data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find the value of \\(\\tan \\left(\\tan^{-1} \\frac{1}{2} + \\tan^{-1} \\frac{1}{2 \\times 2^2} + \\tan^{-1} \\frac{1}{2 \\times 3^2} + \\cdots + \\tan^{-1} \\frac{1}{2 \\times 2009^2}\\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2009}{2010}"
} | {
"index": 11100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of \\(m + n\\) where \\(m\\) and \\(n\\) are integers defined as follows: The positive difference between the roots of the quadratic equation \\(5x^2 - 9x - 14 = 0\\) can be expressed as \\(\\frac{\\sqrt{m}}{n}\\), with \\(m\\) not divisible by the square of any ... | math | {
"style": "rule",
"ground_truth": "366"
} | {
"index": 11101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of \\[\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of cos $$\\frac {π}{11}$$cos $$\\frac {2π}{11}$$cos $$\\frac {3π}{11}$$cos $$\\frac {4π}{11}$$cos $$\\frac {5π}{11}$$\\=\\_\\_\\_\\_\\_\\_．"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{32}"
} | {
"index": 11103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the algebraic cofactor of element $a$ in the determinant \\\\( \\begin{vmatrix} 3 & a & 5 \\\\\\\\ 0 & -4 & 1 \\\\\\\\ -2 & 1 & 3\\\\end{vmatrix} \\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 11104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the arithmetic series $1-3+5-7+9-11+\\cdots +2021-2023+2025$."
}
] | math | {
"style": "rule",
"ground_truth": "1013"
} | {
"index": 11105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the definite integral $\\int_{0}^{1} ( \\sqrt{1-(x-1)^{2}}-{x}^{2})dx$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4} - \\frac{1}{3}"
} | {
"index": 11106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the expression $$ f\\left( \\frac{1}{2000} \\right)+f\\left( \\frac{2}{2000} \\right)+...+ f\\left( \\frac{1999}{2000} \\right)+f\\left( \\frac{2000}{2000} \\right)+f\\left( \\frac{2000}{1999} \\right)+...+f\\left( \\frac{2000}{1} \\right) $$ assuming $f(x)... | math | {
"style": "rule",
"ground_truth": "1999.5"
} | {
"index": 11107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the expression $\\frac{3\\sqrt{12}}{\\sqrt{3} + \\sqrt{4} + \\sqrt{6}}$.\nA) $\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}$\nB) $4 - \\sqrt{3} - 2\\sqrt{2}$\nC) $\\sqrt{3} - 2\\sqrt{2} + \\sqrt{6}$\nD) $\\frac{1}{2}(\\sqrt{3} + \\sqrt{6} - 2\\sqrt{2})$\nE) $\\frac{1}{3}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}"
} | {
"index": 11108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the expression \\(\\cos ^{4} \\frac{7 \\pi}{24}+\\sin ^{4} \\frac{11 \\pi}{24}+\\sin ^{4} \\frac{17 \\pi}{24}+\\cos ^{4} \\frac{13 \\pi}{24}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the expression \\(\\sum_{i=0}^{1009}(2 k+1)-\\sum_{i=1}^{1009} 2 k\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 11110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the first term in the geometric sequence $a,b,c,32,64$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the following expression:\n\n$$\\left| \\, |{ -|{-1 + 1}| - 1 }| + 1\\right|.$$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the following expressions without using a calculator:\n\\\\((1)\\\\lg 5^{2}+ \\frac {2}{3}\\lg 8+\\lg 5\\lg 20+(\\lg 2)^{2}\\\\)\n\\\\((2)\\\\) Let \\(2^{a}=5^{b}=m\\), and \\(\\frac {1}{a}+ \\frac {1}{b}=2\\), find \\(m\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {10}"
} | {
"index": 11113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the function \\( f(x) \\) at the point \\( x_{0} = 4500 \\), if \\( f(0) = 1 \\) and for any \\( x \\) the equality \\( f(x + 3) = f(x) + 2x + 3 \\) holds."
}
] | math | {
"style": "rule",
"ground_truth": "6750001"
} | {
"index": 11114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the function \\( f(x) \\) at the point \\( x_{0}=2000 \\), given \\( f(0)=1 \\) and for any \\( x \\) the equality \\( f(x+4)=f(x)+3x+4 \\) holds."
