data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Through points \\(A(0, 14)\\) and \\(B(0, 4)\\), two parallel lines are drawn. The first line, passing through point \\(A\\), intersects the hyperbola \\(y = \\frac{1}{x}\\) at points \\(K\\) and \\(L\\). The second line, passing through point \\(B\\), intersects the hyperbola ... | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 9300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( Z \\) lies on \\( XY \\) and the three circles have diameters \\( XZ \\), \\( ZY \\), and \\( XY \\). If \\( XZ = 12 \\) and \\( ZY = 8 \\), calculate the ratio of the area of the shaded region to the area of the unshaded region. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"split": "train",
"index": 9301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "577"
} | {
"split": "train",
"index": 9302,
"solution": "S is the set of positive integers with exactly three ones in its binary representation. The number of such integers with at most $d$ total bits is \\binom{d}{3}$, and noting that \\binom{9}{3}=84$ and \\binom{10}{3}=120$, we want the 16th smallest integer of the for... |
deepscaler | [
{
"role": "user",
"content": "A bus can hold a maximum of 38 students. What is the minimum number of buses needed to transport 411 students? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 9303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which\n\\[\\sqrt{1 + \\sqrt{21 + 12 \\sqrt{3}}} = \\sqrt{a} + \\sqrt{b}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,3)"
} | {
"split": "train",
"index": 9304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all the functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \\]\nfor all real numbers $x$ and $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x^2 \\text{ and } f(x) = 0"
} | {
"split": "train",
"index": 9305,
"solution": "\nTo determine all functions \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfying the functional equation:\n\n\\[\nf(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), we will go through the following steps:\n\n### Step 1: S... |
deepscaler | [
{
"role": "user",
"content": "A whole number, $N$, is chosen so that $\\frac{N}{3}$ is strictly between 7.5 and 8. What is the value of $N$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 9306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function defined on $\\mathbb{R}$ with a period of $2$. In the interval $[-1,1)$, $f(x)$ is defined as follows:\n$$\nf(x)=\\begin{cases} \nx + a, & -1 \\leqslant x < 0, \\\\\n\\left| \\frac{2}{5} - x \\right|, & 0 \\leqslant x < 1,\n\\end{cases}\n$$\nwhere $a \\... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 9307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that has exactly eight distinct positive divisors, where all divisors are powers of prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 9308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit numbers have digits whose sum is a perfect square less than or equal to 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 9309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\sqrt{25\\sqrt{15\\sqrt{9}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{15}"
} | {
"split": "train",
"index": 9310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $\\frac{1}{n}$ over all positive integers $n$ with the property that the decimal representation of $\\frac{1}{n}$ terminates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{5}{2}}"
} | {
"split": "train",
"index": 9311,
"solution": "The decimal representation of $\\frac{1}{n}$ terminates if and only if $n=2^{i} 5^{j}$ for some nonnegative integers $i, j$, so our desired sum is $$\\sum_{i \\geq 0} \\sum_{j \\geq 0} 2^{-i} 5^{-j}=\\sum_{i \\geq 0} 2^{-i} \\sum_{j \\geq 0} 5^{-j}=\\left(1-2^{-1}\\... |
deepscaler | [
{
"role": "user",
"content": "A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called \"good\" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 9312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A gasoline tank is $\\frac78$ full. After $12$ gallons have been used, it is half full. How many gallons does this tank hold when it is full? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 9313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ be a prime. We arrange the numbers in ${\\{1,2,\\ldots ,p^2} \\}$ as a $p \\times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can c... | math | {
"style": "rule",
"ground_truth": "2(p!)^2"
} | {
"split": "train",
"index": 9314,
