data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "On the same road, two trucks are driving in the same direction. Initially, Truck A is 4 kilometers ahead of Truck B. The speed of Truck A is 45 kilometers per hour, and the speed of Truck B is 60 kilometers per hour. How far apart are the two trucks 1 minute before Truck B catches up to Truck A, i... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "6200",
"split": "train"
} |
math | [
{
"content": "The number $1025$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "6201",
"split": "train"
} |
math | [
{
"content": "Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "936",
"style": "rule"
} | {
"index": "6202",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C\\_1$ are given by $\\begin{cases}x=\\sqrt{2}\\sin(\\alpha+\\frac{\\pi}{4}) \\\\ y=\\sin 2\\alpha+1\\end{cases}$ (where $\\alpha$ is a parameter). In the polar coordinate system with $O$ as the pole and the posit... | {
"ground_truth": "\\frac{\\sqrt{7}}{2}-1",
"style": "rule"
} | {
"index": "6203",
"split": "train"
} |
math | [
{
"content": "If $4x + 14 = 8x - 48$, what is the value of $2x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "6204",
"split": "train"
} |
math | [
{
"content": "Given the sequence 2, $\\frac{5}{3}$, $\\frac{3}{2}$, $\\frac{7}{5}$, $\\frac{4}{3}$, ..., then $\\frac{17}{15}$ is the \\_\\_\\_\\_\\_ term in this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6205",
"split": "train"
} |
math | [
{
"content": "One of the roots of $z^2 = -55 + 48i$ is $3 + 8i.$ What is the other root? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3 - 8i",
"style": "rule"
} | {
"index": "6206",
"split": "train"
} |
math | [
{
"content": "Find all real values of $x$ that satisfy \\[\\frac{1}{x^2+1} > \\frac{3}{x} + \\frac{17}{10}.\\](Give your answer in interval notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-2, 0)",
"style": "rule"
} | {
"index": "6207",
"split": "train"
} |
math | [
{
"content": "What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $.3\\overline{25}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{161}{495}",
"style": "rule"
} | {
"index": "6208",
"split": "train"
} |
math | [
{
"content": "Let \\( N_{0} \\) be the set of non-negative integers, and \\( f: N_{0} \\rightarrow N_{0} \\) be a function such that \\( f(0)=0 \\) and for any \\( n \\in N_{0} \\), \\( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \\) and \\( f(2n) > f(n) \\). Determine how many elements in \\( f(N_{0}) \\) are less... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "6209",
"split": "train"
} |
math | [
{
"content": "The units digit of $3^{1001} 7^{1002} 13^{1003}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "6210",
"split": "train"
} |
math | [
{
"content": "Calculate:\n$$\n\\left(10^{4}-9^{4}+8^{4}-7^{4}+\\cdots+2^{4}-1^{4}\\right)+\\left(10^{2}+9^{2}+5 \\times 8^{2}+5 \\times 7^{2}+9 \\times 6^{2}+9 \\times 5^{2}+13 \\times 4^{2}+13 \\times 3^{2}\\right) =\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7615",
"style": "rule"
} | {
"index": "6211",
"split": "train"
} |
math | [
{
"content": "Compute $\\arccos (\\cos 7).$ All functions are in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7 - 2 \\pi",
"style": "rule"
} | {
"index": "6212",
"split": "train"
} |
math | [
{
"content": "Rolling a die twice, let the points shown the first and second times be \\(a\\) and \\(b\\) respectively. Find the probability that the quadratic equation \\(x^{2} + a x + b = 0\\) has two distinct real roots both less than -1. (Answer with a number). Please output the final answer within \\boxed{... | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "6213",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be nonzero vectors such that\n\\[\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = \\|\\mathbf{a} + \\mathbf{b}\\|.\\]Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"index": "6214",
"split": "train"
} |
math | [
{
"content": "The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules. Please output the final a... | {
"ground_truth": "1243.64",
"style": "rule"
} | {
"index": "6215",
"split": "train"
} |
math | [
{
"content": "Given the complex number \\( z \\) satisfying\n$$\n\\left|\\frac{z^{2}+1}{z+\\mathrm{i}}\\right|+\\left|\\frac{z^{2}+4 \\mathrm{i}-3}{z-\\mathrm{i}+2}\\right|=4,\n$$\nfind the minimum value of \\( |z - 1| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "6216",
"split": "train"
} |
math | [
{
