data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a cube \\( ABCD-A_1B_1C_1D_1 \\) with edge length 1, a point \\( M \\) is taken on the diagonal \\( A_1D \\) and a point \\( N \\) is taken on \\( CD_1 \\) such that the line segment \\( MN \\) is parallel to the diagonal plane \\( A_1ACC_1 \\), find the minimum value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube \\(ABCD-A_1B_1C_1D_1\\) with side length 1, and \\(E\\) as the midpoint of \\(D_1C_1\\), find the following:\n1. The distance between skew lines \\(D_1B\\) and \\(A_1E\\).\n2. The distance from \\(B_1\\) to plane \\(A_1BE\\).\n3. The distance from \\(D_1C\\) to pla... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube of side length $8$ and balls of clay of radius $1.5$, determine the maximum number of balls that can completely fit inside the cube when the balls are reshaped but not compressed."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube with a side length of \\(4\\), if a solid cube of side length \\(1\\) is removed from each corner, calculate the total number of edges of the resulting structure."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "7\\pi - 4\\sqrt{3}\\pi"
} | {
"index": 13205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 13206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \\cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 13207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube, calculate the total number of pairs of diagonals on its six faces, where the angle formed by each pair is $60^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 13208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cubic function $f(x) = ax^3 + bx^2 + cx + d$ ($a \\neq 0$), the definition of an inflection point is provided: Let $f'(x)$ be the derivative of $y = f(x)$, and let $f''(x)$ be the derivative of $f'(x)$. If the equation $f''(x) = 0$ has a real number solution $x_0$, then... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 13209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cubic function $f(x)=\\frac{a}{3}x^{3}+bx^{2}+cx+d$ ($a < b$) is monotonically increasing on $\\mathbb{R}$, then the minimum value of $\\frac{a+2b+3c}{b-a}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "8+6 \\sqrt{2}"
} | {
"index": 13210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cubic function $f(x)=ax^{3}+bx^{2}+cx+d(a\\neq 0)$, define the \"nice point\" of the function as the point $(x_{0},f(x_{0}))$ where $x_{0}$ is a real root of the equation $f''(x)=0$. It has been observed that every cubic function has a \"nice point,\" a symmetry center,... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 13211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\\alpha, \\beta, \\gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum \n\\[\n\\frac{1}{1-\\alpha} + \\frac{1}{1-\\beta} + \\frac{1}{1-\\gamma}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 13212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cubic polynomial \\( q(x) \\) that satisfies \\( q(3) = 2 \\), \\( q(8) = 20 \\), \\( q(18) = 12 \\), and \\( q(25) = 32 \\). Find the sum \\( q(4) + q(5) + \\ldots + q(26) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "391"
} | {
"index": 13213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cyclic quadrilateral \\(ABCD\\) with a circumradius of \\(200\\sqrt{2}\\) and sides \\(AB = BC = CD = 200\\). Find the length of side \\(AD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 13214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cylinder of fixed volume $V,$ the total surface area (including the two circular ends) is minimized for a radius of $R$ and height $H.$ Find $\\frac{H}{R}.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 13215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} \\sqrt{3}"
} | {
"index": 13216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a deck consisting of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning set."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"index": 13218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a deck of cards consisting of four red cards numbered 1, 2, 3, 4; four green cards numbered 1, 2, 3, 4; and four yellow cards numbered 1, 2, 3, 4, calculate the probability of drawing a winning pair, where a winning pair consists of two cards of the same color or two card... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 13219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a deck of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning pair."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a decreasing arithmetic sequence $\\{a_n\\}$, if $a_1 + a_{100} = 0$, find the value of $n$ when the sum of the first $n$ terms, $S_n$, is maximized."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 13221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a decreasing arithmetic sequence $\\{a_n\\}$, where $a_3 = -1$, and $a_1$, $a_4$, $-a_6$ form a geometric sequence. Find the value of $S_7$, where $S_n$ represents the sum of the first $n$ terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 13222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$."
}
] | math | {
"style": "rule",
"ground_truth": "(0.03)^2 \\times 0.97 + (0.03)^3"
} | {
"index": 13223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a device purchased at $40 less a 15% discount, and a goal of achieving a 25% selling profit based on the cost after allowing a 10% discount on the final sales price, determine the marked price of the device in dollars."