"solution": "\nLet \\( p \\) be a prime. We arrange the numbers in \\( \\{1, 2, \\ldots, p^2\\} \\) as a \\( p \\times p \\) matrix \\( A = (a_{ij}) \\). We can select any row or column and add 1 to every number in it, or subtract 1 from every number in it. We call the arrangeme... |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be a regular 2006-sided polygon. If a diagonal of \\( P \\), whose endpoints divide the boundary of \\( P \\) into two parts each containing an odd number of sides, is called a \"good diagonal\". Note that each side of \\( P \\) is considered a \"good diagonal\". ... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 9315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest natural number whose digits in decimal representation are either 0 or 1 and which is divisible by 225? (China Junior High School Mathematics League, 1989) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100"
} | {
"split": "train",
"index": 9316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of the line determined by any two solutions to the equation $\\frac{2}{x}+\\frac{3}{y} = 0$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac 32"
} | {
"split": "train",
"index": 9317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the inequality \\( \\ln(ax) + ax \\leq x + e^x \\) holds for all \\( x \\) and given real numbers \\( a \\), what is the maximum value of \\( a \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 9318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How long should a covered lip pipe be to produce the fundamental pitch provided by a standard tuning fork? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.189"
} | {
"split": "train",
"index": 9319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[\\left( \\frac{1 + i}{1 - i} \\right)^{1000}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 9320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to the standard convention for exponentiation, \n\\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\\]\nIf the order in which the exponentiations are performed is changed, how many other values are possible? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 9321,
"solution": "To solve this problem, we need to evaluate the expression $2^{2^{2^{2}}}$ with different parenthesizations and determine how many distinct values can be obtained. We will use the notation $a \\uparrow b$ to denote $a^b$ for clarity.\n\nThe possible parenthesizations... |
deepscaler | [
{
"role": "user",
"content": "You are given a $10 \\times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 9322,
"solution": "Since each row has only two squares, it is impossible for two marked squares to be in the same row. Therefore, exactly nine of the ten rows contain marked squares. Consider two cases: Case 1: The first or last row is empty. These two cases are symmetrical, so assume... |
deepscaler | [
{
"role": "user",
"content": "Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:\n- Less than 1 year: 3 marks\n- 1 to less than 2 years: 6 marks\n- 2 to less than 3 years: 5 marks\n- 3 to less than 4 years: 4 marks\n- 4 to les... | math | {
"style": "rule",
"ground_truth": "24.14\\%"
} | {
"split": "train",
"index": 9323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses ... | math | {
"style": "rule",
"ground_truth": "1993"
} | {
"split": "train",
"index": 9324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has saved up $5235_9$ dollars for a trip to Japan. A round-trip airline ticket costs $1250_8$ dollars. In base ten, how many dollars will he have left for lodging and food? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3159"
} | {
"split": "train",
"index": 9325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can \\(a, b, c\\), and \\(d\\) be chosen from the set \\(\\{0,1,2, \\ldots, 9\\}\\) so that \\(a<b<c<d\\) and \\(a+b+c+d\\) is a multiple of three? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 9326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The rectangular flag shown is divided into seven stripes of equal height. The height of the flag is $h$ and the length of the flag is twice its height. The total area of the four shaded regions is $1400 \\mathrm{~cm}^{2}$. What is the height of the flag? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "35 \\mathrm{~cm}"
} | {
"split": "train",
"index": 9327,
"solution": "Since the flag shown is rectangular, then its total area is its height multiplied by its width, or $h \\times 2h=2h^{2}$. Since the flag is divided into seven stripes of equal height and each stripe has equal width, then the area of each stripe is the same. Since th... |
deepscaler | [
{
"role": "user",
"content": "Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ =\f \f \f \f \f \f $|1 1|$ $|1 3|$ \f \f \f \f \f \f\n, and the determinant $D_3=$ \f \f \f \f \f \f \f \f \f\n|1 1 1|\n|1 3 3|\n|1 3 5| \f \f \f \f \f \f \f \f \f\n.\nIn general, for positive integer n, let t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 9328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 9329,