"content": "What is the remainder when the sum of the first 102 counting numbers is divided by 5250? (The first counting number is 1.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "6217",
"split": "train"
} |
math | [
{
"content": "In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\\angle ABO = \\text{arc } CD = 60^\\circ$. Find the length of $BC$.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair O, A, B, C, D;\n\nO = (0,0);\nA = dir(30);\nC = dir(160);\nB = (2*C + A)/3;\nD = -A;\n\ndraw(Ci... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "6218",
"split": "train"
} |
math | [
{
"content": "In the given $5 \\times 5$ grid, there are 6 letters. Divide the grid along the lines to form 6 small rectangles (including squares) of different areas, so that each rectangle contains exactly one letter, and each letter is located in a corner square of its respective rectangle. If each of these s... | {
"ground_truth": "34216",
"style": "rule"
} | {
"index": "6219",
"split": "train"
} |
math | [
{
"content": "Find the greatest common divisor of $8!$ and $9!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40320",
"style": "rule"
} | {
"index": "6220",
"split": "train"
} |
math | [
{
"content": "A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.\nA) $\\frac{25}{128}$\nB) $\\frac{25(5 + 2\\sqrt{5})}{128}$\nC) $\\frac{25(5-2\\sqrt{5})}{128}$\n... | {
"ground_truth": "\\frac{25(5-2\\sqrt{5})}{128}",
"style": "rule"
} | {
"index": "6221",
"split": "train"
} |
math | [
{
"content": "On an island, there are only knights, who always tell the truth, and liars, who always lie. One fine day, 30 islanders sat around a round table. Each of them can see everyone except himself and his neighbors. Each person in turn said the phrase: \"Everyone I see is a liar.\" How many liars were si... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "6222",
"split": "train"
} |
math | [
{
"content": "The least common multiple of $a$ and $b$ is $18$, and the least common multiple of $b$ and $c$ is $20$. Find the least possible value of the least common multiple of $a$ and $c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "6223",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the ellipse such that $PF\\_2$ is perpendicular to the $x$-axis. If $|F\\_1F\\_2| = 2|PF\\_2|$, calculate the eccentricity of the ellipse. Please o... | {
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}",
"style": "rule"
} | {
"index": "6224",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a\\_n\\}$ that satisfies the condition: when $n \\geqslant 2$ and $n \\in \\mathbb{N}^+$, we have $a\\_n + a\\_{n-1} = (-1)^n \\times 3$. Calculate the sum of the first 200 terms of the sequence $\\{a\\_n\\}$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "6225",
"split": "train"
} |
math | [
{
"content": "$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\\angle{ACE}$ is $90^{\\circ}$ . What is the average of the measures, in degrees, of $\\angle{ABC}$ and $\\angle{CDE}$ ?\n\n*2018 CCA Math Bonanza Lightning Round #1.3* Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "6226",
"split": "train"
} |
math | [
{
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2008,0),(2008,2009),$ and $(0,2009)$. What is the probability that $x > 2y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{502}{2009}",
"style": "rule"
} | {
"index": "6227",
"split": "train"
} |
math | [
{
"content": "A knight begins on the lower-left square of a standard chessboard. How many squares could the knight end up at after exactly 2009 legal knight's moves? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "6228",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. Determine the size of the largest subset of $\\{ -n, -n+1, \\dots, n-1, n\\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\left\\lceil \\frac{n}{2} \\right\\rceil",
"style": "rule"
} | {
"index": "6229",
"split": "train"
} |
math | [
{
"content": "An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "640",
"style": "rule"
} | {
"index": "6230",
"split": "train"
} |
math | [
{
"content": "The area of triangle $\\triangle OFA$, where line $l$ has an inclination angle of $60^\\circ$ and passes through the focus $F$ of the parabola $y^2=4x$, and intersects with the part of the parabola that lies on the x-axis at point $A$, is equal to $\\frac{1}{2}\\cdot OA \\cdot\\frac{1}{2} \\cdot O... | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "6231",
"split": "train"
} |
math | [
{
"content": "Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, and arrived at the ai... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "6232",
"split": "train"
} |
math | [