}
] | math | {
"style": "rule",
"ground_truth": "47.22"
} | {
"index": 13224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a die is rolled consecutively three times, determine the probability that the sequence of numbers facing upwards forms an arithmetic progression."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 13225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fair 8-sided die is rolled until an odd number appears, determine the probability that the even numbers 2, 4, and 6 appear in strictly ascending order at least once before the first occurrence of any odd number."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{512}"
} | {
"index": 13226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fair coin is tossed, the probability of heads or tails is both $\\frac{1}{2}$. Construct a sequence $\\{a_n\\}$, where $a_n = \\begin{cases} 1, \\text{if heads on the nth toss} \\\\ -1, \\text{if tails on the nth toss} \\end{cases}$. Let $S_n = a_1 + a_2 + ... + a_n$. F... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{128}"
} | {
"index": 13227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{19}{36}"
} | {
"index": 13228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a family of sets \\(\\{A_{1}, A_{2}, \\ldots, A_{n}\\}\\) that satisfies the following conditions:\n(1) Each set \\(A_{i}\\) contains exactly 30 elements;\n(2) For any \\(1 \\leq i < j \\leq n\\), the intersection \\(A_{i} \\cap A_{j}\\) contains exactly 1 element;\n(3) T... | math | {
"style": "rule",
"ground_truth": "871"
} | {
"index": 13229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite arithmetic sequence \\( a_{1}, a_{2}, \\cdots a_{k} \\), and the conditions:\n$$\n\\begin{array}{c}\na_{4}+a_{7}+a_{10}=17, \\\\\na_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 .\n\\end{array}\n$$\nIf \\( a_{k}=13 \\), then \\( k \\) is... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite arithmetic sequence \\(\\left\\{a_{n}\\right\\}\\) with the first term equal to 1 and the last term \\(a_{n} = 1997\\) (where \\(n > 3\\)), and the common difference being a natural number, find the sum of all possible values of \\(n\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3501"
} | {
"index": 13231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite increasing sequence \\(a_{1}, a_{2}, \\ldots, a_{n}\\) of natural numbers (with \\(n \\geq 3\\)), and the recurrence relation \\(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\\) holds for all \\(\\kappa \\leq n-2\\). The sequence must contain \\(a_{k} = 2022\\). Determine the ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 13232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite sequence $D$: $a\\_1$, $a\\_2$, ..., $a\\_n$, where $S\\_n$ represents the sum of the first $n$ terms of the sequence $D$, define $\\frac{S\\_1 + S\\_2 + ... + S\\_n}{n}$ as the \"De-Guang sum\" of $D$. If a 99-term sequence $a\\_1$, $a\\_2$, ..., $a\\_99$ has a ... | math | {
"style": "rule",
"ground_truth": "998"
} | {
"index": 13233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite sequence $S=(2, 2x, 2x^2,\\ldots ,2x^{200})$ of $n=201$ real numbers, let $A(S)$ be the sequence $\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{200}+a_{201}}{2}\\right)$ of $n-1=200$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, ... | math | {
"style": "rule",
"ground_truth": "2^{3/8} - 1"
} | {
"index": 13234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$\nof $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le n-1$,... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"index": 13235,
