ the sum of these divisors. For example, $d(2018) = 4$ because 2018 has four diviso... | math | {
"ground_truth": "14, 15, 47",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate the expression $\\sqrt{16-8\\sqrt{3}}+\\sqrt{16+8\\sqrt{3}}$.\nA) $8\\sqrt{2}$\nB) $8\\sqrt{3}$\nC) $12\\sqrt{3}$\nD) $4\\sqrt{6}$\nE) $16$",
"role": "user"
}
] | math | {
"ground_truth": "8\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\\equiv a \\pmod n$ , $x^3 \\equiv a \\pmod n... | math | {
"ground_truth": "1154",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\\cdot2$ . If each term of the sequence can be written in the form $a\\cdot r_1^n+b\\... | math | {
"ground_truth": "\\frac{\\sqrt{17}}{2}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$\\boxed{N1}$ Let $n$ be a positive integer, $g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ o... | math | {
"ground_truth": "n = 2^a \\cdot 3^b \\cdot t^2",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\\ 2)$ with the slope $ a$ and the parabola $ y\\equal... | math | {
"ground_truth": "a = 2",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds:\r $ x_1x_2(1\\minus{}x_3)\\equal{}x_4x_5 \r\nx_2x_3(1\... | math | {
"ground_truth": "(ab, cd, 0, ac, bd, 0)",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cone has a base radius of 4 cm and a height of 10 cm. What is the length of the longest segment that would fit inside the cone?",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{116}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A parabolic arch has a height of $20$ inches and a span of $30$ inches. Determine the height of the arch at $3$ inches from the center.",
"role": "user"
}
] | math | {
"ground_truth": "19.2",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the values of $n$ for which the equation $4x + 6y + 2z = n$ has exactly 10 solutions in positive integers $x$, $y$, and $z$.",
"role": "user"
}
] | math | {
"ground_truth": "33",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a small seminar at a college, it is noted that each student averages two days absent out of twenty. What is the probability that out of any three students chosen at random, ex... | math | {
"ground_truth": "2.7\\%",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Coordinate System and Parametric Equations\n\nGiven that the pole of the polar coordinate system coincides with the origin of the Cartesian coordinate system, and the polar axis ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the solution set of the inequality $\\frac{x+2}{x+1} < 0$ is $\\{x|a < x < b\\}$, and the point $A(a,b)$ lies on the line $mx+ny+1=0$, where $mn > 0$, then the minimum valu... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the equation $(\\frac{1}{4})^{x}+(\\frac{1}{2})^{x}+a=0$ has a positive solution, find the range of values for the real number $a$.",
"role": "user"
}
] | math | {
"ground_truth": "(-2,0)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
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