}
] | math | {
"style": "rule",
"ground_truth": "1499001"
} | {
"index": 11115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of the product \\( \\cos \\frac{\\pi}{15} \\cos \\frac{2 \\pi}{15} \\cos \\frac{3 \\pi}{15} \\cdots \\cos \\frac{7 \\pi}{15} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"index": 11116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value of: \\(\\frac{\\left(\\sqrt{3} \\cdot \\tan 12^{\\circ} - 3\\right) \\cdot \\csc 12^{\\circ}}{4 \\cos ^{2} 12^{\\circ} - 2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-4 \\sqrt{3}"
} | {
"index": 11117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \\frac{9}{2} = 0$ is true for all values of $z$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\\equal{}\\minus{}x^4\\plus{}2x^2$ and the line $ y\\equal{}k$ are all equal."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 11120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the values of $a$ and $b$ such that $a + b^2$ can be calculated, where $x = a \\pm b i$ are the solutions to the equation $5x^2 + 7 = 2x - 10$. Express your answer as a fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{25}"
} | {
"index": 11121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the values of $a$, $b$, and $c$ such that the equation $\\sin^2 x + \\sin^2 3x + \\sin^2 4x + \\sin^2 5x = 3$ can be reduced to an equivalent form involving $\\cos ax \\cos bx \\cos cx = 0$ for some positive integers $a$, $b$, and $c$, and then find $a + b + c$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the values of the real number \\( a \\) such that all the roots of the polynomial in the variable \\( x \\),\n\n\\[ x^{3}-2x^{2}-25x+a \\]\n\nare integers."
}
] | math | {
"style": "rule",
"ground_truth": "-50"
} | {
"index": 11123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the vector $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}} \\mathbf{v} = \\begin{pmatrix} \\frac{38}{5} \\\\ \\frac{19}{5} \\end{pmatrix}\\]and\n\\[\\operatorname{proj}_{\\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}} \\mathbf{v} = \\b... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ 5 \\end{pmatrix}"
} | {
"index": 11124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "250\\pi"
} | {
"index": 11125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of a regular tetrahedron with the side of its base equal to $\\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 11126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of a regular triangular pyramid where the planar angle at the apex is $90^{\\circ}$ and the side of the base is 3 cm."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{2}}{8}"
} | {
"index": 11127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region given by the inequality\n\\[\n|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 6.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region given by the inequality\n\\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 4.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"index": 11129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[ |x + 2y + z| + |x + 2y - z| \\le 12 \\]\nand $x, y \\geq 0$, $z \\geq -2$."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[ |x + y + 2z| + |x + y - 2z| \\le 12 \\]\nand $x, y, z \\ge 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[|x + 2y + z| + |x + 2y - z| \\le 12\\]\nand \\(x, y, z \\ge 0.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[|x + y + 2z| + |x + y - 2z| \\le 12\\]\nand $x, y, z \\ge 0$."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[|x + y + z| + |x + y - z| \\le 8\\]and $x,$ $y,$ $z \\ge 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 11134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[|x - y + z| + |x - y - z| \\le 10\\]and $x, y, z \\ge 0$."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the region in space defined by \n\\[ |z + x + y| + |z + x - y| \\leq 10 \\] \nand \\(x, y, z \\geq 0\\)."
}
] | math | {
"style": "rule",
"ground_truth": "62.5"
} | {
"index": 11136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the set of points $(x, y, z)$ satisfying $$\\begin{array}{r} x, y, z \\geq 0 \\\\ x+y \\leq 1 \\\\ y+z \\leq 1 \\\\ z+x \\leq 1 \\end{array}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 11137,
"solution": "Without loss of generality, assume that $x \\geq y$ - half the volume of the solid is on this side of the plane $x=y$. For each value of $c$ from 0 to $\\frac{1}{2}$, the region of the intersection of this half of the solid with the plane $y=c$ is a trapezoid. The trapezoid has height... |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the solid $T$ consisting of all points $(x, y, z)$ such that $|x| + |y| \\leq 2$, $|x| + |z| \\leq 2$, and $|y| + |z| \\leq 2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"index": 11138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64\\pi}{105}"
} | {
"index": 11139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the volume of the three-dimensional solid given by the inequality $\\sqrt{x^{2}+y^{2}}+$ $|z| \\leq 1$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi / 3"
} | {
"index": 11140,
"solution": "$2 \\pi / 3$. The solid consists of two cones, one whose base is the circle $x^{2}+y^{2}=1$ in the $x y$-plane and whose vertex is $(0,0,1)$, and the other with the same base but vertex $(0,0,-1)$. Each cone has a base area of $\\pi$ and a height of 1, for a volume of $\\pi / 3$, so t... |
DeepScaleR | [
{
"role": "user",
"content": "Find the x-coordinate of point Q, given that point P has coordinates $(\\frac{3}{5}, \\frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\\angle POQ = \\frac{3\\pi}{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\sqrt{2}}{10}"
} | {
"index": 11141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find two lines of symmetry of the graph of the function $y=x+\\frac{1}{x}$. Express your answer as two equations of the form $y=a x+b$."
