data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Liam has $x$ candies, Mia has three times as many candies as Liam, Noah has four times as many candies as Mia, and Olivia has six times the number of candies Noah has. If in total Liam, Mia, Noah, and Olivia have 468 candies, what is the value of $x?$ Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{117}{22}",
"style": "rule"
} | {
"index": "32500",
"split": "train"
} |
math | [
{
"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 $\\{a_n\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "32501",
"split": "train"
} |
math | [
{
"content": "Determine the periodicity of the following functions. If the function is periodic, find its smallest positive period:\n(1) \\( y = \\tan x - \\cot x \\);\n(2) \\( y = \\sin (\\cos x) \\);\n(3) \\( y = \\sin x^{2} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "32502",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ with the property that in the set $\\{70, 71, 72,... 70 + n\\}$ you can choose two different numbers whose product is the square of an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "32503",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^{2}}{m} + \\frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \\pm \\sqrt{2}x$, calculate the hyperbola's eccentricity. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "32504",
"split": "train"
} |
math | [
{
"content": "Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9\\pi",
"style": "rule"
} | {
"index": "32505",
"split": "train"
} |
math | [
{
"content": "Given that the coefficient of the second term in the expansion of $((x+2y)^{n})$ is $8$, find the sum of the coefficients of all terms in the expansion of $((1+x)+(1+x)^{2}+…+(1+x)^{n})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32506",
"split": "train"
} |
math | [
{
"content": "An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "32507",
"split": "train"
} |
math | [
{
"content": "In the trapezoid \\(ABCD\\), if \\(AB = 8\\), \\(DC = 10\\), the area of \\(\\triangle AMD\\) is 10, and the area of \\(\\triangle BCM\\) is 15, then the area of trapezoid \\(ABCD\\) is \\(\\quad\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "32508",
"split": "train"
} |
math | [
{
"content": "What is the value of the sum $\\frac{3}{4}+\\frac{3^2}{4^2}+\\frac{3^3}{4^3}+ \\ldots +\\frac{3^{15}}{4^{15}}$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3177884751}{1073741824}",
"style": "rule"
} | {
"index": "32509",
"split": "train"
} |
math | [
{
"content": "Determine all four-digit numbers $\\overline{abcd}$ which are perfect squares and for which the equality holds: $\\overline{ab}=3 \\cdot \\overline{cd} + 1$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2809",
"style": "rule"
} | {
"index": "32510",
"split": "train"
} |
math | [
{
"content": "Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15. Pl... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "32511",
"split": "train"
} |
math | [
{
"content": "When 2007 bars of soap are packed into \\( N \\) boxes, where \\( N \\) is a positive integer, there is a remainder of 5. How many possible values of \\( N \\) are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "32512",
"split": "train"
} |
math | [
{
"content": "Let \\( g(x) = 10x + 5 \\). Find the sum of all \\( x \\) that satisfy the equation \\( g^{-1}(x) = g((3x)^{-2}) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "32513",
"split": "train"
} |
math | [
{
"content": "Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch? Please output the final an... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "32514",
"split": "train"
} |
math | [
{
"content": "In a class, there are 4 lessons in one morning, and each lesson needs a teacher to teach it. Now, from 6 teachers A, B, C, D, E, F, 4 teachers are to be arranged to teach one lesson each. The first lesson can only be taught by either A or B, and the fourth lesson can only be taught by either A or ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "32515",
"split": "train"
} |
math | [
{
"content": "The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \\in (0, +\\infty)$. The value of the real number $m$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32516",
"split": "train"
} |
math | [
{
"content": "Given that \\(15^{-1} \\equiv 31 \\pmod{53}\\), find \\(38^{-1} \\pmod{53}\\), as a residue modulo 53. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "32517",
"split": "train"
} |
math | [
{