"solution": "1. **Understanding the Sequence Transformation**:\n Given a sequence $S = (a_1, a_2, \\ldots, a_n)$, the sequence $A(S)$ is defined as:\n \\[\n A(S) = \\left(\\frac{a_1 + a_2}{2}, \\frac{a_2 + a_3}{2}, \\ldots, \\frac{a_{n-1} + a_n}{2}\\right)\n \\]\n This transformation red... |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite sequence $\\{a\\_1\\}, \\{a\\_2\\}, \\ldots \\{a\\_m\\} (m \\in \\mathbb{Z}^+)$ that satisfies the conditions: $\\{a\\_1\\} = \\{a\\_m\\}, \\{a\\_2\\} = \\{a\\_{m-1}\\}, \\ldots \\{a\\_m\\} = \\{a\\_1\\}$, it is called a \"symmetric sequence\" with the additional... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 13236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a finite sequence \\(P = \\left(p_{1}, p_{2}, \\cdots, p_{n}\\right)\\), the Caesar sum (named after a mathematician Caesar) is defined as \\(\\frac{s_{1}+s_{2}+\\cdots+s_{n}}{n}\\), where \\(s_{k} = p_{1} + p_{2} + \\cdots + p_{k}\\) for \\(1 \\leq k \\leq n\\). If a seq... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"index": 13237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed circle $\\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\\odot Q$ is externally tangent to circle $\\odot P$. Circle $\\odot Q$ intersects $l$ at points $M$ and $N$. For any d... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed circle \\\\(F_{1}:(x+2)^{2}+y^{2}=24\\\\) and a moving circle \\\\(N\\\\) passing through point \\\\(F_{2}(2,0)\\\\) and tangent to circle \\\\(F_{1}\\\\), denote the locus of the center of circle \\\\(N\\\\) as \\\\(E\\\\).\n\n\\\\((I)\\\\) Find the equation of t... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"index": 13239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed integer \\( n \\) where \\( n \\geq 2 \\):\n\na) Determine the smallest constant \\( c \\) such that the inequality \\(\\sum_{1 \\leq i < j \\leq n} x_i x_j (x_i^2 + x_j^2) \\leq c \\left( \\sum_{i=1}^n x_i \\right)^4\\) holds for all nonnegative real numbers \\( ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 13240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \\perp l$, with the foot of the perpendicular being $Q$, and $\\left( \\overrightarrow{PC}+\\frac{1}{2}\\overrightarrow{PQ} \\right)\\cdot \\left( \\overrightarrow{PC}-\\frac{1}... | math | {
"style": "rule",
"ground_truth": "12-4\\sqrt{3}"
} | {
"index": 13241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed point A (3, 4), and point P is a moving point on the parabola $y^2=4x$, the distance from point P to the line $x=-1$ is denoted as $d$. Find the minimum value of $|PA|+d$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 13242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a fixed point P (-2, 0) and a line $l: (1+3\\lambda)x + (1+2\\lambda)y - (2+5\\lambda) = 0$, where $\\lambda \\in \\mathbb{R}$, find the maximum distance $d$ from point P to line $l$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 13243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a four-digit number $\\overline{A B C D}$ that satisfies the following properties: $\\overline{A B}$, $\\overline{B C}$, and $\\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $8... | math | {
"style": "rule",
"ground_truth": "13462"
} | {
"index": 13244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a four-digit number $\\overline{ABCD}$ such that $\\overline{ABCD} + \\overline{AB} \\times \\overline{CD}$ is a multiple of 1111, what is the minimum value of $\\overline{ABCD}$?"
}
] | math | {
"style": "rule",
"ground_truth": "1729"
} | {
"index": 13245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a four-digit number \\(\\overline{abcd}\\), when divided by 2, 3, 4, 5, 6, and 7, the remainders are all different and none of them are 0. Find the minimum value of \\(\\overline{abcd}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1259"
} | {
"index": 13246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a four-digit positive integer $\\overline{abcd}$, if $a+c=b+d=11$, then this number is called a \"Shangmei number\". Let $f(\\overline{abcd})=\\frac{{b-d}}{{a-c}}$ and $G(\\overline{abcd})=\\overline{ab}-\\overline{cd}$. For example, for the four-digit positive integer $3... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 13247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a frustum of a cone with the areas of its two bases being $4\\pi$ and $25\\pi$ respectively, and the height of the frustum is 4, find the volume and the lateral surface area of the frustum."
