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 maximum and minimum values of the function $f(x)=x^{3}-12x$ on the interval $[-3,5]$."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 10000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum and minimum values of the function $f(x)=x^{3}-2x^{2}+5$ on the interval $[-2,2]$."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"index": 10001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum constant $k$ such that $\\frac{k a b c}{a+b+c} \\leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 10002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum constant \\( k \\) such that for \\( x, y, z \\in \\mathbb{R}_+ \\), the following inequality holds:\n$$\n\\sum \\frac{x}{\\sqrt{y+z}} \\geqslant k \\sqrt{\\sum x},\n$$\n\nwhere \" \\( \\sum \\) \" denotes a cyclic sum."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}}"
} | {
"index": 10003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors."
}
] | math | {
"style": "rule",
"ground_truth": "11250"
} | {
"index": 10004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of lengt... | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"index": 10005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of Permutation of set { $1,2,3,...,2014$ } such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$"
}
] | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"index": 10006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of different integers that can be selected from the set $ \\{1,2,...,2013\\}$ so that no two exist that their difference equals to $17$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 10007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of elements in a set $S$ that satisfies the following conditions:\n(1) Every element in $S$ is a positive integer not exceeding 100.\n(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest com... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:\na) No $x_i - x_j , 1 \\le i < j \\le m$ is divisible by $11$ , and\nb) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \\cdot \\cdot \\cdot + x_1x_2... x_{m-1}$ is divisible by $11$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of points $X_{i}$ such that for each $i$, $\\triangle A B X_{i} \\cong \\triangle C D X_{i}$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10010,
"solution": "One of the sides $A X_{i}$ or $B X_{i}$ is equal to $C D$, thus $X_{i}$ is on one of the circles of radius $C D$ and center $A$ or $B$. In the same way $X_{i}$ is on one of circles of radius $A B$ with center $C$ or $D$. The intersection of these four circles has no more than 8 points... |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \\times 2$ rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum of\n\\[\n\\sqrt{x + 31} + \\sqrt{17 - x} + \\sqrt{x}\n\\]\nfor $0 \\le x \\le 17$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum of\n\\[\\sqrt{x + 27} + \\sqrt{13 - x} + \\sqrt{x}\\]for $0 \\le x \\le 13.$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \\geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 10015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum of the expression\n$$\n|| \\ldots|| x_{1}-x_{2}\\left|-x_{3}\\right|-\\ldots\\left|-x_{2023}\\right|,\n$$\nwhere \\( x_{1}, x_{2}, \\ldots, x_{2023} \\) are distinct natural numbers between 1 and 2023."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 10016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\\{1,2, \\cdots, 1000\\}$, there exist two subsets that have at least $r$ common elements."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 10017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum positive integer \\( n \\) such that\n\\[ \nn^{2} \\leq 160 \\times 170 \\times 180 \\times 190\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "30499"
} | {
"index": 10018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum possible area of a section passing through the diagonal $XY$ of an arbitrary parallelepiped with edges $a \\leq b \\leq c$. The section is a parallelogram $Z X T Y$ with vertices lying on opposite edges of the parallelepiped. The area of the parallelogram equal... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum possible number of diagonals of equal length in a convex hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 10020,
"solution": "First, we will prove that 7 is possible. Consider the following hexagon \\(A B C D E F\\) whose vertices are located at \\(A(0,0), B\\left(\\frac{1}{2}, 1-\\frac{\\sqrt{3}}{2}\\right), C\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right), D(0,1), E\\left(-\\frac{1}{2}, \\frac{\\sqrt{3}}... |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\nf(n) = \\left\\lfloor \\frac{(n-1)^2}{2} \\right\\rfloor\n\\]"
} | {
"index": 10021,
"solution": "Consider the first few cases for $n$ with the entire $n$ numbers forming an arithmetic sequence \\[(1, 2, 3, \\ldots, n)\\] If $n = 3$ , there will be one ascending triplet (123). Let's only consider the ascending order for now.\nIf $n = 4$ , the first 3 numbers give 1 triplet, the ad... |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum possible value of $H \\cdot M \\cdot M \\cdot T$ over all ordered triples $(H, M, T)$ of integers such that $H \\cdot M \\cdot M \\cdot T=H+M+M+T$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10022,
"solution": "If any of $H, M, T$ are zero, the product is 0. We can do better (examples below), so we may now restrict attention to the case when $H, M, T \\neq 0$. When $M \\in\\{-2,-1,1,2\\}$, a little casework gives all the possible $(H, M, T)=(2,1,4),(4,1,2),(-1,-2,1),(1,-2,-1)$. If $M=-2$, i.... |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum possible volume of a cylinder inscribed in a cone with a height of 27 and a base radius of 9."