"content": "Let \\(ABC\\) be a triangle with \\(AB=8, AC=12\\), and \\(BC=5\\). Let \\(M\\) be the second intersection of the internal angle bisector of \\(\\angle BAC\\) with the circumcircle of \\(ABC\\). Let \\(\\omega\\) be the circle centered at \\(M\\) tangent to \\(AB\\) and \\(AC\\). The tangents to \... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "32518",
"split": "train"
} |
math | [
{
"content": "Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy $$x+y+z=-\\frac{\\sqrt{3}}{2}-i \\sqrt{5}$$ and $$x y z=\\sqrt{3}+i \\sqrt{5}.$$ If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then $$\\left(x_... | {
"ground_truth": "1516",
"style": "rule"
} | {
"index": "32519",
"split": "train"
} |
math | [
{
"content": "In square $ABCD$, points $E$ and $H$ lie on $\\overline{AB}$ and $\\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\\overline{BC}$ and $\\overline{CD}$, respectively, and points $I$ and $J$ lie on $\\overline{EH}$ so that $\\overline{FI} \\perp \\overline{EH}$ and $\\over... | {
"ground_truth": "8-4\\sqrt{2}",
"style": "rule"
} | {
"index": "32520",
"split": "train"
} |
math | [
{
"content": "Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three fact... | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "32521",
"split": "train"
} |
math | [
{
"content": "If the solution set of the inequality $tx^2-6x+t^2<0$ with respect to $x$ is $(-\\infty, a) \\cup (1, +\\infty)$, then the value of $a$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "32522",
"split": "train"
} |
math | [
{
"content": "The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were randomly selected, wi... | {
"ground_truth": "\\dfrac{2}{5}",
"style": "rule"
} | {
"index": "32523",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^2 + b^2 = 2017c^2$, calculate the value of $\\frac{\\tan C}{\\tan A} + \\frac{\\tan C}{\\tan B}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1008}",
"style": "rule"
} | {
"index": "32524",
"split": "train"
} |
math | [
{
"content": "Let \\( g(x) \\) be the function defined on \\(-2 \\le x \\le 2\\) by the formula \n\\[ g(x) = 2 - \\sqrt{4 - x^2}. \\]\nIf a graph of \\( x = g(y) \\) is overlaid on the graph of \\( y = g(x) \\), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded... | {
"ground_truth": "2.28",
"style": "rule"
} | {
"index": "32525",
"split": "train"
} |
math | [
{
"content": "If $z=1+i$, then $|{iz+3\\overline{z}}|=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "32526",
"split": "train"
} |
math | [
{
"content": "In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and additionally ensure my bike key is next to my mailbox key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32527",
"split": "train"
} |
math | [
{
"content": "What is the tens digit of the smallest positive integer that is divisible by each of 20, 16, and 2016? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "32528",
"split": "train"
} |
math | [
{
"content": "A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1462",
"style": "rule"
} | {
"index": "32529",
"split": "train"
} |
math | [
{
"content": "In a hypothetical math competition, contestants are given the problem to find three distinct positive integers $X$, $Y$, and $Z$ such that their product $X \\cdot Y \\cdot Z = 399$. What is the largest possible value of the sum $X+Y+Z$? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "32530",
"split": "train"
} |
math | [
{
"content": "For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$. \n$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$. \n$(2)$ The inequality $t\\leqslant \\left(m-1\\right)^{2}+\\left(n-1\\right)^{2}+\\left(m-n\\right)^... | {
"ground_truth": "\\frac{9}{8}",
"style": "rule"
} | {
"index": "32531",
"split": "train"
} |
math | [
{
"content": "The graph of the equation $y = \\frac{x}{x^3 + Ax^2 + Bx + C}$, where $A,B,C$ are integers, is shown below. Find $A + B + C$. [asy]\nimport graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.52,xmax=4.62,ymin=-3.66,ymax=3.94;\n\npen ... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32532",
"split": "train"
} |
math | [
{
"content": "Find $\\overrightarrow{a}+2\\overrightarrow{b}$, where $\\overrightarrow{a}=(2,0)$ and $|\\overrightarrow{b}|=1$, and then calculate the magnitude of this vector. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "32533",
"split": "train"
} |
math | [
{
"content": "Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "32534",
"split": "train"
} |
math | [
{
"content": "In the sequence $00$ , $01$ , $02$ , $03$ , $\\cdots$ , $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$ , $39$ , or $28$ , but not by $30$ or $20$ ). W... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "32535",