{
"content": "Twelve million added to twelve thousand equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12012000",
"style": "rule"
} | {
"index": "6233",
"split": "train"
} |
math | [
{
"content": "A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\\sqrt7$ .\n\nNotes: ''Using a ruler construct points $\\ldots$ '' means: Newly constructed poi... | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "6234",
"split": "train"
} |
math | [
{
"content": "Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\\frac{z}{40}$ and $\\frac{40}{\\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$? Please output the final answer within \\box... | {
"ground_truth": "572",
"style": "rule"
} | {
"index": "6235",
"split": "train"
} |
math | [
{
"content": "Given the equations \\( x^{2} - xy + x = 2018 \\) and \\( y^{2} - xy - y = 52 \\), find the sum of all possible values of \\( x - y \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6236",
"split": "train"
} |
math | [
{
"content": "Triangle \\(ABC\\) has sides \\(AB = 14\\), \\(BC = 13\\), and \\(CA = 15\\). It is inscribed in circle \\(\\Gamma\\), which has center \\(O\\). Let \\(M\\) be the midpoint of \\(AB\\), let \\(B'\\) be the point on \\(\\Gamma\\) diametrically opposite \\(B\\), and let \\(X\\) be the intersection o... | {
"ground_truth": "\\frac{65}{12}",
"style": "rule"
} | {
"index": "6237",
"split": "train"
} |
math | [
{
"content": "Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is Please output the final answer with... | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "6238",
"split": "train"
} |
math | [
{
"content": "A pen and its ink refill together cost $\\;\\$1.10$. The pen costs $\\;\\$1$ more than the ink refill. What is the cost of the pen in dollars? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.05",
"style": "rule"
} | {
"index": "6239",
"split": "train"
} |
math | [
{
"content": "Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\\pi$, and the arc of the semicircle on segment DF has length $10\\pi$. Determine the radius of the semicircle on segment EF. Please output the f... | {
"ground_truth": "\\sqrt{136}",
"style": "rule"
} | {
"index": "6240",
"split": "train"
} |
math | [
{
"content": "What is the largest divisor by which $29 \\cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "6241",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\\begin{cases} x=2t \\\\ y=-2-t \\end{cases}$ (where $t$ is the parameter). In the polar coordinate system (using the same length unit as the rectangular coordinate system and with the origin $O$ as the ... | {
"ground_truth": "\\frac{12\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "6242",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin^2x+\\sin x\\cos x$, when $x=\\theta$ the function $f(x)$ attains its minimum value, find the value of $\\dfrac{\\sin 2\\theta+2\\cos \\theta}{\\sin 2\\theta -2\\cos 2\\theta}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{1}{3}",
"style": "rule"
} | {
"index": "6243",
"split": "train"
} |
math | [
{
"content": "Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.\n\n$$\ny = \\arcsin x, \\quad y = \\arccos x, \\quad y = 0\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "6244",
"split": "train"
} |
math | [
{
"content": "Solve the equations: \n(1) $x(x+4)=-5(x+4)$ \n(2) $(x+2)^2=(2x-1)^2$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "6245",
"split": "train"
} |
math | [
{
"content": "1. Let \\( x_i \\in \\{0,1\\} \\) (for \\( i=1,2,\\cdots,n \\)). If the value of the function \\( f=f(x_1, x_2, \\cdots, x_n) \\) only takes 0 or 1, then \\( f \\) is called an n-ary Boolean function. We denote\n \\[\n D_{n}(f)=\\left\\{(x_1, x_2, \\cdots, x_n) \\mid f(x_1, x_2, \\cdots, x_n)=... | {
"ground_truth": "565",
"style": "rule"
} | {
"index": "6246",
"split": "train"
} |
math | [
{
"content": "In an apartment building, each floor in every entrance has the same number of apartments (more than one). Additionally, each entrance has the same number of floors. The number of floors is more than the number of apartments per floor, but less than the number of entrances. How many floors are in t... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "6247",
"split": "train"
} |
math | [
{
"content": "There are a certain number of identical plastic bags that can be nested within each other. If all the other bags are inside one of the bags, we call this situation a \"bag of bags.\" Calculate the number of ways to make a \"bag of bags\" from 10 bags.\n\nExplanation: Use parentheses to denote a ba... | {
