data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Six distinct integers are picked at random from $\\{1,2,3,\\ldots,10\\}$. What is the probability that, among those selected, the second smallest is $3$?\n$\\textbf{(A)}\\ \\frac{1}{60}\\qquad \\textbf{(B)}\\ \\frac{1}{6}\\qquad \\textbf{(C)}\\ \\frac{1}{3}\\qquad \\textbf{(D)}\\ \\frac{1}{2}\\qqu... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "39300",
"split": "train"
} |
math | [
{
"content": "A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$? Please output the fina... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "39301",
"split": "train"
} |
math | [
{
"content": "Given two non-empty sets P and Q, define P+Q as the set \\{x | x = a + b, a \\in P, b \\in Q\\}. If P = \\{0, 2, 4\\} and Q = \\{1, 2, 3\\}, the sum of all elements in P+Q is \\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "39302",
"split": "train"
} |
math | [
{
"content": "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\... | {
"ground_truth": "\\frac{159}{323}",
"style": "rule"
} | {
"index": "39303",
"split": "train"
} |
math | [
{
"content": "Two subsets of the set $S=\\{a, b, c, d, e\\}$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? Please output the final answer within \\boxed... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "39304",
"split": "train"
} |
math | [
{
"content": "During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \\mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \\mathrm{m/s}$? (Air resistance is neglected.) Please output the final answer within \\b... | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "39305",
"split": "train"
} |
math | [
{
"content": "Consider two sets of consecutive integers. Let $A$ be the least common multiple (LCM) of the integers from $15$ to $25$ inclusive. Let $B$ be the least common multiple of $A$ and the integers $26$ to $45$. Compute the value of $\\frac{B}{A}$.\nA) 4536\nB) 18426\nC) 3 * 37 * 41 * 43 \nD) 1711\nE) 5... | {
"ground_truth": "3 \\cdot 37 \\cdot 41 \\cdot 43",
"style": "rule"
} | {
"index": "39306",
"split": "train"
} |
math | [
{
"content": "What is the sum of the positive odd divisors of $60$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39307",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $2^{x-3}=4^{x+1}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "39308",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of\n\\[\\frac{(x^2 + 3x + 1)(y^2 + 3y + 1)(z^2 + 3z + 1)}{xyz}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "39309",
"split": "train"
} |
math | [
{
"content": "Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three times. Please outpu... | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "39310",
"split": "train"
} |
math | [
{
"content": "Given that a flower bouquet contains pink roses, red roses, pink tulips, and red tulips, and that one fourth of the pink flowers are roses, one third of the red flowers are tulips, and seven tenths of the flowers are red, calculate the percentage of the flowers that are tulips. Please output the f... | {
"ground_truth": "46\\%",
"style": "rule"
} | {
"index": "39311",
"split": "train"
} |
math | [
{
"content": "Find $|3-2i|\\cdot |3+2i|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39312",
"split": "train"
} |
math | [
{
"content": "The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "39313",
"split": "train"
} |
math | [
{
"content": "In square \\(R S T U\\), a quarter-circle arc with center \\(S\\) is drawn from \\(T\\) to \\(R\\). A point \\(P\\) on this arc is 1 unit from \\(TU\\) and 8 units from \\(RU\\). What is the length of the side of square \\(RSTU\\)? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39314",
"split": "train"
} |
math | [
{
"content": "Let $r(x)$ be a monic quintic polynomial such that $r(1) = 3$, $r(2) = 7$, $r(3) = 13$, $r(4) = 21$, and $r(5) = 31$. Find $r(6)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "158",
"style": "rule"
} | {
"index": "39315",
"split": "train"
} |
math | [
{
"content": "Compute the prime factorization of $1007021035035021007001$ . (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\\ldots p_k^{e_k}$ where $p_1,\\ldots,p_k$ are distinct prime numbers and $e_1,\\ldots,e_k$ are positive integers.) Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "7^7 \\times 11^7 \\times 13^7",
"style": "rule"
} | {
"index": "39316",
"split": "train"
} |
math | [
{