"split": "train"
} |
math | [
{
"content": "Given that the length of the major axis of the ellipse is 4, the left vertex is on the parabola \\( y^2 = x - 1 \\), and the left directrix is the y-axis, find the maximum value of the eccentricity of such an ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "32536",
"split": "train"
} |
math | [
{
"content": "It is known that the number of birch trees in a certain mixed forest plot ranges from $13\\%$ to $14\\%$ of the total number of trees. Find the minimum possible total number of trees in this plot. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "32537",
"split": "train"
} |
math | [
{
"content": "A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circum... | {
"ground_truth": ".2",
"style": "rule"
} | {
"index": "32538",
"split": "train"
} |
math | [
{
"content": "Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?\nA) $\\frac{1}{4}$\nB) $\\frac{57}{64}$\nC) $\\frac{199}{256}$\nD) $\\frac{57}{256}$\nE) $\\frac{63}{64}$ Please output the final answe... | {
"ground_truth": "\\frac{199}{256}",
"style": "rule"
} | {
"index": "32539",
"split": "train"
} |
math | [
{
"content": "Find all the solutions to\n\\[\\sqrt[3]{15x - 1} + \\sqrt[3]{13x + 1} = 4 \\sqrt[3]{x}.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0, \\frac{1}{14}, -\\frac{1}{12}",
"style": "rule"
} | {
"index": "32540",
"split": "train"
} |
math | [
{
"content": "Ten 6-sided dice are rolled. What is the probability that exactly three of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": ".155",
"style": "rule"
} | {
"index": "32541",
"split": "train"
} |
math | [
{
"content": "Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\pi",
"style": "rule"
} | {
"index": "32542",
"split": "train"
} |
math | [
{
"content": "Given that the solution set of the inequality $x^{2}-2x+1-m^{2} \\lt 0$ is $A$; \n$(1)$ Find $A$; \n$(2)$ If $0 \\lt m \\lt 1$, and $A=\\{x\\left|\\right.a \\lt x \\lt b\\}$, find the minimum value of $\\frac{1}{{8a+2b}}-\\frac{1}{{3a-3b}}$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "32543",
"split": "train"
} |
math | [
{
"content": "Determine the value of the expression\n\\[\\log_2 (27 + \\log_2 (27 + \\log_2 (27 + \\cdots))),\\]assuming it is positive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32544",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-112",
"style": "rule"
} | {
"index": "32545",
"split": "train"
} |
math | [
{
"content": "Express as a common fraction: $(0.\\overline{09})(0.\\overline{7})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{99}",
"style": "rule"
} | {
"index": "32546",
"split": "train"
} |
math | [
{
"content": "In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelog... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32547",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120 < n \\leq 130",
"style": "rule"
} | {
"index": "32548",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x) = \\frac{2+x}{1+x} \\), let \\( f(1) + f(2) + \\cdots + f(1000) = m \\) and \\( f\\left(\\frac{1}{2}\\right) + f\\left(\\frac{1}{3}\\right) + \\cdots + f\\left(\\frac{1}{1000}\\right) = n \\). What is the value of \\( m + n \\)? Please output the final answer within \\b... | {
"ground_truth": "2998.5",
"style": "rule"
} | {
"index": "32549",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-3)(x+3) = 21x - 63.$$If $p > q$, what is the value of $p - q$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "32550",
"split": "train"
} |
math | [
{
"content": "Let $S_{7}$ denote all the permutations of $1,2, \\ldots, 7$. For any \\pi \\in S_{7}$, let $f(\\pi)$ be the smallest positive integer $i$ such that \\pi(1), \\pi(2), \\ldots, \\pi(i)$ is a permutation of $1,2, \\ldots, i$. Compute \\sum_{\\pi \\in S_{7}} f(\\pi)$. Please output the final answer w... | {
"ground_truth": "29093",
"style": "rule"
} | {
"index": "32551",
"split": "train"
} |
math | [
{
"content": "Let $E(n)$ denote the sum of the even digits of $n$. Modify $E(n)$ such that if $n$ is prime, $E(n)$ is counted as zero, and if $n$ is not prime, $E(n)$ is counted twice. Calculate $E'(1)+E'(2)+E'(3)+\\cdots+E'(200)$.\nA) 1200\nB) 1320\nC) 1360\nD) 1400\nE) 1500 Please output the final answer with... | {
"ground_truth": "1360",
"style": "rule"
} | {
"index": "32552",
"split": "train"
} |
math | [
{