}
] | math | {
"style": "rule",
"ground_truth": "$y=(1+\\sqrt{2}) x$ and $y=(1-\\sqrt{2}) x$"
} | {
"index": 11142,
"solution": "The graph of the function $y=x+\\frac{1}{x}$ is a hyperbola. We can see this more clearly by writing it out in the standard form $x^{2}-x y+1=0$ or $\\left(\\frac{y}{2}\\right)^{2}-\\left(x-\\frac{1}{2} y\\right)^{2}=1$. The hyperbola has asymptotes given by $x=0$ and $y=x$, so the li... |
DeepScaleR | [
{
"role": "user",
"content": "Find whole numbers $\\heartsuit$ and $\\clubsuit$ such that $\\heartsuit \\cdot \\clubsuit = 48$ and $\\heartsuit$ is even, then determine the largest possible value of $\\heartsuit + \\clubsuit$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find, as a function of $\\, n, \\,$ the sum of the digits of \\[9 \\times 99 \\times 9999 \\times \\cdots \\times \\left( 10^{2^n} - 1 \\right),\\] where each factor has twice as many digits as the previous one."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ 9 \\cdot 2^n \\]"
} | {
"index": 11144,
"solution": "The answer is $9 \\cdot 2^n$ .\nLet us denote the quantity $\\prod_{k=0}^n \\bigl( 10^{2^k}-1 \\bigr)$ as $P_n$ . We wish to find the sum of the digits of $P_n$ .\nWe first note that \\[P_{n-1} < \\prod_{k=0}^{n-1} 10^{2^k} = 10^{2^n-1},\\] so $P_{n-1}$ is a number of at most $2^n$ d... |
DeepScaleR | [
{
"role": "user",
"content": "Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ n = 1 \\]"
} | {
"index": 11145,
"solution": "The answer is $n=1$ , which is easily verified to be a valid integer $n$ .\nNotice that \\[2^n+12^n+2011^n\\equiv 2^n+7^n \\pmod{12}.\\] Then for $n\\geq 2$ , we have $2^n+7^n\\equiv 3,5 \\pmod{12}$ depending on the parity of $n$ . But perfect squares can only be $0,1,4,9\\pmod{12}$ ,... |
DeepScaleR | [
{
"role": "user",
"content": "Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\\{1, 2,...,N\\}$ , one can still find $2016$ distinct numbers among the remaining elements with sum $N$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\boxed{6097392}"
} | {
"index": 11146,
"solution": "Since any $2016$ elements are removed, suppose we remove the integers from $1$ to $2016$ . Then the smallest possible sum of $2016$ of the remaining elements is \\[2017+2018+\\cdots + 4032 = 1008 \\cdot 6049 = 6097392\\] so clearly $N\\ge 6097392$ . We will show that $N=6097392$ works... |
DeepScaleR | [
{
"role": "user",
"content": "Find, with proof, the maximum positive integer \\(k\\) for which it is possible to color \\(6k\\) cells of a \\(6 \\times 6\\) grid such that, for any choice of three distinct rows \\(R_{1}, R_{2}, R_{3}\\) and three distinct columns \\(C_{1}, C_{2}, C_{3}\\), there exists an u... | math | {
"style": "rule",
"ground_truth": "\\[\nk = 4\n\\]"
} | {
"index": 11147,
"solution": "The answer is \\(k=4\\). This can be obtained with the following construction: [grid image]. It now suffices to show that \\(k=5\\) and \\(k=6\\) are not attainable. The case \\(k=6\\) is clear. Assume for sake of contradiction that the \\(k=5\\) is attainable. Let \\(r_{1}, r_{2}, r_... |
DeepScaleR | [
{
"role": "user",
"content": "Find, with proof, the number of positive integers whose base- $n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\\pm 1$ from some digit further to the left. (Your answer should be an explicit function ... | math | {
"style": "rule",
"ground_truth": "\\[ 2^{n+1} - 2(n+1) \\]"
} | {
"index": 11148,
"solution": "Let a $k$ -good sequence be a sequence of distinct integers $\\{ a_i \\}_{i=1}^k$ such that for all integers $2\\le i \\le k$ , $a_i$ differs from some preceding term by $\\pm 1$ .\nLemma. Let $a$ be an integer. Then there are $2^{k-1}$ $k$ -good sequences starting on $a$ , and furt... |
DeepScaleR | [
{
"role": "user",
"content": "Find, with proof, the smallest real number $C$ with the following property:\nFor every infinite sequence $\\{x_i\\}$ of positive real numbers such that $x_1 + x_2 +\\cdots + x_n \\leq x_{n+1}$ for $n = 1, 2, 3, \\cdots$, we have\n\\[\\sqrt{x_1}+\\sqrt{x_2}+\\cdots+\\sqrt{x_n} ... | math | {