"content": "Determine the maximum number of elements in the set \\( S \\) that satisfy the following conditions:\n1. Each element in \\( S \\) is a positive integer not exceeding 100;\n2. For any two different elements \\( a \\) and \\( b \\) in \\( S \\), there exists another element \\( c \\) in \\( S \\) s... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "39317",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin (3x+ \\frac {\\pi}{3})+\\cos (3x+ \\frac {\\pi}{6})+m\\sin 3x$ ($m\\in\\mathbb{R}$), and $f( \\frac {17\\pi}{18})=-1$ \n$(1)$ Find the value of $m$; \n$(2)$ In triangle $ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, if... | {
"ground_truth": "-3 \\sqrt {3}",
"style": "rule"
} | {
"index": "39318",
"split": "train"
} |
math | [
{
"content": "Circles with centers at $(5,5)$ and $(20,15)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 \\sqrt{13} - 20",
"style": "rule"
} | {
"index": "39319",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle. Please output the final... | {
"ground_truth": "9\\pi - \\frac{81\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "39320",
"split": "train"
} |
math | [
{
"content": "Find the largest root of the equation $|\\sin (2 \\pi x) - \\cos (\\pi x)| = ||\\sin (2 \\pi x)| - |\\cos (\\pi x)|$, which belongs to the interval $\\left(\\frac{1}{4}, 2\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "39321",
"split": "train"
} |
math | [
{
"content": "A one-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if the regulations stipulate that the stove should not be closer than 25 cm to the main gas pipe. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "9/16",
"style": "rule"
} | {
"index": "39322",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\sin x, \\sqrt{3}\\cos x)$, $\\overrightarrow{b}=(-1,1)$, and $\\overrightarrow{c}=(1,1)$, where $x \\in [0, \\pi]$.\n\n(1) If $(\\overrightarrow{a} + \\overrightarrow{b}) \\parallel \\overrightarrow{c}$, find the value of $x$;\n\n(2) If $\\overrightarrow{a} \\... | {
"ground_truth": "\\frac{\\sqrt{15}}{4}",
"style": "rule"
} | {
"index": "39323",
"split": "train"
} |
math | [
{
"content": "How many employees of the enterprise can be insured by the given insurance company if the enterprise is willing to pay an insurance premium of 5,000,000 rubles? \n\nThe cost of service for one person in the clinic (without hospitalization), with which the insurance company has a contract, averages... | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "39324",
"split": "train"
} |
math | [
{
"content": "Given an ellipse C: $\\frac{x^{2}}{3}+y^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects C at points A and B. If the area of $\\triangle F_{1}AB$ is twice the area of $\\triangle F_{2}AB$, find $m$. Please output the final answer within \\boxe... | {
"ground_truth": "-\\frac{\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "39325",
"split": "train"
} |
math | [
{
"content": "The values of a function $f(x)$ are given in the table below. \\begin{tabular}{|r||c|c|c|c|c|c|} \\hline $x$ & 1 & 2 & 3 & 5 & 8 & 13 \\\\ \\hline $f(x)$ & 3 & 13 & 8 & 1 & 0 & 5 \\\\ \\hline \\end{tabular}If $f^{-1}$ exists, what is $f^{-1}\\left(\\frac{f^{-1}(5) +f^{-1}(13)}{f^{-1}(1)}\\right)$... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39326",
"split": "train"
} |
math | [
{
"content": "There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples remain. How many ap... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "39327",
"split": "train"
} |
math | [
{
"content": "$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?\n\n(D. Fomin, Leningrad) Please out... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "39328",
"split": "train"
} |
math | [
{
"content": "Given that a point on the terminal side of angle \\(\\alpha\\) has coordinates \\((\\sin \\frac{2\\pi }{3},\\cos \\frac{2\\pi }{3})\\), find the smallest positive angle for \\(\\alpha\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11\\pi }{6}",
"style": "rule"
} | {
"index": "39329",
"split": "train"
} |
math | [
{
"content": "Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "39330",
"split": "train"
} |
math | [
{
"content": "Four tour guides are leading eight tourists. Each tourist must choose one of the guides, however, each guide must take at least two tourists. How many different groupings of guides and tourists are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "39331",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39332",
"split": "train"
} |
math | [
{
"content": "A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \\mathrm{~cm}^{2}$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39333",
"split": "train"
} |
math | [
{