"ground_truth": "16796",
"style": "rule"
} | {
"index": "6248",
"split": "train"
} |
math | [
{
"content": "Find the greatest value of $b$ such that $-b^2+7b-10 \\ge 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "6249",
"split": "train"
} |
math | [
{
"content": "When $k$ candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, $3 k$ candies were distributed among seven people in this way, then how many candies would ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "6250",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $a\\cos B - b\\cos A = \\frac{1}{2}c$. When $\\tan(A-B)$ takes its maximum value, the value of angle $B$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "6251",
"split": "train"
} |
math | [
{
"content": "In a Cartesian coordinate system, the parametric equation for curve $C_1$ is \n$$\n\\left\\{ \\begin{aligned} \nx &= 2\\cos\\alpha, \\\\\ny &= \\sqrt{2}\\sin\\alpha \n\\end{aligned} \\right.\n$$\nwith $\\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as th... | {
"ground_truth": "\\frac{\\sqrt{7} - 1}{2}",
"style": "rule"
} | {
"index": "6252",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\\frac{\\pi}{6}$, and the distance from origin to this line is $\\frac{\\sqrt{3}}{2}$.\n\n(1) Find the equation of the ellipse.\n(2) Suppose... | {
"ground_truth": "k = \\frac{7}{6}",
"style": "rule"
} | {
"index": "6253",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 9x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "6254",
"split": "train"
} |
math | [
{
"content": "Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\\frac{n-1}{n}$. (In particular, the 1 is never... | {
"ground_truth": "681751",
"style": "rule"
} | {
"index": "6255",
"split": "train"
} |
math | [
{
"content": "How many times does the digit 8 appear in the list of all integers from 1 to 800? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "161",
"style": "rule"
} | {
"index": "6256",
"split": "train"
} |
math | [
{
"content": "$ABC$ is an equilateral triangle with sides equal to 2cm. $\\overline{BC}$ is extended its own length to $D$, and $E$ is the midpoint of $\\overline{AB}$. Suppose $\\overline{ED}$ meets $\\overline{AC}$ at $F$. Find the area of the quadrilateral $BEFC$ in square centimeters.\n\n[asy]\nsize(200);\n... | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "6257",
"split": "train"
} |
math | [
{
"content": "The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "6258",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a rectangle. We consider the points $E\\in CA,F\\in AB,G\\in BC$ such that $DC\\perp CA,EF\\perp AB$ and $EG\\perp BC$ . Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "6259",
"split": "train"
} |
math | [
{
"content": "What is the largest $2$-digit prime factor of the integer $n = {200\\choose 100}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "6260",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ and $\\beta$ are the roots of $x^2 - 2x - 1 = 0,$ find $5 \\alpha^4 + 12 \\beta^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "6261",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the sizes of angles $A$, $B$, $C$ form an arithmetic sequence. Let vector $\\overrightarrow{m}=(\\sin \\frac {A}{2},\\cos \\frac {A}{2})$, $\\overrightarrow{n}=(\\cos \\frac {A}{2},- \\sqrt {3}\\cos... | {
"ground_truth": "\\frac {3+ \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": "6262",
"split": "train"
} |
math | [
{
"content": "For \\( n=1,2,3, \\ldots \\), let \\( a_{n}=n^{2}+100 \\), and let \\( d_{n} \\) denote the greatest common divisor of \\( a_{n} \\) and \\( a_{n+1} \\). Find the maximum value of \\( d_{n} \\) as \\( n \\) ranges over all positive integers. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "6263",
"split": "train"
} |
math | [
{
"content": "Find the maximum number of points $X_{i}$ such that for each $i$, $\\triangle A B X_{i} \\cong \\triangle C D X_{i}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6264",
"split": "train"
} |
math | [
{
"content": "Sasha has $10$ cards with numbers $1, 2, 4, 8,\\ldots, 512$ . He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “ $+$ ” sign, and before the other cards... | {
"ground_truth": "1023",
"style": "rule"
} | {
"index": "6265",
"split": "train"
} |
math | [
{
"content": "A Tim number is a five-digit positive integer with the following properties:\n1. It is a multiple of 15.\n2. Its hundreds digit is 3.\n3. Its tens digit is equal to the sum of its first three (leftmost) digits.\n\nHow many Tim numbers are there? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6266",