}
] | math | {
"style": "rule",
"ground_truth": "324\\pi"
} | {
"index": 10023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum real number $\\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \\geqslant \\lambda(x-a)^{3}$ for all $x \\geqslant 0$. Additionally, determine when equality holds in the gi... | math | {
"style": "rule",
"ground_truth": "-1/27"
} | {
"index": 10024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum real number \\( M \\) such that for all real numbers \\( x \\) and \\( y \\) satisfying \\( x + y \\geqslant 0 \\), the following inequality holds:\n$$\n\\left(x^{2}+y^{2}\\right)^{3} \\geqslant M\\left(x^{3}+y^{3}\\right)(xy - x - y)."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 10025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum real number \\( k \\) such that for any positive numbers \\( a \\) and \\( b \\), the following inequality holds:\n$$\n(a+b)(ab+1)(b+1) \\geqslant k \\, ab^2.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "27/4"
} | {
"index": 10026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum real number \\( k \\) such that for any simple graph \\( G \\) with \\( n \\) vertices (\\( n \\geq 3 \\)), the inequality \\( x^3 \\geq k y^2 \\) holds, where \\( x \\) is the number of edges of \\( G \\) and \\( y \\) is the number of triangles in \\( G \\)."... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 10027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\n\\frac{3x + 4y + 6}{\\sqrt{x^2 + 4y^2 + 4}}\n\\]\nover all real numbers \\(x\\) and \\(y\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{61}"
} | {
"index": 10028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\n\\frac{\\sin^4 x + \\cos^4 x + 2}{\\sin^6 x + \\cos^6 x + 2}\n\\]\nover all real values \\(x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{9}"
} | {
"index": 10029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\begin{vmatrix} 1 & 1 & 1 \\\\ 1 & 1 + \\sin \\theta & 1 \\\\ 1 + \\cos \\theta & 1 & 1 \\end{vmatrix},\\]as $\\theta$ ranges over all angles."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 10030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos \\theta_1 \\sin \\theta_2 + \\cos \\theta_2 \\sin \\theta_3 + \\cos \\theta_3 \\sin \\theta_4 + \\cos \\theta_4 \\sin \\theta_5 + \\cos \\theta_5 \\sin \\theta_1,\\]over all real numbers $\\theta_1,$ $\\theta_2,$ $\\theta_3,$ $\\theta_4,$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 10031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\frac{2x + 3y + 4}{\\sqrt{x^2 + 4y^2 + 2}}\\]\nover all real numbers \\( x \\) and \\( y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"index": 10032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\frac{2x + 3y + 4}{\\sqrt{x^2 + y^2 + 4}}\\]\nover all real numbers $x$ and $y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"index": 10033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\frac{x + 2y + 3}{\\sqrt{x^2 + y^2 + 1}}\\]over all real numbers $x$ and $y.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"index": 10034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\sin \\frac{\\theta}{2} \\cdot (1 + \\cos \\theta)\\]for $0 < \\theta < \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{9}"
} | {
"index": 10035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[f(x) = 3x - x^3\\]for $0 \\le x \\le \\sqrt{3}.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of\n\\[y = \\tan \\left( x + \\frac{2 \\pi}{3} \\right) - \\tan \\left( x + \\frac{\\pi}{6} \\right) + \\cos \\left( x + \\frac{\\pi}{6} \\right)\\]for $-\\frac{5 \\pi}{12} \\le x \\le -\\frac{\\pi}{3}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\sqrt{3}}{6}"
} | {
"index": 10037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $$ \\int^1_0|f'(x)|^2|f(x)|\\frac1{\\sqrt x}dx $$ over all continuously differentiable functions $f:[0,1]\\to\\mathbb R$ with $f(0)=0$ and $$ \\int^1_0|f'(x)|^2dx\\le1. $$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 10038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $\\lambda ,$ such that for $\\forall x,y\\in\\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\\lambda y^2\\leqslant 1.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 10039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 10040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 10041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $\\cos x + 2 \\sin x,$ over all angles $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 10042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of $m$ for a sequence $P_{0}, P_{1}, \\cdots, P_{m+1}$ of points on a grid satisfying certain conditions."