"content": "If the system of equations\n\n\\begin{align*}\n8x - 6y &= c, \\\\\n10y - 15x &= d.\n\\end{align*}\n\nhas a solution $(x,y)$ where $x$ and $y$ are both nonzero, find $\\frac{c}{d},$ assuming $d$ is nonzero. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{5}",
"style": "rule"
} | {
"index": "32553",
"split": "train"
} |
math | [
{
"content": "A certain number is written in the base-12 numeral system. For which divisor \\( m \\) is the following divisibility rule valid: if the sum of the digits of the number is divisible by \\( m \\), then the number itself is divisible by \\( m \\)? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32554",
"split": "train"
} |
math | [
{
"content": "Given that at the beginning of the year, 40% of students answered \"Yes\", 40% answered \"No\", and 20% were undecided, while at the end of the year, 60% answered \"Yes\", 30% answered \"No\", and 10% remained undecided, find the difference between the maximum and minimum possible values of y%, th... | {
"ground_truth": "60\\%",
"style": "rule"
} | {
"index": "32555",
"split": "train"
} |
math | [
{
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\\left[0, \\frac{\\pi}{4}\\right]$. Please output the final answer within \\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32556",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis, using the same unit of length in both coordinate systems. Given that circle $C$ has a center at point ($2$, $\\frac{7π}{6}$) i... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "32557",
"split": "train"
} |
math | [
{
"content": "There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? Please output the final a... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32558",
"split": "train"
} |
math | [
{
"content": "Given that sinα + cosα = $\\frac{7}{5}$, find the value of tanα. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "32559",
"split": "train"
} |
math | [
{
"content": "Let \\( S = \\{1, 2, 3, \\ldots, 30\\} \\). Determine the number of vectors \\((x, y, z, w)\\) with \\(x, y, z, w \\in S\\) such that \\(x < w\\) and \\(y < z < w\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90335",
"style": "rule"
} | {
"index": "32560",
"split": "train"
} |
math | [
{
"content": "A line parallel to leg \\(AC\\) of right triangle \\(ABC\\) intersects leg \\(BC\\) at point \\(K\\) and the hypotenuse \\(AB\\) at point \\(N\\). On leg \\(AC\\), a point \\(M\\) is chosen such that \\(MK = MN\\). Find the ratio \\(\\frac{AM}{MC}\\) if \\(\\frac{BK}{BC} = 14\\). Please output the... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "32561",
"split": "train"
} |
math | [
{
"content": "A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32562",
"split": "train"
} |
math | [
{
"content": "$\\triangle DEF$ is inscribed inside $\\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\\triangle DEC, \\triangle BFD, \\triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\\stackrel{\\frown}{BF} = \\stackrel{\\fr... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "32563",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\( \\triangle ABC \\) with sides \\( a, b, c \\) and corresponding medians \\( m_a, m_b, m_c \\), and angle bisectors \\( w_a, w_b, w_c \\). Let \\( w_a \\cap m_b = P \\), \\( w_b \\cap m_c = Q \\), and \\( w_c \\cap m_a = R \\). Denote the area of \\( \\triangle PQR \\) by \\( \... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "32564",
"split": "train"
} |
math | [
{
"content": "A science student is asked to find the coefficient of the $x^2$ term in the expansion of $(x^2-3x+2)^4$. The coefficient is \\_\\_\\_\\_\\_\\_. (Answer with a number) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248",
"style": "rule"
} | {
"index": "32565",
"split": "train"
} |
math | [
{
"content": "Given that $\\tan \\alpha = -\\frac{1}{3}$ and $\\cos \\beta = \\frac{\\sqrt{5}}{5}$, with $\\alpha, \\beta \\in (0, \\pi)$, find:\n1. The value of $\\tan(\\alpha + \\beta)$;\n2. The maximum value of the function $f(x) = \\sqrt{2} \\sin(x - \\alpha) + \\cos(x + \\beta)$. Please output the final an... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "32566",
"split": "train"
} |
math | [
{
"content": "Let a line passing through the origin \\\\(O\\\\) intersect a circle \\\\((x-4)^{2}+y^{2}=16\\\\) at point \\\\(P\\\\), and let \\\\(M\\\\) be the midpoint of segment \\\\(OP\\\\). Establish a polar coordinate system with the origin \\\\(O\\\\) as the pole and the positive half-axis of \\\\(x\\\\)... | {
"ground_truth": "3+ \\dfrac {3}{2} \\sqrt {3}",
"style": "rule"
} | {
"index": "32567",
"split": "train"
} |
math | [
{