"style": "rule",
"ground_truth": "$C=1+\\sqrt{2}$"
} | {
"index": 11149,
"solution": "\nTo find the smallest real number \\( C \\) such that the inequality \n\n\\[\n\\sqrt{x_1} + \\sqrt{x_2} + \\cdots + \\sqrt{x_n} \\leq C \\sqrt{x_1 + x_2 + \\cdots + x_n}\n\\]\n\nholds for every infinite sequence \\(\\{x_i\\}\\) of positive real numbers satisfying \n\n\\[\nx_1 + x_2 +... |
DeepScaleR | [
{
"role": "user",
"content": "Find: $\\frac{12}{30}-\\frac{1}{7}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{35}"
} | {
"index": 11150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find: $\\frac{2}{7}+\\frac{8}{10}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{38}{35}"
} | {
"index": 11151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:\n\n[list]\n\n [*]If at some stage the cards are in $m$ piles... | math | {
"style": "rule",
"ground_truth": "2^{n-2}"
} | {
"index": 11152,
"solution": "\nFiona starts with \\( n \\) cards labeled from 1 to \\( n \\) lined up in increasing order. Her task is to perform a series of pile rearrangements until she consolidates them into a single pile. Given the specific operation she can perform, we explore how many distinct final single ... |
DeepScaleR | [
{
"role": "user",
"content": "Fiona is people-watching again. She spies a group of ten high schoolers and starts playing a game by herself, in which she looks at a pair of people from the group of ten and tries to guess whether they like or dislike each other. How many pairs of friends can she observe befor... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 11153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "First $a$ is chosen at random from the set $\\{1,2,3,\\cdots,99,100\\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 11154,
"solution": "1. **Identify the Cyclic Nature of Units Digits for Powers of 3 and 7**:\n - The units digit of powers of $3$ cycles through $1, 3, 9, 7$. This can be verified by calculating $3^1, 3^2, 3^3, 3^4$, etc., and observing the units digits.\n - Similarly, the units digit of powers of $7... |
DeepScaleR | [
{
"role": "user",
"content": "First, a number \\( a \\) is randomly selected from the set \\(\\{1,2,3, \\cdots, 99,100\\}\\), then a number \\( b \\) is randomly selected from the same set. Calculate the probability that the last digit of \\(3^{a} + 7^{b}\\) is 8."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 11155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "First, factorize 42 and 30 into prime factors, then answer the following questions: \n(1) 42= , 30= . \n(2) The common prime factors of 42 and 30 are . \n(3) The unique prime factors of 42 and 30 are . \n(4) The greatest common divisor (GCD) of 42 and 30 is ... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 11156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "First, find the derivative of the following functions and calculate the derivative at \\\\(x=\\pi\\\\).\n\n\\\\((1) f(x)=(1+\\sin x)(1-4x)\\\\) \\\\((2) f(x)=\\ln (x+1)-\\dfrac{x}{x+1}\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{(\\pi+1)^{2}}"
} | {
"index": 11157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "First, select $n$ numbers from the set $ \\{1, 2, \\cdots, 2020\\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 11158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, ge... | math | {
"style": "rule",
"ground_truth": "4035"
} | {
"index": 11160,
"solution": "The only possible way for Ritmo to get 2017 cubic centimeters is to have his measurements rounded to $1,1,2017$ centimeters respectively. Therefore the largest value of $V$ is achieved when the dimensions are $(1.5-\\epsilon)(1.5-\\epsilon)(2017.5-\\epsilon)=4539.375-\\epsilon^{\\prim... |
DeepScaleR | [
{
"role": "user",
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, ge... | math | {
"style": "rule",
"ground_truth": "7174"
} | {
"index": 11161,