"content": "Find the sum of all the solutions to $2^{|x|} + 3|x| = 18.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "39334",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": "39335",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that\n\\begin{align*}\na^2 + b^2 + c^2 + d^2 &= 4, \\\\\n(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16.\n\\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39336",
"split": "train"
} |
math | [
{
"content": "Compute $2 \\times 2^5 - 8^{58} \\div 8^{56}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "39337",
"split": "train"
} |
math | [
{
"content": "Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5040",
"style": "rule"
} | {
"index": "39338",
"split": "train"
} |
math | [
{
"content": "In the figure shown, the ratio of $BD$ to $DC$ is $4$ to $3$. The area of $\\triangle ABD$ is $24$ square centimeters. What is the area of $\\triangle ADC$? [asy] size(85); defaultpen(linewidth(1)+fontsize(10));\n\npair A = (0,5.5), B=(0,0), D = (2,0), C = (3,0);\n\ndraw(A--B--C--A--D); label(\"A\... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39339",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\( ABCD \\), point \\( X \\) is taken on the base \\( BC \\) such that segments \\( XA \\) and \\( XD \\) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \\( AB \\) has a length of 5. Find \\( XC \\cdot BX \\). Please output the final a... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "39340",
"split": "train"
} |
math | [
{
"content": "180 grams of 920 purity gold was alloyed with 100 grams of 752 purity gold. What is the purity of the resulting alloy? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "860",
"style": "rule"
} | {
"index": "39341",
"split": "train"
} |
math | [
{
"content": "If the line \\(x=\\frac{\\pi}{4}\\) intersects the curve \\(C: (x-\\arcsin a)(x-\\arccos a) + (y-\\arcsin a)(y+\\arccos a)=0\\), determine the minimum value of the chord length as \\(a\\) varies. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "39342",
"split": "train"
} |
math | [
{
"content": "Let \\( z \\) be a complex number. If \\( \\frac{z-2}{z-\\mathrm{i}} \\) (where \\( \\mathrm{i} \\) is the imaginary unit) is a real number, then the minimum value of \\( |z+3| \\) is \\(\\quad\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "39343",
"split": "train"
} |
math | [
{
"content": "For a positive integer \\( n \\), consider the function\n\n\\[ \nf(n)=\\frac{4n+\\sqrt{4n^2-1}}{\\sqrt{2n+1}+\\sqrt{2n-1}} \n\\]\n\nCalculate the value of\n\n\\[ \nf(1)+f(2)+f(3)+\\cdots+f(40) \n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "364",
"style": "rule"
} | {
"index": "39344",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, there is a curve $C_{1}: x+y=4$, and another curve $C_{2}$ defined by the parametric equations $\\begin{cases} x=1+\\cos \\theta, \\\\ y=\\sin \\theta \\end{cases}$ (with $\\theta$ as the parameter). A polar coordinate system is established with the origin... | {
"ground_truth": "\\dfrac{1}{4}(\\sqrt{2}+1)",
"style": "rule"
} | {
"index": "39345",
"split": "train"
} |
math | [
{
"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|. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\dfrac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "39346",
"split": "train"
} |
math | [
{
"content": "There are 10 horizontal roads and 10 vertical roads in a city, and they intersect at 100 crossings. Bob drives from one crossing, passes every crossing exactly once, and return to the original crossing. At every crossing, there is no wait to turn right, 1 minute wait to go straight, and 2 minutes ... | {
"ground_truth": "90 \\leq S<100",
"style": "rule"
} | {
"index": "39347",
"split": "train"
} |
math | [
{
"content": "In the triangular pyramid $A-BCD$, where $AB=AC=BD=CD=BC=4$, the plane $\\alpha$ passes through the midpoint $E$ of $AC$ and is perpendicular to $BC$, calculate the maximum value of the area of the section cut by plane $\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "39348",
"split": "train"
} |
math | [
{
"content": "Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "39349",
"split": "train"
} |
math | [
{
"content": "Compute $\\tan \\left(\\operatorname{arccot} \\frac{3}{5}\\right)$ and verify if $\\sin \\left(\\operatorname{arccot} \\frac{3}{5}\\right) > \\frac{1}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "39350",
"split": "train"
} |
math | [
{
"content": "Four-digit \"progressive numbers\" are arranged in ascending order, determine the 30th number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1359",
"style": "rule"
} | {
"index": "39351",
"split": "train"
} |
math | [
{
"content": "In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "39352",