}
] | math | {
"style": "rule",
"ground_truth": "n(n-1)"
} | {
"index": 10043,
"solution": "We will show that the desired maximum value for $m$ is $n(n-1)$. First, let us show that $m \\leq n(n-1)$ always holds for any sequence $P_{0}, P_{1}, \\cdots, P_{m+1}$ satisfying the conditions of the problem. Call a point a turning point if it coincides with $P_{i}$ for some $i$ wit... |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of \\( x + y \\), given that \\( x^2 + y^2 - 3y - 1 = 0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{26}+3}{2}"
} | {
"index": 10044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the area of triangle $\\triangle ABC$ that satisfies $AB=4$ and $AC=2BC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"index": 10045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression\n\n$$\n\\frac{a}{x} + \\frac{a+b}{x+y} + \\frac{a+b+c}{x+y+z}\n$$\n\nwhere \\( a, b, c \\in [2,3] \\), and the triplet of numbers \\( x, y, z \\) is some permutation of the triplet \\( a, b, c \\)."
}
] | math | {
"style": "rule",
"ground_truth": "15/4"
} | {
"index": 10046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression\n\n$$\n\\left(x_{1}-x_{2}\\right)^{2}+\\left(x_{2}-x_{3}\\right)^{2}+\\ldots+\\left(x_{2010}-x_{2011}\\right)^{2}+\\left(x_{2011}-x_{1}\\right)^{2}\n$$\n\nwhere \\(x_{1}, \\ldots, x_{2011} \\in [0, 1]\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 10047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( (\\sin 2x + \\sin 3y + \\sin 4z)(\\cos 2x + \\cos 3y + \\cos 4z) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 10048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( (\\sin 3x + \\sin 2y + \\sin z)(\\cos 3x + \\cos 2y + \\cos z) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 10049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( (\\sin x + \\sin 2y + \\sin 3z)(\\cos x + \\cos 2y + \\cos 3z) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 10050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( x + y \\) if \\( (2 \\sin x - 1)(2 \\cos y - \\sqrt{3}) = 0 \\), \\( x \\in [0, \\frac{3\\pi}{2}] \\), \\( y \\in [\\pi, 2\\pi] \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\pi}{3}"
} | {
"index": 10051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( x^{2} + y^{2} \\) if \\( |x-y| \\leq 2 \\) and \\( |3x + y| \\leq 6 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\((\\sin 3x + \\sin 2y + \\sin z)(\\cos 3x + \\cos 2y + \\cos z)\\). \n\n(Possible points: 15)"
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 10053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\((\\sqrt{36-4 \\sqrt{5}} \\sin x-\\sqrt{2(1+\\cos 2 x)}-2) \\cdot (3+2 \\sqrt{10-\\sqrt{5}} \\cos y-\\cos 2 y)\\). If the answer is not an integer, round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 10054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\((\\sqrt{8-4 \\sqrt{3}} \\sin x - 3 \\sqrt{2(1+\\cos 2x)} - 2) \\cdot (3 + 2 \\sqrt{11 - \\sqrt{3}} \\cos y - \\cos 2y)\\). If the answer is a non-integer, round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 10055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression \\(\\cos(x+y)\\) given that \\(\\cos x - \\cos y = \\frac{1}{4}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "31/32"
} | {
"index": 10056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the expression for \\( a, b > 0 \\):\n\n$$\n\\frac{|4a - 10b| + |2(a - b\\sqrt{3}) - 5(a\\sqrt{3} + b)|}{\\sqrt{a^2 + b^2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{87}"
} | {
"index": 10057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the following expression:\n$$\n|\\cdots|\\left|x_{1}-x_{2}\\right|-x_{3}\\left|-\\cdots-x_{1990}\\right|,\n$$\nwhere \\( x_{1}, x_{2}, \\cdots, x_{1990} \\) are distinct natural numbers from 1 to 1990."