"solution": "1. **Simplify the Numerator and Denominator**: \n The given expression is:\n \\[\n \\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}\n \\]\n We can factor out the common terms in both the numerator and the denominator:\n \\[\n \\frac{2 \\cdot 2^3}{2 \\cdot 2^{-3}} = ... |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = x^2 + mx + n$.\n1. If $f(x)$ is an even function and its minimum value is 1, find the expression for $f(x)$.\n2. Based on (1), for the function $g(x) = \\frac{6x}{f(x)}$, solve the inequality $g(2^x) > 2^x$ for $x$.\n3. Let $h(x) = |f(x)|$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin(\\omega x + \\varphi)$ ($\\omega > 0$, $|\\varphi| < \\frac{\\pi}{2}$) has a minimum positive period of $\\pi$, and its graph is translated to the right by $\\frac{\\pi}{6}$ units to obtain the graph of the function $g(x) = \\sin(\\omega x)$, de... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 9331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the tr... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 9332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $u$ and $v$ be integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 9333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adja... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 9334,
"solution": "To solve this problem, we need to consider the number of ways to move from the top face to the bottom face via a sequence of adjacent faces, visiting each face at most once, and without moving from the bottom ring to the top ring. We will use casework based on the n... |
deepscaler | [
{
"role": "user",
"content": "$\\triangle BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. The magnitude of $\\angle DAB$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60^{\\circ}"
} | {
"split": "train",
"index": 9335,
"solution": "1. **Identify Key Properties**: Given $\\triangle BAD$ is right-angled at $B$, and $C$ is a point on $AD$ such that $AC = CD$ and $AB = BC$. Since $AC = CD$, point $C$ is the midpoint of $AD$.\n\n2. **Analyze Triangle ABC**: Since $AB = BC$ and $AC = BC$, triangle $... |
deepscaler | [
{
"role": "user",
"content": "A telephone station serves 400 subscribers. For each subscriber, the probability of calling the station within an hour is 0.01. Find the probabilities of the following events: \"within an hour, 5 subscribers will call the station\"; \"within an hour, no more than 4 subscribers ... | math | {
"style": "rule",
"ground_truth": "0.7619"
} | {
"split": "train",
"index": 9336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 9337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive solutions to\n\\[\\sin (\\arccos( \\tan (\\arcsin x) ) ) ) = x.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 9338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\\alpha+\\beta$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 9339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $S = 6 \\times 10,000 + 5 \\times 1000 + 4 \\times 10 + 3 \\times 1$, calculate the value of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65043"
} | {
"split": "train",
"index": 9340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. What is the $y$-intercept of the line? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 9341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$ , the decimal representation of the number $c^n+2014$ has digits all less than $5$ .\n\n*Proposed by Evan Chen* Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, six squares form a \\( 2 \\times 3 \\) grid. The middle square in the top row is marked with an \\( R \\). Each of the five remaining squares is to be marked with an \\( R \\), \\( S \\), or \\( T \\). In how many ways can the grid be completed so that it includ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 9343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 9344,
"solution": "If there are two edges on a single diameter, there would be six diameters. There are four ways to put the third number, and four equilateral triangles. There are $4+ 2 \\cdot 2\\cdot 6 = 28$ ways. Then, if one length was $\\sqrt{3}$ but no side on the diameter, ther... |
deepscaler | [
{
"role": "user",
"content": "A convex quadrilateral is divided by its diagonals into four triangles; the areas of three of them are \\(10 \\, \\text{cm}^2\\), \\(20 \\, \\text{cm}^2\\), and \\(30 \\, \\text{cm}^2\\), and each is less than the area of the fourth triangle. Find the area of the given quadrila... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 9345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8r^2"
} | {
"split": "train",
"index": 9346,