"split": "train"
} |
math | [
{
"content": "Let the real numbers \\(a_{1}, a_{2}, \\cdots, a_{100}\\) satisfy the following conditions: (i) \\(a_{1} \\geqslant a_{2} \\geqslant \\cdots \\geqslant a_{100} \\geqslant 0\\); (ii) \\(a_{1}+a_{2} \\leqslant 100\\); (iii) \\(a_{3}+a_{4} + \\cdots + a_{100} \\leqslant 100\\). Find the maximum value... | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "6267",
"split": "train"
} |
math | [
{
"content": "The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{13}",
"style": "rule"
} | {
"index": "6268",
"split": "train"
} |
math | [
{
"content": "How many four-character license plates consist of a consonant, followed by a vowel, followed by a consonant, and then a digit? (For this problem, consider Y a vowel.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24{,}000",
"style": "rule"
} | {
"index": "6269",
"split": "train"
} |
math | [
{
"content": "Oleg drew an empty $50 \\times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the product of th... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "6270",
"split": "train"
} |
math | [
{
"content": "A certain company implements an annual salary system, where an employee's annual salary consists of three components: basic salary, housing allowance, and medical expenses, as specified below:\n\n| Item | Salary in the First Year (in ten thousand yuan) | Calculation Method After One Year ... | {
"ground_truth": "20\\%",
"style": "rule"
} | {
"index": "6271",
"split": "train"
} |
math | [
{
"content": "Simplify completely: $$\\sqrt[3]{30^3+40^3+50^3}$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "6272",
"split": "train"
} |
math | [
{
"content": "Let the sequence $\\{x_n\\}$ be defined by $x_1 \\in \\{5, 7\\}$ and, for $k \\ge 1, x_{k+1} \\in \\{5^{x_k} , 7^{x_k} \\}$ . For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and $7^{7^7}$ . Determine the sum of all possible v... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "6273",
"split": "train"
} |
math | [
{
"content": "Given a circle $⊙O$ with equation $x^{2}+y^{2}=6$, let $P$ be a moving point on the circle. Draw a line $PM$ perpendicular to the x-axis at point $M$, and let $N$ be a point on $PM$ such that $\\overrightarrow{PM}= \\sqrt {2} \\overrightarrow{NM}$.\n\n(I) Find the equation for the trajectory of po... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "6274",
"split": "train"
} |
math | [
{
"content": "The class monitor wants to buy soda in batches for all 50 students and teachers in the class for the sports day. According to the store's policy, every 5 empty bottles can be exchanged for one soda bottle, so there is no need to buy 50 bottles of soda. Then, the minimum number of soda bottles that... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "6275",
"split": "train"
} |
math | [
{
"content": "What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "6276",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD_{}^{}$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ into 168 congruent segments with points $A_{}^{}=P_0, P_1, \\ldots, P_{168}=B$, and divide $\\overline {CB}$ into 168 congruent segments with points $C_{}^{}=Q_0, Q_1, \\ldots... | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "6277",
"split": "train"
} |
math | [
{
"content": "Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes. Please output the final answer ... | {
"ground_truth": "252",
"style": "rule"
} | {
"index": "6278",
"split": "train"
} |
math | [
{
"content": "The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in... | {
"ground_truth": "2,6,10",
"style": "rule"
} | {
"index": "6279",
"split": "train"
} |
math | [
{
"content": "A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "6280",
"split": "train"
} |
math | [
{
"content": "A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?\nA) $2,111,111,111,111... | {
"ground_truth": "2,222,222,222,222",
"style": "rule"
} | {
"index": "6281",
"split": "train"
} |
math | [
{
"content": "Suppose that $x$ and $y$ are nonzero real numbers such that $\\frac{4x-3y}{x+4y} = 3$. Find the value of $\\frac{x-4y}{4x+3y}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{63}",
"style": "rule"
} | {
"index": "6282",
"split": "train"
} |
math | [
{
"content": "Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6283",
"split": "train"
} |
math | [
{
"content": "Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the great... | {
"ground_truth": "36.8",
"style": "rule"
} | {
"index": "6284",
"split": "train"
} |
math | [
{