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"index": 10058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the following expression:\n$$\n|\\cdots||| x_{1}-x_{2}\\left|-x_{3}\\right|-x_{4}\\left|-\\cdots-x_{1990}\\right|,\n$$\n\nwhere \\( x_{1}, x_{2}, \\cdots, x_{1990} \\) are distinct natural numbers from 1 to 1990."
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"index": 10059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the function\n\n$$\nf(x)=\\sin (x+\\sin x)+\\sin (x-\\sin x)+\\left(\\frac{\\pi}{2}-2\\right) \\sin (\\sin x)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi - 2}{\\sqrt{2}}"
} | {
"index": 10060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the function\n$$\nf(x) = \\sqrt{3} \\sin 2x + 2 \\sin x + 4 \\sqrt{3} \\cos x.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"index": 10061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the function $y=\\frac{x}{{{e}^{x}}}$ on the interval $[0,2]$.\n\nA) When $x=1$, $y=\\frac{1}{e}$\nB) When $x=2$, $y=\\frac{2}{{{e}^{2}}}$\nC) When $x=0$, $y=0$\nD) When $x=\\frac{1}{2}$, $y=\\frac{1}{2\\sqrt{e}}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"index": 10062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the function $y=\\sin^2x+3\\sin x\\cos x+4\\cos^2x$ for $0 \\leqslant x \\leqslant \\frac{\\pi}{2}$ and the corresponding value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 10063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the function \\( f(x) \\), which is defined as the minimum of the three functions \\( 4x + 1 \\), \\( x + 2 \\), and \\( -2x + 4 \\) for each real number \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 10064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the parameter \\( b \\) for which the inequality \\( b \\sqrt{b}\\left(x^{2}-10 x+25\\right)+\\frac{\\sqrt{b}}{\\left(x^{2}-10 x+25\\right)} \\leq \\frac{1}{5} \\cdot \\sqrt[4]{b^{3}} \\cdot \\left| \\sin \\frac{\\pi x}{10} \\right| \\) has at least on... | math | {
"style": "rule",
"ground_truth": "1/10000"
} | {
"index": 10065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the positive real number \\( A \\) such that for any real numbers \\( x, y, z \\), the inequality\n$$\nx^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \\geq 0\n$$\nholds."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 10066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the product \\(x^{2} y^{2} z^{2} u\\) given the condition that \\(x, y, z, u \\geq 0\\) and:\n\n\\[ 2x + xy + z + yz u = 1 \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1/512"
} | {
"index": 10067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\\geq C(x+y)$ , and $ x^{2}+y^{2}+xy+1\\geq C(x+y)$ for all reals $x,y$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 10068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the mean of all solutions for $x$ when $x^3 + 3x^2 - 10x = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 10069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the measure of angle \\( B \\widehat{A} D \\), given that \\( D \\widehat{A C}=39^{\\circ} \\), \\( A B = A C \\), and \\( A D = B D \\)."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 10070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the measure of the angle\n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2903^{\\circ}+\\sin 2904^{\\circ}+\\cdots+\\sin 6503^{\\circ}\\right)^{\\cos 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}}\\right)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 10071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the measure of the angle\n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2905^{\\circ}+\\sin 2906^{\\circ}+\\cdots+\\sin 6505^{\\circ}\\right)^{\\cos } 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}\\right)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 10072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the measure of the angle \n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2907^{\\circ}+\\sin 2908^{\\circ}+\\cdots+\\sin 6507^{\\circ}\\right)^{\\cos 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}}\\right)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 10073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 10074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum angle formed by any triple among five points on the plane such that the minimum angle is greater than or equal to $36^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ 36^{\\circ} \\]"