"split": "train"
} |
math | [
{
"content": "There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly taken out, a white ball is put back in. What is the probability that all red balls are taken out exactly by the 4th draw? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "353/5000",
"style": "rule"
} | {
"index": "39353",
"split": "train"
} |
math | [
{
"content": "Compute \\[\n\\left\\lfloor \\frac{2007! + 2004!}{2006! + 2005!}\\right\\rfloor.\n\\](Note that $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2006",
"style": "rule"
} | {
"index": "39354",
"split": "train"
} |
math | [
{
"content": "Given a set of data $x_1, x_2, x_3, x_4, x_5$ with a mean of 8 and variance of 2, find the mean and variance of a new set of data: $4x_1+1, 4x_2+1, 4x_3+1, 4x_4+1, 4x_5+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39355",
"split": "train"
} |
math | [
{
"content": "Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "39356",
"split": "train"
} |
math | [
{
"content": "The minute hand on a clock points at the 12. After rotating $120^{\\circ}$ clockwise, which number will it point at? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39357",
"split": "train"
} |
math | [
{
"content": "A circle with center $O$ has area $156\\pi$. Triangle $ABC$ is equilateral, $\\overline{BC}$ is a chord on the circle, $OA = 4\\sqrt{3}$, and point $O$ is outside $\\triangle ABC$. What is the side length of $\\triangle ABC$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "$6$",
"style": "rule"
} | {
"index": "39358",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=4\\sin x\\cos \\left(x- \\frac {\\pi}{3}\\right)- \\sqrt {3}$. \n(I) Find the smallest positive period and zeros of $f(x)$. \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[ \\frac {\\pi}{24}, \\frac {3\\pi}{4}\\right]$. Please output the fina... | {
"ground_truth": "- \\sqrt {2}",
"style": "rule"
} | {
"index": "39359",
"split": "train"
} |
math | [
{
"content": "In quadrilateral \\( \\square ABCD \\), point \\( M \\) lies on diagonal \\( BD \\) with \\( MD = 3BM \\). Let \\( AM \\) intersect \\( BC \\) at point \\( N \\). Find the value of \\( \\frac{S_{\\triangle MND}}{S_{\\square ABCD}} \\). Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "39360",
"split": "train"
} |
math | [
{
"content": "How many ways are there to arrange the letters of the word $\\text{CA}_1\\text{N}_1\\text{A}_2\\text{N}_2\\text{A}_3\\text{T}_1\\text{T}_2$, where there are three A's, two N's, and two T's, with each A, N, and T considered distinct? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "5040",
"style": "rule"
} | {
"index": "39361",
"split": "train"
} |
math | [
{
"content": "Parallelepiped $ABCDEFGH$ is generated by vectors $\\overrightarrow{AB},$ $\\overrightarrow{AD},$ and $\\overrightarrow{AE},$ as shown below.\n\n[asy]\nimport three;\n\nsize(220);\ncurrentprojection = orthographic(0.5,0.3,0.2);\n\ntriple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);\ntriple ... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39362",
"split": "train"
} |
math | [
{
"content": "A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y+3x-4=0",
"style": "rule"
} | {
"index": "39363",
"split": "train"
} |
math | [
{
"content": "What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 16$ and $(x-5)^2 + 4y^2 = 64$?\nA) $\\frac{5\\sqrt{110}}{6}$ \nB) $\\frac{5\\sqrt{119}}{6}$ \nC) $\\frac{10\\sqrt{119}}{6}$ \nD) $\\frac{5\\sqrt{125}}{6}$ Please output the final answer within ... | {
"ground_truth": "\\frac{5\\sqrt{119}}{6}",
"style": "rule"
} | {
"index": "39364",
"split": "train"
} |
math | [
{
"content": "Twelve tiles numbered $1$ through $12$ are turned up at random, and an 8-sided die (sides numbered from 1 to 8) is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{48}",
"style": "rule"
} | {
"index": "39365",
"split": "train"
} |
math | [
{
"content": "Point A is a fixed point on a circle with a circumference of 3. If a point B is randomly selected on the circumference of the circle, the probability that the length of the minor arc is less than 1 is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "39366",
"split": "train"
} |
math | [
{
"content": "Given integer $n\\geq 2$. Find the minimum value of $\\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\\cdots$, ${a_n}$ and ${b}$,\n$$\\lambda\\sum\\limits_{i=1}^n\\sqrt{|a_i-b|}+\\sqrt{n\\left|\\sum\\limits_{i=1}^na_i\\right|}\\geqslant\\sum\\limits_{i=1}^n\\sqrt{|a_i|}.$$ Please ou... | {
"ground_truth": "\\frac{n-1 + \\sqrt{n-1}}{\\sqrt{n}}",
"style": "rule"
} | {
"index": "39367",