"solution": "Let's analyze the problem and derive the solution step-by-step.\n\n1. **Understanding the Geometry**:\n - The circle is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$. This implies that the radius $r$ of the circle is the distance from the center of the... |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $252^2 - 248^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 9347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\alpha \\in (0, \\pi)$, if $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{3}}{3}$, calculate the value of $\\cos^2 \\alpha - \\sin^2 \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 9348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$ . Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\\ (b)$ . $(a)$ The point of intersection $Q$ of the line... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 9349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\overline{AB}$ is a diameter of a circle. Tangents $\\overline{AD}$ and $\\overline{BC}$ are drawn so that $\\overline{AC}$ and $\\overline{BD}$ intersect in a point on the circle. If $\\overline{AD}=a$ and $\\overline{BC}=b$, $a \\not= b$, the diameter of the circle is: Let'... | math | {
"style": "rule",
"ground_truth": "\\sqrt{ab}"
} | {
"split": "train",
"index": 9350,
"solution": "Let's analyze the problem and derive the solution step-by-step.\n\n1. **Identify the Geometry and Key Points**:\n - $\\overline{AB}$ is the diameter of the circle.\n - $\\overline{AD}$ and $\\overline{BC}$ are tangents to the circle at points $A$ and $B$ respect... |
deepscaler | [
{
"role": "user",
"content": "Given the arithmetic sequence $\\left\\{ a_n \\right\\}$ where each term is positive, the sum of the first $n$ terms is $S_n$. When $n \\in N^*, n \\geqslant 2$, it holds that $S_n = \\frac{n}{n-1}\\left( a_n^2 - a_1^2 \\right)$. Find the value of $S_{20} - 2S_{10}$. Let's thin... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 9351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\sin^2 0^\\circ + \\sin^2 10^\\circ + \\sin^2 20^\\circ + \\dots + \\sin^2 180^\\circ.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine identical bowling balls weigh the same as five identical canoes. If four of the canoes weigh a total of 120 pounds, how many pounds does one bowling ball weigh? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"split": "train",
"index": 9353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 9354,
"solution": "1. **Identify the number of games played within and outside the division**:\n Each team plays $N$ games against each of the three other teams in its division, resulting in $3N$ games within the division.\n Each team plays $M$ games against each of the four teams... |
deepscaler | [
{
"role": "user",
"content": "The base of a pyramid is a square. The height of the pyramid intersects the diagonal of the base. Find the maximum volume of such a pyramid if the perimeter of the diagonal cross-section that contains the height of the pyramid is 5. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 9355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \\ldots < d_k = n$ , $k \\geq 7$ , and for which the following equalities hold: $$ d_7 = 2d_5 + 1\\text{ and }d_7 = 3d_4 - 1 $$ *Proposed by Mykyta Kharin* Let's think step b... | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"split": "train",
"index": 9356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, side $AC = 900$ and side $BC = 600$. Points $K$ and $L$ are located on segment $AC$ and segment $AB$ respectively, such that $AK = CK$, and $CL$ is the angle bisector of $\\angle C$. Let point $P$ be the intersection of line segments $BK$ and $CL$. Point $M$ ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 9357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Given $\\tan \\frac{\\alpha}{2} = \\frac{1}{2}$, find the value of $\\sin\\left(\\alpha + \\frac{\\pi}{6}\\right)$.\n2. Given $\\alpha \\in \\left(\\pi, \\frac{3\\pi}{2}\\right)$ and $\\cos\\alpha = -\\frac{5}{13}$, $\\tan \\frac{\\beta}{2} = \\frac{1}{3}$, find the value of... | math | {
"style": "rule",
"ground_truth": "-\\frac{17\\sqrt{13}}{65}"
} | {
"split": "train",
"index": 9358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \\sqrt 3$ . We construct four semicircles $\\omega_1$ , $\\omega_2$ , $\\omega_3$ , $\\omega_4$ whose diameters are the segments $AB$ , $BC$ , $CD$ , $DA$ . It is given that $\\omega_i$ and $\\omega_{i+1}$ i... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 9359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately? Let's think step ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 9360,
"solution": "1. **Understanding the discount offer**: The store offers one free window for every four purchased. This effectively means a discount of $100$ dollars for every five windows bought, as the fifth window is free.\n\n2. **Calculating individual purchases**:\n - **Dav... |
deepscaler | [
{
"role": "user",
"content": "Given that the focus of the parabola $x^{2}=2py$ coincides with the lower focus of the ellipse $\\frac{x^{2}}{3}+\\frac{y^{2}}{4}=1$, find the value of $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a > c$. Given that $\\overrightarrow{BA} \\cdot \\overrightarrow{BC} = 2$ and $\\cos B = \\frac{1}{3}$, with $b=3$, find:\n\n$(1)$ The values of $a$ and $c$; $(2)$ The value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"split": "train",
"index": 9362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6,-12"
} | {