}
] | math | {
"style": "rule",
"ground_truth": "35\\pi"
} | {
"index": 13249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f$ for which \\[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\\] holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\\ldots,f(999)?$"
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"index": 13250,
"solution": "\\begin{align*}f(2158 - x) = f(x) &= f(3214 - (2158 - x)) &= f(1056 + x)\\\\ f(398 - x) = f(x) &= f(2158 - (398 - x)) &= f(1760 + x)\\end{align*}\n Since $\\mathrm{gcd}(1056, 1760) = 352$ we can conclude that (by the Euclidean algorithm)\n\\[f(x) = f(352 + x)\\]\n So we need only to c... |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \\infty)$, find the value of the real number $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x) = 2\\sin x\\cos x + 2\\cos\\left(x + \\frac{\\pi}{4}\\right)\\cos\\left(x - \\frac{\\pi}{4}\\right)$,\n1. Find the interval(s) where $f(x)$ is monotonically decreasing.\n2. If $\\alpha \\in (0, \\pi)$ and $f\\left(\\frac{\\alpha}{2}\\right) = \\frac{\\sqr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}"
} | {
"index": 13252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x) = \\cos x \\sin \\left( x + \\frac{\\pi}{3} \\right) - \\sqrt{3} \\cos^2 x + \\frac{\\sqrt{3}}{4}$, where $x \\in \\mathbb{R}$,\n(1) Find the smallest positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing;\n(2) Find the maxim... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x) = \\frac {1}{2}\\sin x - \\frac {\\sqrt {3}}{2}\\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\\frac {1}{2}, 1]$. Find the maximum value of $b-a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"index": 13254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x) = m\\ln{x} + nx$ whose tangent at point $(1, f(1))$ is parallel to the line $x + y - 2 = 0$, and $f(1) = -2$, where $m, n \\in \\mathbb{R}$,\n(Ⅰ) Find the values of $m$ and $n$, and determine the intervals of monotonicity for the function $f(x)$;\n(Ⅱ) Let... | math | {
"style": "rule",
"ground_truth": "\\frac{e(e - 2)}{e - 1}"
} | {
"index": 13255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ $(x \\in \\mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \\frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \\dots, 168... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"index": 13256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ ($x \\in \\mathbb{R}$) that satisfies $f(-x) = 8 - f(4 + x)$, and a function $g(x) = \\frac{4x + 3}{x - 2}$, determine the value of $(x_1 + y_1) + (x_2 + y_2) + \\ldots + (x_{168} + y_{168})$ where $P_i(x_i, y_i)$ ($i = 1, 2, \\ldots, 168$) are the commo... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"index": 13257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $R$ such that $f(2x+2)=-f(2x)$. If $f(x)=4x+3$ when $x\\in(\\frac{1}{2},\\frac{5}{4})$, then $f(2023)=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 13258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \\in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$."
}
] | math | {
"style": "rule",
"ground_truth": "506"
} | {
"index": 13259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $R$ such that $f(x) + x^{2}$ is an odd function and $f(x) + x^{3}$ is an even function, then $f(2)$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 13260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$ satisfies $f(x+6)=f(x)$. When $-3<x\\leq-1$, $f(x)=-(x+2)^2$, and when $-1\\leq x\\leq 3$, $f(x)=x$. Calculate the value of $f(1) + f(2) + \\dots + f(2015)$."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 13261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 13262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$, for any real numbers $x_1$, $x_2$, it satisfies $f(x_1+x_2)=f(x_1)+f(x_2)+2$. The sequence $\\{a_n\\}$ satisfies $a_1=0$, and for any $n\\in\\mathbb{N}^*$, $a_n=f(n)$. Find the value of $f(2010)$."
}
] | math | {
"style": "rule",
"ground_truth": "4018"
} | {
"index": 13263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ satisfies $f(x) + f(4-x) = 4$, $f(x+2) - f(-x) = 0$, and $f(1) = a$, calculate the value of $f(1) + f(2) + f(3) + \\cdots + f(51)$."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 13264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ such that for any $x$, $f(x+2)=f(x+1)-f(x)$, and $f(1)=\\log_3-\\log_2$, $f(2)=\\log_3+\\log_5$, calculate the value of $f(2010)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ that always satisfies the following conditions on its domain $\\mathbb{R}$: \n① $f(x) = f(-x)$, ② $f(2+x) = f(2-x)$, when $x \\in [0, 4)$, $f(x) = -x^2 + 4x$. \n(1) Find $f(8)$. \n(2) Find the number of zeros of $f(x)$ in $[0, 2015]$."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 13266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies $f(x+3)=-f(x)$, when $x\\in \\left[-3,0\\right)$, $f(x)=2^{x}+\\sin \\frac{πx}{3}$, determine the value of $f(2023)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4} + \\frac{\\sqrt{3}}{2}"