"split": "train"
} |
math | [
{
"content": "Stock investor Li Jin bought shares of a certain company last Saturday for $27 per share. The table below shows the price changes of the stock within the week.\n\n| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |\n|-----------------|--------|---------|-----------|--... | {
"ground_truth": "24.5",
"style": "rule"
} | {
"index": "39368",
"split": "train"
} |
math | [
{
"content": "Given that the moving point $P$ satisfies $|\\frac{PA}{PO}|=2$ with two fixed points $O(0,0)$ and $A(3,0)$, let the locus of point $P$ be curve $\\Gamma$. The equation of $\\Gamma$ is ______; the line $l$ passing through $A$ is tangent to $\\Gamma$ at points $M$, where $B$ and $C$ are two points o... | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "39369",
"split": "train"
} |
math | [
{
"content": "Given a regular triangular prism $ABC-A_1B_1C_1$ with side edges equal to the edges of the base, the sine of the angle formed by $AB_1$ and the lateral face $ACCA_1$ is equal to _______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{4}",
"style": "rule"
} | {
"index": "39370",
"split": "train"
} |
math | [
{
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$; $\\stackrel \\frown {AB} = 100^\\circ$ and $\\stackrel \\frown {BC} = 80^\\circ$.\nA point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\\a... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39371",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \\geq 2 a$. Compute the sum of all loose positive integers less than 100. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "39372",
"split": "train"
} |
math | [
{
"content": "Given \\\\(-\\pi < x < 0\\\\), \\\\(\\sin x + \\cos x = \\frac{1}{5}\\\\).\n\\\\((1)\\\\) Find the value of \\\\(\\sin x - \\cos x\\\\);\n\\\\((2)\\\\) Find the value of \\\\(\\frac{3\\sin^2 \\frac{x}{2} - 2\\sin \\frac{x}{2}\\cos \\frac{x}{2} + \\cos^2 \\frac{x}{2}}{\\tan x + \\frac{1}{\\tan x}}\... | {
"ground_truth": "-\\frac{108}{125}",
"style": "rule"
} | {
"index": "39373",
"split": "train"
} |
math | [
{
"content": "What is the minimum number of participants that could have been in the school drama club if fifth-graders constituted more than $25\\%$, but less than $35\\%$; sixth-graders more than $30\\%$, but less than $40\\%$; and seventh-graders more than $35\\%$, but less than $45\\%$ (there were no partic... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39374",
"split": "train"
} |
math | [
{
"content": "You have 5 identical buckets, each with a maximum capacity of some integer number of liters, and a 30-liter barrel containing an integer number of liters of water. All the water from the barrel was poured into the buckets, with the first bucket being half full, the second one-third full, the third... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "39375",
"split": "train"
} |
math | [
{
"content": "Given two unit vectors \\\\( \\overrightarrow{a} \\\\) and \\\\( \\overrightarrow{b} \\\\). If \\\\( |3 \\overrightarrow{a} - 2 \\overrightarrow{b}| = 3 \\\\), find the value of \\\\( |3 \\overrightarrow{a} + \\overrightarrow{b}| \\\\). Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2 \\sqrt {3}",
"style": "rule"
} | {
"index": "39376",
"split": "train"
} |
math | [
{
"content": "Given that a bin contains 10 kg of peanuts, 2 kg of peanuts are removed and 2 kg of raisins are added and thoroughly mixed in, and then 2 kg of this mixture are removed and 2 kg of raisins are added and thoroughly mixed in again, determine the ratio of the mass of peanuts to the mass of raisins in... | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "39377",
"split": "train"
} |
math | [
{
"content": "Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number of toothpicks used.... | {
"ground_truth": "850",
"style": "rule"
} | {
"index": "39378",
"split": "train"
} |
math | [
{
"content": "What is the greatest integer less than or equal to \\[\\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15624",
"style": "rule"
} | {
"index": "39379",
"split": "train"
} |
math | [
{
"content": "Read the following material: The overall idea is a common thinking method in mathematical problem solving: Here is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Regard \"$x^{2}+2x$\" as a whole, let $x^{2}+2x=y$, then the original expression $=y^{2}+2y+1=\\left(y+1\... | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "39380",
"split": "train"
} |
math | [
{
"content": "The digits of 2021 can be rearranged to form other four-digit whole numbers between 1000 and 3000. Find the largest possible difference between two such four-digit whole numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1188",