"split": "train",
"index": 9363,
"solution": "We must have $m^{2}+6 m+28=n^{2}$, where $n$ is an integer. Rewrite this as $(m+3)^{2}+19=$ $n^{2} \\Rightarrow n^{2}-(m+3)^{2}=19 \\Rightarrow(n-m-3)(n+m+3)=19$. Let $a=n-m-3$ and $b=n+m+3$, so we want $a b=19$. This leaves only 4 cases: - $a=1, b=19$. Solve the sy... |
deepscaler | [
{
"role": "user",
"content": "Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever p... | math | {
"style": "rule",
"ground_truth": "2982"
} | {
"split": "train",
"index": 9364,
"solution": "Knot can only lose if all of his cards are lower than 10; if not he can win by playing the lowest card that beats Gammadorf's card, or if this is not possible, his lowest card, each turn. There are $\\binom{7}{5}=21$ losing hands, so he has $\\binom{15}{5}-\\binom{7... |
deepscaler | [
{
"role": "user",
"content": "Circle I is externally tangent to Circle II and passes through the center of Circle II. Given that the area of Circle I is increased to 16 square inches, determine the area of Circle II, in square inches. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 9365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight balls are randomly and independently painted either black or white with equal probability. What is the probability that each ball is painted a color such that it is different from the color of at least half of the other 7 balls?\nA) $\\frac{21}{128}$\nB) $\\frac{28}{128}$... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{128}"
} | {
"split": "train",
"index": 9366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 9367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least odd prime factor of $2047^4 + 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 9368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an angle α with its vertex at the origin of coordinates, its initial side coinciding with the non-negative half-axis of the x-axis, and two points on its terminal side A(1,a), B(2,b), and cos(2α) = 2/3, determine the value of |a-b|. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 9369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \\mathrm{~km}$ on the rear wheel or $25000 \\mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the fr... | math | {
"style": "rule",
"ground_truth": "18750"
} | {
"split": "train",
"index": 9370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the five faces of the triangular prism shown here will be used as the base of a new pyramid. The numbers of exterior faces, vertices and edges of the resulting shape (the fusion of the prism and pyramid) are added. What is the maximum value of this sum?\n\n[asy]\ndraw((0... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 9371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 1}\\left(\\frac{e^{\\sin \\pi x}-1}{x-1}\\right)^{x^{2}+1}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi^2"
} | {
"split": "train",
"index": 9372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial of degree $13$ is divided by $d(x)$ to give a quotient of degree $7$ and a remainder of $3x^3+4x^2-x+12$. What is $\\deg d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 9373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Say that an integer $A$ is delicious if there exist several consecutive integers, including $A$, that add up to 2024. What is the smallest delicious integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2023"
} | {
"split": "train",
"index": 9374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "162,500"
} | {
"split": "train",
"index": 9375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factorize the number \\( 989 \\cdot 1001 \\cdot 1007 + 320 \\) into prime factors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "991 * 997 * 1009"
} | {
"split": "train",
"index": 9376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $5 \\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 9377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of $\\frac{20 \\cdot 21 \\cdot 22 \\cdot 23 \\cdot 24 \\cdot 25}{1000}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 9378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $x$ and $y$ satisfy\n\\begin{align*}\nx^2 + y^2 &= 2023 \n(x-2)(y-2) &= 3.\n\\end{align*}\nFind the largest possible value of $|x-y|$ .\n\n*Proposed by Howard Halim* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13\\sqrt{13}"
} | {
"split": "train",
"index": 9379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One out of every five Americans suffers from allergies. In a random sample of 250 Americans, how many would you expect to suffer from allergies? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 9380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{4}"
} | {
"split": "train",
"index": 9381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 9382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a natural number \\( N \\) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \\( N \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12005"