"solution": "It is not difficult to see that the maximum possible value of $V$ can be achieved when the dimensions are $(0.5+\\epsilon) \\times(0.5+\\epsilon) \\times\\left(8070-\\epsilon^{\\prime}\\right)=2017.5-\\epsilon^{\\prime \\prime}$ for some very small reals $\\epsilon, \\epsilon^{\\prime... |
DeepScaleR | [
{
"role": "user",
"content": "Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original po... | math | {
"style": "rule",
"ground_truth": "2.2"
} | {
"index": 11162,
"solution": "To solve this problem, we need to calculate the expected number of balls that return to their original positions after two successive transpositions by Chris and Silva. We will use the linearity of expectation to find the expected number of balls in their original positions.\n\n1. **I... |
DeepScaleR | [
{
"role": "user",
"content": "Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 11163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{126}"
} | {
"index": 11164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?\n\n*Author: Ray Li*"
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 11165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five brothers divided their father's inheritance equally. The inheritance included three houses. Since it was not possible to split the houses, the three older brothers took the houses, and the younger brothers were given money: each of the three older brothers paid $2,000. How... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"index": 11166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older b... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 11167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five candidates are to be selected to perform four different jobs, where one candidate can only work as a driver and the other four can do all the jobs. Determine the number of different selection schemes."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 11168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five cards are lying on a table as shown.\n\\[\\begin{matrix} & \\qquad & \\boxed{\\tt{P}} & \\qquad & \\boxed{\\tt{Q}} \\\\ \\\\ \\boxed{\\tt{3}} & \\qquad & \\boxed{\\tt{4}} & \\qquad & \\boxed{\\tt{6}} \\end{matrix}\\]\nEach card has a letter on one side and a whole number ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11169,
"solution": "To solve this problem, we need to test Jane's statement: \"If a vowel is on one side of any card, then an even number is on the other side.\" We need to find a counterexample to disprove this statement.\n\n1. **Understanding the statement**: The statement can be interpreted using logi... |
DeepScaleR | [
{
"role": "user",
"content": "Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \\(a, b, c, d,\\) and \\(e\\) respectively. We know that \\(a + 2 = b - 2 = 2c = \\frac{d}{2} = e^2\\).\n\nGina picks up the card whic... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 11170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \\(a, b, c, d\\), and \\(e\\) respectively. We know that:\n\n1. \\(c = b \\cdot e\\)\n2. \\(a + b = d\\)\n3. \\(e - d = a\\)\n\nFranki... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 11171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from ... | math | {
"style": "rule",
"ground_truth": "50308"
} | {
"index": 11172,
"solution": "For a given card, let $p(n)$ denote the probability that it is in its original position after $n$ swaps. Then $p(n+1)=p(n) \\cdot \\frac{3}{5}+(1-p(n)) \\cdot \\frac{1}{10}$, by casework on whether the card is in the correct position or not after $n$ swaps. In particular, $p(0)=1, p(1... |
DeepScaleR | [
{
"role": "user",
"content": "Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11173,
"solution": "The only things we can do is leave cards where they are or switch them with adjacent cards. There is 1 way to leave them all where they are, 4 ways to switch just one adjacent pair, and 3 ways to switch two different adjacent pairs, for 8 possibilities total."