} | {
"index": 13267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies the functional equation $f(x) = f(x+1) - f(x+2)$ for all $x \\in \\mathbb{R}$. When $x \\in (0,3)$, $f(x) = x^2$. Express the value of $f(2014)$ using the functional equation."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies: (1) For any positive real number $x$, it always holds that $f(2x) = 2f(x)$; (2) When $x \\in (1, 2)$, $f(x) = 2 - x$. If $f(a) = f(2020)$, determine the smallest positive real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies: For any $x \\in (0, +\\infty)$, it always holds that $f(2x) = 2f(x)$; (2) When $x \\in (1, 2]$, $f(x) = 2 - x$. If $f(a) = f(2020)$, find the smallest positive real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)$ whose domain is $D$, if for any $x_1, x_2 \\in D$, it holds that $f(x_1) \\leq f(x_2)$ whenever $x_1 < x_2$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on the interval $[0,1]$ and sati... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"index": 13271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f(x)=\\log _{a}\\left(\\sqrt {x^{2}+1}+x\\right)+\\dfrac{1}{a^{x}-1}+\\dfrac{3}{2}$, where $a > 0$ and $a \\neq 1$. If $f\\left(\\log _{3}b\\right)=5$ for $b > 0$ and $b \\neq 1$, find the value of $f\\left(\\log _{\\frac{1}{3}}b\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 13272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $f: \\{1, 2, 3\\} \\rightarrow \\{1, 2, 3\\}$ that satisfies $f(f(x)) = f(x)$, determine the total number of such functions."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function $y=f(x)$ defined on the domain $I$, if there exists an interval $[m,n] \\subseteq I$ that simultaneously satisfies the following conditions: $①f(x)$ is a monotonic function on $[m,n]$; $②$when the domain is $[m,n]$, the range of $f(x)$ is also $[m,n]$, then we ... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {3}}{3}"
} | {
"index": 13274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function \\( f: \\mathbf{R} \\rightarrow \\mathbf{R} \\) such that for any real numbers \\( x \\) and \\( y \\), \\( f(2x) + f(2y) = f(x+y) f(x-y) \\). Additionally, \\( f(\\pi) = 0 \\) and \\( f(x) \\) is not identically zero. What is the period of \\( f(x) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 13275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function \\( f: \\mathbf{R} \\rightarrow \\mathbf{R} \\) that satisfies the condition: for any real numbers \\( x \\) and \\( y \\),\n\\[ f(2x) + f(2y) = f(x+y) f(x-y) \\]\n\nand given that \\( f(\\pi) = 0 \\) and \\( f(x) \\) is not identically zero, determine the peri... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 13276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function \\\\(f(x)\\\\) defined on \\\\(\\mathbb{R}\\\\) that satisfies: the graph of \\\\(y=f(x-1)\\\\) is symmetric about the point \\\\((1,0)\\\\), and when \\\\(x \\geqslant 0\\\\), it always holds that \\\\(f(x+2)=f(x)\\\\). When \\\\(x \\in [0,2)\\\\), \\\\(f(x)=e... | math | {
"style": "rule",
"ground_truth": "1-e"
} | {
"index": 13277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function defined on $\\mathbb{R}$, $f(x)=A\\sin (\\omega x+\\varphi)$ where $A > 0$, $\\omega > 0$, and $|\\varphi| \\leqslant \\frac {\\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\\frac {\\pi}{2}$. Aft... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {741}}{32}- \\frac {3}{32}"
} | {
"index": 13278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function defined on the set of positive integers as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{if } n \\geq 1000 \\\\\nf[f(n + 7)], & \\text{if } n < 1000 \n\\end{cases} \\]\nFind the value of \\( f(90) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 13279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function defined on the set of positive integers as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{if } n \\geq 1000 \\\\\nf[f(n + 7)], & \\text{if } n < 1000 \n\\end{cases} \\]\nWhat is the value of \\( f(90) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 13280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function f(n) defined on the set of positive integers, where f(1) = 2: For even n, f(n) = f(n-1) + 2; For odd n > 1, f(n) = f(n-2) + 2. Calculate the value of f(2017)."