"content": "A positive real number \\( x \\) is such that\n$$\n\\sqrt[3]{1-x^{3}} + \\sqrt[3]{1+x^{3}} = 1.\n$$\n\nFind \\( x^{2} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt[3]{28}}{3}",
"style": "rule"
} | {
"index": "6285",
"split": "train"
} |
math | [
{
"content": "Find the largest prime factor of $9879$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "6286",
"split": "train"
} |
math | [
{
"content": "The equation of the ellipse shown below can be written as\n\\[\\frac{(x - h)^2}{a^2} + \\frac{(y - k)^2}{b^2} = 1.\\]Find $h + k + a + b.$\n\n[asy]\nunitsize(0.3 cm);\n\nint i, n = 10;\n\nfor (i = -n; i <= n; ++i) {\n draw((i,-n)--(i,n),gray(0.7));\n draw((-n,i)--(n,i),gray(0.7));\n}\n\ndraw((0,... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "6287",
"split": "train"
} |
math | [
{
"content": "Let $\\sigma_1 : \\mathbb{N} \\to \\mathbb{N}$ be a function that takes a natural number $n$ , and returns the sum of the positive integer divisors of $n$ . For example, $\\sigma_1(6) = 1 + 2 + 3 + 6 = 12$ . What is the largest number n such that $\\sigma_1(n) = 1854$ ? Please output the fin... | {
"ground_truth": "1234",
"style": "rule"
} | {
"index": "6288",
"split": "train"
} |
math | [
{
"content": "The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}",
"style": "rule"
} | {
"index": "6289",
"split": "train"
} |
math | [
{
"content": "How many integers are there between 0 and \\( 10^5 \\) having the digit sum equal to 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "495",
"style": "rule"
} | {
"index": "6290",
"split": "train"
} |
math | [
{
"content": "A circle is tangent to two adjacent sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\) and cuts off a segment of length 8 cm from vertices \\(B\\) and \\(D\\) at the points of tangency. On the other two sides, the circle cuts off segments of 4 cm and 2 cm respectively from the vertices at the points... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6291",
"split": "train"
} |
math | [
{
"content": "Find the value of the expression $\\frac{3\\sqrt{12}}{\\sqrt{3} + \\sqrt{4} + \\sqrt{6}}$.\nA) $\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}$\nB) $4 - \\sqrt{3} - 2\\sqrt{2}$\nC) $\\sqrt{3} - 2\\sqrt{2} + \\sqrt{6}$\nD) $\\frac{1}{2}(\\sqrt{3} + \\sqrt{6} - 2\\sqrt{2})$\nE) $\\frac{1}{3}(2\\sqrt{2} + \\sqrt... | {
"ground_truth": "\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}",
"style": "rule"
} | {
"index": "6292",
"split": "train"
} |
math | [
{
"content": "Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "6293",
"split": "train"
} |
math | [
{
"content": "A certain school conducted a survey on the daily reading time of students during the summer vacation, as shown in the table below:\n\n| | Mean | Variance | Number of Students |\n|---------|------|----------|--------------------|\n| Grade 10 | $2.7$ | $1$ | $800$ |\n| Grade 11 | $3.1$ | $2$... | {
"ground_truth": "1.966",
"style": "rule"
} | {
"index": "6294",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, there is a circle $C_{1}$: $(x+ \\sqrt {3})^{2}+y^{2}=4$, and a curve $C_{2}$ with parametric equations $\\begin{cases}x=2+2\\cos θ \\\\\\\\ y=2\\sin θ\\end{cases} (θ \\text{ is the parameter})$. Establish a polar coordinate system with $O$ as the pole... | {
"ground_truth": "\\dfrac {3}{2}",
"style": "rule"
} | {
"index": "6295",
"split": "train"
} |
math | [
{
"content": "There is a room with four doors. Determine the number of different ways for someone to enter and exit this room. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6296",
"split": "train"
} |
math | [
{
"content": "A gardener plans to place potted plants along both sides of a 150-meter-long path (including at both ends), with one pot every 2 meters. In total, \\_\\_\\_\\_\\_\\_ pots are needed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "152",
"style": "rule"
} | {
"index": "6297",
"split": "train"
} |
math | [
{
"content": "How many even integers between 4000 and 7000 have four different digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "728",
"style": "rule"
} | {
"index": "6298",
"split": "train"
} |
math | [
{
"content": "Let \\([x]\\) represent the integral part of the decimal number \\(x\\). Given that \\([3.126] + \\left[3.126 + \\frac{1}{8}\\right] + \\left[3.126 + \\frac{2}{8}\\right] + \\ldots + \\left[3.126 + \\frac{7}{8}\\right] = P\\), find the value of \\(P\\). Please output the final answer within \\boxe... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "6299",
"split": "train"
} |
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