"content": "When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "32568",
"split": "train"
} |
math | [
{
"content": "We are approaching a 120-meter high skyscraper on a horizontal road. After traveling 300 meters, we see the building at an angle of elevation that is $45^\\circ$ greater than at the start of our journey. How close have we approached the skyscraper? Please output the final answer within \\boxed{}."... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32569",
"split": "train"
} |
math | [
{
"content": "If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 35, what is the probability that this number will be divisible by 11?\nA) $\\frac{1}{4}$\nB) $\\frac{1}{8}$\nC) $\\frac{1}{5}$\nD) $\\frac{1}{10}$\nE) $\\frac{1}{15}$ Please output t... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "32570",
"split": "train"
} |
math | [
{
"content": "What is the positive difference between the two largest prime factors of $159137$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "32571",
"split": "train"
} |
math | [
{
"content": "Calculate: \n\n$$\\frac{\\left(1+\\frac{1}{2}\\right)^{2} \\times\\left(1+\\frac{1}{3}\\right)^{2} \\times\\left(1+\\frac{1}{4}\\right)^{2} \\times\\left(1+\\frac{1}{5}\\right)^{2} \\times \\cdots \\times\\left(1+\\frac{1}{10}\\right)^{2}}{\\left(1-\\frac{1}{2^{2}}\\right) \\times\\left(1-\\frac{1... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "32572",
"split": "train"
} |
math | [
{
"content": "Given \\( x_{i}=\\frac{i}{101} \\), find the value of \\( S=\\sum_{i=1}^{101} \\frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "32573",
"split": "train"
} |
math | [
{
"content": "Determine the area enclosed by the curves \\( y = \\sin x \\) and \\( y = \\left(\\frac{4}{\\pi}\\right)^{2} \\sin \\left(\\frac{\\pi}{4}\\right) x^{2} \\) (the latter is a quadratic function). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}\\left(1 + \\frac{\\pi}{12}\\right)",
"style": "rule"
} | {
"index": "32574",
"split": "train"
} |
math | [
{
"content": "In a class, each student has either 5 or 6 friends (friendship is mutual), and any two friends have a different number of friends. What is the minimum number of students, greater than 0, that can be in the class? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32575",
"split": "train"
} |
math | [
{
"content": "A function \\( g(x) \\) is defined for all real numbers \\( x \\). For all non-zero values \\( x \\), we have\n\\[ 3g(x) + g\\left(\\frac{1}{x}\\right) = 7x + 6. \\]\nLet \\( T \\) denote the sum of all of the values of \\( x \\) for which \\( g(x) = 2005 \\). Compute the integer nearest to \\( T ... | {
"ground_truth": "763",
"style": "rule"
} | {
"index": "32576",
"split": "train"
} |
math | [
{
"content": "Max bought a new dirt bike and paid $10\\%$ of the cost upfront, which was $\\$150$. What was the price of the bike? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\$ 1500",
"style": "rule"
} | {
"index": "32577",
"split": "train"
} |
math | [
{
"content": "Suppose that $a$ and $b$ are integers with $4<a<b<22$. If the average (mean) of the numbers $4, a, b, 22$ is 13, how many possible pairs $(a, b)$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "32578",
"split": "train"
} |
math | [
{
"content": "A valid license plate in Xanadu consists of two letters followed by three digits. How many valid license plates are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "676,\\!000",
"style": "rule"
} | {
"index": "32579",
"split": "train"
} |
math | [
{
"content": "The distribution of populations in a group of counties is shown in this pie chart. What percent of the counties have fewer than 100,000 residents?\n\n[asy]\ndraw(Circle((0,0),25),linewidth(1));\ndraw((-24,-7)--(0,0)--(24,-7),linewidth(1));\ndraw((0,0)--(7,-24),linewidth(1));\nlabel(\"59\\%\",(0,12... | {
"ground_truth": "84\\%",
"style": "rule"
} | {
"index": "32580",
"split": "train"
} |
math | [
{
"content": "Let $g(x) = 3x^4 + 2x^3 - x^2 - 4x + s$. For what value of $s$ is $g(-1) = 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "32581",
"split": "train"
} |
math | [
{
"content": "Function $f(x, y): \\mathbb N \\times \\mathbb N \\to \\mathbb Q$ satisfies the conditions:\n\n(i) $f(1, 1) =1$ ,\n\n(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \\in \\mathbb N$ , and\n\n(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \\in \\mathbb N$ .\n\nFind $f(1990, ... | {
"ground_truth": "\\frac{30! \\cdot 1989!}{2020!}",
"style": "rule"
} | {
"index": "32582",
"split": "train"
} |
math | [
{