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"index": 13281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a function f(x) defined on ℝ that satisfies f(x-2)=f(-2-x), and when x ≥ -2, f(x)=2^x-3. If the function f(x) has a zero point in the interval (k,k+1) (k ∈ ℤ), determine the value of k."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 13282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a general triangle \\(ABC\\) with points \\(K, L, M, N, U\\) on its sides:\n\n- Point \\(K\\) is the midpoint of side \\(AC\\).\n- Point \\(U\\) is the midpoint of side \\(BC\\).\n- Points \\(L\\) and \\(M\\) lie on segments \\(CK\\) and \\(CU\\) respectively, such that \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric progression $\\{a_n\\}$ with the first term $a_1=2$ and the sum of the first $n$ terms as $S_n$, and the equation $S_5 + 4S_3 = 5S_4$ holds, find the maximum term of the sequence $\\left\\{ \\frac{2\\log_{2}a_n + 1}{\\log_{2}a_n - 6} \\right\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric progression \\( b_1, b_2, \\ldots, b_{3000} \\) with all positive terms and a total sum \\( S \\). It is known that if every term with an index that is a multiple of 3 (i.e., \\( b_3, b_6, \\ldots, b_{3000} \\)) is increased by 50 times, the sum \\( S \\) incr... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{8}"
} | {
"index": 13285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"index": 13286,
"solution": "1. **Identify the terms of the geometric progression**: Let the first term be $a$ and the common ratio be $r$. The terms of the progression are $a, ar, ar^2, ar^3, ar^4$. The sum of these terms is given by:\n \\[\n a + ar + ar^2 + ar^3 + ar^4 = a(1 + r + r^2 + r^3 + r^4) = 211\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with a common ratio $q \\in (1,2)$, and that $a_{n}$ is a positive integer for $1 \\leq n \\leq 6$, find the minimum value of $a_{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"index": 13287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, calculate the value of $S_{40}$."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 13288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then $S_{40}$ is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 13289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ composed of positive numbers, with a common ratio $q=2$, and $a_1 \\cdot a_2 \\cdot a_3 \\cdot \\ldots \\cdot a_{30} = 2^{45}$, calculate the product of the terms $a_1 \\cdot a_4 \\cdot a_7 \\cdot \\ldots \\cdot a_{28}$."
}
] | math | {
"style": "rule",
"ground_truth": "2^5"
} | {
"index": 13290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ satisfies $a_1=3$, and $a_1+a_3+a_5=21$. Calculate the value of $a_3+a_5+a_7$."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 13291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ satisfies $a_2a_5=2a_3$, and $a_4$, $\\frac{5}{4}$, $2a_7$ form an arithmetic sequence, the maximum value of $a_1a_2a_3…a_n$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 13292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ where each term is positive, and $a_1a_3=4$, $a_7a_9=25$, find $a_5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 13293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ where the sum of the first $n$ terms $S_n = a \\cdot 3^n - 2$, find the value of $a_2$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=2$, if $\\frac {S_{6}}{S_{2}}=21$, then the sum of the first five terms of the sequence $\\{\\frac {1}{a_n}\\}$ is \n\nA) $\\frac {1}{2}$ or $\\frac {11}{32}$\n\nB) $\\frac {1}{2}$ or $\\... | math | {
"style": "rule",
"ground_truth": "\\frac {31}{32}"
} | {
"index": 13295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with $a_1=1$, $0<q<\\frac{1}{2}$, and for any positive integer $k$, $a_k - (a_{k+1}+a_{k+2})$ is still an element of the sequence, find the common ratio $q$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 13296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q > 1$, $a_1 = 2$, and $a_1, a_2, a_3 - 8$ form an arithmetic sequence. The sum of the first n terms of the sequence $\\{b_n\\}$ is denoted as $S_n$, and $S_n = n^2 - 8n$.\n(I) Find the general term formulas for both s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{162}"
} | {
"index": 13297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q > 1$, the sum of the first $n$ terms is $S_n$, and $S_3 = 7$. Also, $a_1, a_2, a_3 - 1$ form an arithmetic sequence. For the sequence $\\{b_n\\}$, the sum of the first $n$ terms is $T_n$, and $6T_n = (3n+1)b_n + 2$, ... | math | {
"style": "rule",
"ground_truth": "3318"
} | {
"index": 13298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \\cdot a_5 = -15$, find the value of $a_{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{25}{3}"
} | {
"index": 13299,
"solution": ""
} |
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