} | {
"index": 10075,
"solution": "We will show that $36^{\\circ}$ is the desired answer for the problem. First, we observe that if the given 5 points form a regular pentagon, then the minimum of the angles formed by any triple among the five vertices is $36^{\\circ}$, and therefore, the answer we seek must be bigger t... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum area of the part bounded by the parabola $ y\\equal{}a^3x^2\\minus{}a^4x\\ (a>0)$ and the line $ y\\equal{}x$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 10076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum distance between point $P$ on the line $y = 2x + 1$ and point $Q$ on the curve $y = x + \\ln{x}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 10077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum distance from a point on the curve y=e^{x} to the line y=x."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 10078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$ , $b$ , $c$ , and $d$ (not necessarily distinct) such that $a+b+c=d$ ."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum number of planes that divide a cube into at least 300 parts."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum of the expression \\(6 t^{2} + 3 s^{2} - 4 s t - 8 t + 6 s + 5\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{7}"
} | {
"index": 10081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum of the function\n\\[\\frac{xy}{x^2 + y^2}\\]in the domain $\\frac{2}{5} \\le x \\le \\frac{1}{2}$ and $\\frac{1}{3} \\le y \\le \\frac{3}{8}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{13}"
} | {
"index": 10082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum point of the function $f(x)=x+2\\cos x$ on the interval $[0, \\pi]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"index": 10083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum positive integer $k$ such that $f(n+k) \\equiv f(n)(\\bmod 23)$ for all integers $n$."
}
] | math | {
"style": "rule",
"ground_truth": "2530"
} | {
"index": 10084,
"solution": "Note that $\\phi(23)=22$ and $\\phi(22)=10$, so if $\\operatorname{lcm}(23,22,10)=2530 \\mid k$ then $f(n+k) \\equiv f(n)(\\bmod 23)$ is always true. We show that this is necessary as well. Choosing $n \\equiv 0(\\bmod 23)$, we see that $k \\equiv 0(\\bmod 23)$. Thus $n+k \\equiv n(\\... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\\Bbb{Z}$ of all integers to $\\{1, 2, \\ldots k\\}$ with the property that $f(x) \\neq f(y)$ whenever $|x-y| \\in \\{5, 7, 12\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10085,
"solution": "\nWe are tasked with finding the minimum positive integer \\( k \\) such that there exists a function \\( f: \\mathbb{Z} \\to \\{1, 2, \\ldots, k\\} \\) satisfying the condition that \\( f(x) \\neq f(y) \\) whenever \\( |x - y| \\in \\{5, 7, 12\\} \\).\n\n### Analyzing the Problem\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum positive integer $n\\ge 3$, such that there exist $n$ points $A_1,A_2,\\cdots, A_n$ satisfying no three points are collinear and for any $1\\le i\\le n$, there exist $1\\le j \\le n (j\\neq i)$, segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10086,