"content": "The number of cans in the layers of a display in a supermarket form an arithmetic sequence. The bottom layer has 28 cans; the next layer has 25 cans and so on until there is one can at the top of the display. How many cans are in the entire display? Please output the final answer within \\boxed{}.... | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "32583",
"split": "train"
} |
math | [
{
"content": "A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "32584",
"split": "train"
} |
math | [
{
"content": "How many positive perfect squares less than \\(10^8\\) are multiples of 36? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1666",
"style": "rule"
} | {
"index": "32585",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 18 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $G$ is the midpoint of $CD$. Determine the area of triangle $EFG$. Please output the fin... | {
"ground_truth": "52.5",
"style": "rule"
} | {
"index": "32586",
"split": "train"
} |
math | [
{
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "32587",
"split": "train"
} |
math | [
{
"content": "Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid below so that a path is formed from the upper left-hand corner $A$ to the lower righthand corner $B$. In a path, consecutive dominoes must touch at their sides and not just their corners. No domino may... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "32588",
"split": "train"
} |
math | [
{
"content": "A palindrome is a string that does not change when its characters are written in reverse order. Let S be a 40-digit string consisting only of 0's and 1's, chosen uniformly at random out of all such strings. Let $E$ be the expected number of nonempty contiguous substrings of $S$ which are palindrom... | {
"ground_truth": "113",
"style": "rule"
} | {
"index": "32589",
"split": "train"
} |
math | [
{
"content": "Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[3.14] = 3$. Then, find the value of $\\left[\\frac{2017 \\times 3}{11}\\right] + \\left[\\frac{2017 \\times 4}{11}\\right] + \\left[\\frac{2017 \\times 5}{11}\\right] + \\left[\\frac{2017 \\times 6}{11}\\right] + \\left[\\frac... | {
"ground_truth": "6048",
"style": "rule"
} | {
"index": "32590",
"split": "train"
} |
math | [
{
"content": "In right $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 12$, $BC = 35$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\overline{AI}$ and $\\overline{BI}$... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32591",
"split": "train"
} |
math | [
{
"content": "The probability that event $A$ occurs is $\\frac{3}{4}$; the probability that event B occurs is $\\frac{2}{3}$. \nLet $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "[\\frac{5}{12},\\frac{2}{3}]",
"style": "rule"
} | {
"index": "32592",
"split": "train"
} |
math | [
{
"content": "Given that $α\\in\\mathbb{R}$ and $\\sin α + 2\\cos α = \\frac{\\sqrt{10}}{2}$, find the value of $\\tan α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "32593",
"split": "train"
} |
math | [
{
"content": "Let $a \\oslash b = (\\sqrt{2a+b})^3$. If $4 \\oslash x = 27$, find the value of $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32594",
"split": "train"
} |
math | [
{
"content": "In a right triangle $PQR$, where $\\angle P = 90^\\circ$, suppose $\\cos Q = \\frac{4}{5}$. The length of side $PQ$ (adjacent to $\\angle Q$) is $12$. What is the length of $PR$ (hypotenuse)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "32595",
"split": "train"
} |
math | [
{
"content": "When simplified, what is the value of $$(10^{0.5})(10^{0.3})(10^{0.2})(10^{0.1})(10^{0.9})?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "32596",
"split": "train"
} |
math | [
{
"content": "What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32597",
"split": "train"
} |
math | [
{
"content": "Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\\angle A$, $\\angle B$, and $\\angle C$ are $134^{\\circ}$, $106^{\\circ}$, and $134^{\\circ}$ respectively. Find the measure of the internal angle $\\angle E$. Please output the final answer within \\boxed... | {
"ground_truth": "134",
"style": "rule"
} | {
"index": "32598",
"split": "train"
} |
math | [
{
"content": "Petya wrote a natural number \\( A \\) on the board. If you multiply it by 8, you get the square of a natural number. How many such three-digit numbers \\( B \\) exist for which \\( A \\cdot B \\) is also a square of a natural number? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "32599",
"split": "train"
} |
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