} | {
"split": "train",
"index": 9383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 9384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer solutions does the equation have $$ \\left\\lfloor\\frac{x}{10}\\right\\rfloor= \\left\\lfloor\\frac{x}{11}\\right\\rfloor + 1? $$ ( $\\lfloor x \\rfloor$ denotes the integer part of $x$ , for example $\\lfloor 2\\rfloor = 2$ , $\\lfloor \\pi\\... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 9385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ted's favorite number is equal to \\[1\\cdot\\binom{2007}1+2\\cdot\\binom{2007}2+3\\cdot\\binom{2007}3+\\cdots+2007\\cdot\\binom{2007}{2007}.\\] Find the remainder when Ted's favorite number is divided by $25$ . Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 9386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "373"
} | {
"split": "train",
"index": 9387,
"solution": "In both the numerator and the denominator, each factor is of the form $N^4+324=N^4+18^2$ for some positive integer $N.$\n We factor $N^4+18^2$ by completing the square, then applying the difference of squares: \\begin{align*} N^4+18^2&=\\left(N^4+36N^2+18^2\\right)-... |
deepscaler | [
{
"role": "user",
"content": "Given the functions $f(x)=2\\sin \\left(\\pi x+\\frac{\\pi }{3}\\right)$ and $g(x)=2\\cos \\left(\\pi x+\\frac{\\pi }{3}\\right)$, find the area of the triangle formed by the intersection points of the two functions within the interval $\\left[-\\frac{4}{3},\\frac{7}{6}\\right]... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 9388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\\overline{PA}\\perp \\overline{AD}$, $\\overline{PA}\\perp \\overline{AB}$, and $PB = 17$, then what is the volume of $PABCD$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 9389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four of the six numbers 1867, 1993, 2019, 2025, 2109, and 2121 have a mean (average) of 2008. What is the mean (average) of the other two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2051"
} | {
"split": "train",
"index": 9390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $\\left(\\frac{1}{2} \\right)^{3} \\cdot \\left(\\frac{1}{7} \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{56}"
} | {
"split": "train",
"index": 9391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $D$ lies on side $AC$ of isosceles triangle $ABC$ (where $AB = AC$) such that angle $BAC$ is $100^\\circ$ and angle $DBC$ is $30^\\circ$. Determine the ratio of the area of triangle $ADB$ to the area of triangle $CDB$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 9392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of possible values for \\( m \\) such that the lengths of the sides of a triangle are \\( \\ln 20 \\), \\( \\ln 60 \\), and \\( \\ln m \\), and the triangle has a positive area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1196"
} | {
"split": "train",
"index": 9393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the argument of the sum:\n\\[ e^{2\\pi i/40} + e^{6\\pi i/40} + e^{10\\pi i/40} + e^{14\\pi i/40} + e^{18\\pi i/40} + e^{22\\pi i/40} + e^{26\\pi i/40} + e^{30\\pi i/40} + e^{34\\pi i/40} + e^{38\\pi i/40} \\]\nand express it in the form \\( r e^{i \\theta} \\), where \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 9394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain cross country meet between 2 teams of 5 runners each, a runner who finishes in the $n$th position contributes $n$ to his teams score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?\n(A) 10... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 9395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex quadrilateral $ABCD$, $M$ and $N$ are the midpoints of sides $AD$ and $BC$, respectively. Given that $|\\overrightarrow{AB}|=2$, $|\\overrightarrow{MN}|=\\frac{3}{2}$, and $\\overrightarrow{MN} \\cdot (\\overrightarrow{AD} - \\overrightarrow{BC}) = \\frac{3}{2}$, fi... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 9397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "799"
} | {
"split": "train",
"index": 9398,
"solution": "The harmonic mean of $x$ and $y$ is equal to $\\frac{1}{\\frac{\\frac{1}{x}+\\frac{1}{y}}2} = \\frac{2xy}{x+y}$, so we have $xy=(x+y)(3^{20}\\cdot2^{19})$, and by SFFT, $(x-3^{20}\\cdot2^{19})(y-3^{20}\\cdot2^{19})=3^{40}\\cdot2^{38}$. Now, $3^{40}\\cdot2^{38}$ has ... |
deepscaler | [
{
"role": "user",
"content": "An electronic flea lands on a point $k$ on the number line. In the first step, it jumps 1 unit to the left to $k_1$, in the second step it jumps 2 units to the right to $k_2$, in the third step it jumps 3 units to the left to $k_3$, in the fourth step it jumps 4 units to the ri... | math | {
"style": "rule",
"ground_truth": "-30.06"
} | {
"split": "train",
"index": 9399,
"solution": ""
} |
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