"style": "rule"
} | {
"index": "39381",
"split": "train"
} |
math | [
{
"content": "Given the parametric equation of line $l$ as $\\begin{cases} x = \\frac{\\sqrt{2}}{2}t \\\\ y = \\frac{\\sqrt{2}}{2}t + 4\\sqrt{2} \\end{cases}$ (where $t$ is the parameter) and the polar equation of circle $C$ as $\\rho = 2\\cos (\\theta + \\frac{\\pi}{4})$,\n\n(I) Find the rectangular coordinate... | {
"ground_truth": "2\\sqrt {6}",
"style": "rule"
} | {
"index": "39382",
"split": "train"
} |
math | [
{
"content": "A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \\mathrm{~km/h}$. If they run at full speed eithe... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39383",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\(ABCD\\), the bases \\(AD\\) and \\(BC\\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \\(ABD\\) is tangent to lines \\(BC\\) and \\(CD\\). Find the perimeter of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "39384",
"split": "train"
} |
math | [
{
"content": "Given that the broad money supply $\\left(M2\\right)$ balance was 2912000 billion yuan, express this number in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.912 \\times 10^{6}",
"style": "rule"
} | {
"index": "39385",
"split": "train"
} |
math | [
{
"content": "Given the angle $\\frac {19\\pi}{5}$, express it in the form of $2k\\pi+\\alpha$ ($k\\in\\mathbb{Z}$), then determine the angle $\\alpha$ that makes $|\\alpha|$ the smallest. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac {\\pi}{5}",
"style": "rule"
} | {
"index": "39386",
"split": "train"
} |
math | [
{
"content": "Given $α-β=\\frac{π}{3}$ and $tanα-tanβ=3\\sqrt{3}$, calculate the value of $\\cos \\left(\\alpha +\\beta \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{6}",
"style": "rule"
} | {
"index": "39387",
"split": "train"
} |
math | [
{
"content": "Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\\frac{A}{B}$? Please output the final a... | {
"ground_truth": "\\frac{15}{8}",
"style": "rule"
} | {
"index": "39388",
"split": "train"
} |
math | [
{
"content": "The formula $N=8 \\times 10^{8} \\times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars.\nThe lowest income, in dollars, of the wealthiest $800$ individuals is at least: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10^4",
"style": "rule"
} | {
"index": "39389",
"split": "train"
} |
math | [
{
"content": "In the trapezoid \\(ABCD\\) with bases \\(AD \\parallel BC\\), the diagonals intersect at point \\(E\\). Given the areas \\(S(\\triangle ADE) = 12\\) and \\(S(\\triangle BCE) = 3\\), find the area of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "39390",
"split": "train"
} |
math | [
{
"content": "Jeremy made a Venn diagram showing the number of students in his class who own types of pets. There are 32 students in his class. In addition to the information in the Venn diagram, Jeremy knows half of the students have a dog, $\\frac{3}{8}$ have a cat, six have some other pet and five have no pe... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39391",
"split": "train"
} |
math | [
{
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $\\cos A= \\frac {3}{4}$, $\\cos C= \\frac {1}{8}$,\n(I) find the ratio $a:b:c$;\n(II) if $| \\overrightarrow{AC}+ \\overrightarrow{BC}|= \\sqrt {46}$, find the area of $\\triangle ... | {
"ground_truth": "\\frac {15 \\sqrt {7}}{4}",
"style": "rule"
} | {
"index": "39392",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. Pl... | {
"ground_truth": "\\frac{60}{11}",
"style": "rule"
} | {
"index": "39393",
"split": "train"
} |
math | [
{
"content": "Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "397",
"style": "rule"
} | {
"index": "39394",
"split": "train"
} |
math | [
{
"content": "Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39395",
"split": "train"
} |
math | [
{
"content": "Find the sum of the rational roots of $g(x)=x^3-9x^2+16x-4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39396",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 60^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "39397",
"split": "train"
} |
math | [
{
"content": "If $\\frac{1}{x} - \\frac{1}{y} = \\frac{1}{z}$, then $z$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{xy}{y - x}",
"style": "rule"
} | {
"index": "39398",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.\n\n(Here $n!$ denotes $1\\times\\cdots\\times n$ for each positive integer $n$.) Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39399",
"split": "train"
} |
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