} |
DeepScaleR | [
{
"role": "user",
"content": "Five cards with different numbers are given: $-5$, $-4$, $0$, $+4$, $+6$. Two cards are drawn from them. The smallest quotient obtained by dividing the numbers on these two cards is ____."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3}{2}"
} | {
"index": 11174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five consecutive two-digit positive integers, each less than 30, are not prime. What is the largest of these five integers?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 11175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five consecutive two-digit positive integers, each less than 40, are not prime. What is the largest of these five integers?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 11176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five consecutive two-digit positive integers, each less than 50, are not prime. What is the largest of these five integers?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 11177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five couples were at a party. If each person shook hands exactly once with everyone else except his/her spouse, how many handshakes were exchanged? (Note: One obviously doesn't shake hands with oneself.)"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 11178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five dice with faces numbered 1 through 6 are arranged in a configuration where 14 of the visible faces are showing. The visible numbers are 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 4, 5, 6. What is the total number of dots NOT visible in this view?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 11179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five dice with faces numbered 1 through 6 are stacked in a similar manner to the original problem. Ten of the thirty faces are visible, leaving twenty faces hidden. The visible numbers are 1, 2, 2, 3, 3, 3, 4, 4, 5, and 6. What is the total number of dots NOT visible in this vi... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 11181,
"solution": "1. **Identify the possible distributions of awards:** Since each student must receive at least one award and there are five awards in total, the possible distributions of awards among three students are:\n - One student receives 3 awards, and the other two students receive 1 award e... |
DeepScaleR | [
{
"role": "user",
"content": "Five different products, A, B, C, D, and E, are to be arranged in a row on a shelf. Products A and B must be placed together, while products C and D cannot be placed together. How many different arrangements are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 11182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). Wha... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 11183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five doctors A, B, C, D, and E are assigned to four different service positions located in the Sichuan disaster zone, labeled A, B, C, and D. Each position must be filled by at least one doctor. Calculate the total number of ways doctors A and B can serve independently in diffe... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 11184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a $50 \\%$ chance of winning, and the re... | math | {
"style": "rule",
"ground_truth": "\\frac{49}{64}"
} | {
"index": 11185,
"solution": "We make the following claim: if there is a 5-cycle (a directed cycle involving 5 players) in the tournament, then there is a 4-cycle. Proof: Assume that $A$ beats $B, B$ beats $C, C$ beats $D, D$ beats $E$ and $E$ beats $A$. If $A$ beats $C$ then $A, C, D, E$ forms a 4-cycle, and simi... |
DeepScaleR | [
{
"role": "user",
"content": "Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{113}{648}"
} | {
"index": 11186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five fair ten-sided dice are rolled. Calculate the probability that at least four of the five dice show the same value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{5000}"
} | {
"index": 11187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five friends all brought some cakes with them when they met. Each of them gave a cake to each of the others. They then ate all the cakes they had just been given. As a result, the total number of cakes they had between them decreased by half. How many cakes did the five friends... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 11188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five friends did gardening for their local community and earned $15, $22, $28, $35, and $50 respectively. They decide to share their total earnings equally. How much money must the friend who earned $50 contribute to the pool?\nA) $10 \nB) $15 \nC) $20 \nD) $25 \nE) $35... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five friends earned $18, $23, $28, $35, and $45. If they split their earnings equally among themselves, how much will the friend who earned $45 need to give to the others?"
}
] | math | {
"style": "rule",
"ground_truth": "15.2"
} | {
"index": 11190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions \"left\" and \"right\" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11191,
"solution": "To solve this problem, we need to analyze the movements of each friend and determine the original seating arrangement based on the given movements.\n\n1. **Initial Setup**: There are 5 seats, and each friend occupies one seat. Ada leaves, creating one empty seat.\n\n2. **Movements**:\... |
DeepScaleR | [
{
"role": "user",
"content": "Five friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \\mathrm{~kg}$, but Ivan brought only $43 \\mathrm{~kg}$.\n\nWhat is the average amount of iron brought without Ivan?\n\n(Note: By how many kilograms does Ivan's contribution ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 11192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the to... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep coun... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 11194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half?"
}
] | math | {
"style": "rule",
"ground_truth": "8/63"
} | {
"index": 11195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five identical balls roll on a smooth horizontal surface towards each other. The velocities of the first and second are $v_{1}=v_{2}=0.5$ m/s, and the velocities of the others are $v_{3}=v_{4}=v_{5}=0.1$ m/s. The initial distances between the balls are the same, $l=2$ m. All co... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown. The area of $PQRS$ is $4000$. What is the length, $x$, rounded off to the nearest integer? [asy]\nreal x = 1; real w = 2/3;\n\n// Draw outer square and labels\npair s = (0, 0); pair r = (2 * x... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 11197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five identical right-angled triangles can be arranged so that their larger acute angles touch to form a star. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch. How many triangles are needed to form the se... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five identical white pieces and ten identical black pieces are arranged in a row. It is required that the right neighbor of each white piece must be a black piece. The number of different arrangements is ."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 11199,
"solution": ""
} |
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