"solution": "\nTo find the minimum positive integer \\( n \\geq 3 \\) such that there exist \\( n \\) points \\( A_1, A_2, \\ldots, A_n \\) satisfying no three points are collinear and for any \\( 1 \\leq i \\leq n \\), there exists \\( 1 \\leq j \\leq n \\) (with \\( j \\neq i \\)), such that the... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum positive period of the function \\( f(x) = |\\tan 2x| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 10087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum possible value of\n\\[\\frac{a}{b^3+4}+\\frac{b}{c^3+4}+\\frac{c}{d^3+4}+\\frac{d}{a^3+4},\\] \ngiven that $a,b,c,d,$ are nonnegative real numbers such that $a+b+c+d=4$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\\frac{1}{2}\\]"
} | {
"index": 10088,
"solution": "See here: https://artofproblemsolving.com/community/c5t211539f5h1434574_looks_like_mount_inequality_erupted_ \nor:\nhttps://www.youtube.com/watch?v=LSYP_KMbBNc "
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum possible value of $\\left(x^{2}+6 x+2\\right)^{2}$ over all real numbers $x$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10089,
"solution": "This is $\\left((x+3)^{2}-7\\right)^{2} \\geq 0$, with equality at $x+3= \\pm \\sqrt{7}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum possible value of $\\sqrt{58-42 x}+\\sqrt{149-140 \\sqrt{1-x^{2}}}$ where $-1 \\leq x \\leq 1$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{109}"
} | {
"index": 10090,
"solution": "Substitute $x=\\cos \\theta$ and $\\sqrt{1-x^{2}}=\\sin \\theta$, and notice that $58=3^{2}+7^{2}, 42=2 \\cdot 3 \\cdot 7,149=7^{2}+10^{2}$, and $140=2 \\cdot 7 \\cdot 10$. Therefore the first term is an application of Law of Cosines on a triangle that has two sides 3 and 7 with an an... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum possible value of the largest of $x y, 1-x-y+x y$, and $x+y-2 x y$ if $0 \\leq x \\leq y \\leq 1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 10091,
"solution": "I claim the answer is $4 / 9$. Let $s=x+y, p=x y$, so $x$ and $y$ are $\\frac{s \\pm \\sqrt{s^{2}-4 p}}{2}$. Since $x$ and $y$ are real, $s^{2}-4 p \\geq 0$. If one of the three quantities is less than or equal to $1 / 9$, then at least one of the others is at least $4 / 9$ by the pig... |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\\delta=0.3$, given that the standard deviation $\\sigma=1.2$ of the normally distributed population is known."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 10092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value for \\(a, b > 0\\) of the expression\n\n$$\n\\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\\sqrt{4a^2 + b^2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 10093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value for \\(a, b > 0\\) of the expression\n\n$$\n\\frac{|6a - 4b| + |3(a + b\\sqrt{3}) + 2(a\\sqrt{3} - b)|}{\\sqrt{a^2 + b^2}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{39}"
} | {
"index": 10094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n$$\n\\begin{aligned}\nA & =\\sqrt{\\left(1264-z_{1}-\\cdots-z_{n}\\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\\\\n& \\sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\\cdots+\\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\\\\n& \\sqrt{z_{1}^{2}+\\left(948-x_{1}-\\cdots-x_{n... | math | {
"style": "rule",
"ground_truth": "1975"
} | {
"index": 10095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\n\\sqrt{x^2 + (2 - x)^2} + \\sqrt{(2 - x)^2 + (2 + x)^2}\n\\]\nover all real numbers $x$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 10096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\n\\sqrt{x^2 + (2 - x)^2} + \\sqrt{(2 - x)^2 + (2 + x)^2}\n\\]\nover all real numbers \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 10097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\ny^2 + 9y + \\frac{81}{y^3}\n\\]\nfor \\(y > 0\\)."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 10098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the minimum value of\n\\[ x^3 + 9x + \\frac{81}{x^4} \\]\nfor \\( x > 0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 10099,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.