data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38700",
"split": "train"
} |
math | [
{
"content": "Find the area in the plane contained by the graph of\n\\[\n|x + 2y| + |2x - y| \\le 6.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5.76",
"style": "rule"
} | {
"index": "38701",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^2-7x+18$ and let $g(f(x))=2x+3$. What is the sum of all possible values of $g(8)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38702",
"split": "train"
} |
math | [
{
"content": "Determine how many non-similar regular 500-pointed stars exist, given that a regular $n$-pointed star adheres to the rules set in the original problem description. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38703",
"split": "train"
} |
math | [
{
"content": "It can be shown that for any positive integer $n,$\n\\[\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n + 1} & F_n \\\\ F_n & F_{n - 1} \\end{pmatrix},\\]where $F_n$ denotes the $n$th Fibonacci number.\n\nCompute $F_{784} F_{786} - F_{785}^2.$ Please output the final answ... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "38704",
"split": "train"
} |
math | [
{
"content": "$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ? Please output the final ... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "38705",
"split": "train"
} |
math | [
{
"content": "Given the sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, where $a_{n}$ represents the $n$-th digit after the decimal point of $\\sqrt{2}=1.41421356237⋯$ (for example, $a_{1}=4$, $a_{6}=3)$, and $b_{1}=a_{1}$, with ${b_{n+1}}={a_{{b_n}}}$ for all $n\\in N^{*}$, find all values of $n$ that satisfy $b_{n... | {
"ground_truth": "675",
"style": "rule"
} | {
"index": "38706",
"split": "train"
} |
math | [
{
"content": "A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the locomotive is $100$ km... | {
"ground_truth": "20 \\sqrt[3]{5}",
"style": "rule"
} | {
"index": "38707",
"split": "train"
} |
math | [
{
"content": "How many positive four-digit integers of the form $\\_\\_90$ are divisible by 90? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38708",
"split": "train"
} |
math | [
{
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9448",
"style": "rule"
} | {
"index": "38709",
"split": "train"
} |
math | [
{
"content": "What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38710",
"split": "train"
} |
math | [
{
"content": "On a line \\( l \\) in space, points \\( A \\), \\( B \\), and \\( C \\) are sequentially located such that \\( AB = 18 \\) and \\( BC = 14 \\). Find the distance between lines \\( l \\) and \\( m \\) if the distances from points \\( A \\), \\( B \\), and \\( C \\) to line \\( m \\) are 12, 15, an... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38711",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word MADAM. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38712",
"split": "train"
} |
math | [
{
"content": "Simplify the expression: $(a-3+\\frac{1}{a-1})÷\\frac{{a}^{2}-4}{{a}^{2}+2a}⋅\\frac{1}{a-2}$, then choose a suitable number from $-2$, $-1$, $0$, $1$, $2$ to substitute and evaluate. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38713",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0) = 2$, determine the value of $f(4)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "38714",
"split": "train"
} |
math | [
{
"content": "In how many different ways can four students stand in a straight line if two of the students refuse to stand next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38715",
"split": "train"
} |
math | [
{
"content": "Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants t... | {
"ground_truth": "5600",
"style": "rule"
} | {
"index": "38716",
"split": "train"
} |
math | [
{
"content": "Given that $F(c,0)$ is the right focus of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > b > 0)$, a line perpendicular to one asymptote of the hyperbola passes through $F$ and intersects the two asymptotes at points $A$ and $B$. $O$ is the coordinate origin, and the area of $\\triangl... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "38717",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\begin{cases} 2^{x}, & x < 0 \\\\ f(x-1)+1, & x\\geqslant 0 \\end{cases}$, calculate the value of $f(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{5}{2}",
"style": "rule"
} | {
"index": "38718",
"split": "train"
} |
math | [
{
"content": "A man is standing on a platform and sees his train move such that after $t$ seconds it is $2 t^{2}+d_{0}$ feet from his original position, where $d_{0}$ is some number. Call the smallest (constant) speed at which the man have to run so that he catches the train $v$. In terms of $n$, find the $n$th... | {
"ground_truth": "4^{n-1}",
"style": "rule"
} | {
"index": "38719",
"split": "train"
} |
math | [
{
"content": "In the redesigned version of Mathland's licensing system, all automobile license plates still contain four symbols. The first is a vowel (A, E, I, O, U), the second a non-vowel (any consonant), and the last two symbols are now digits (0 through 9). What is the probability that a randomly chosen li... | {
"ground_truth": "\\frac{1}{10,500}",
"style": "rule"
} | {
"index": "38720",
"split": "train"
} |
math | [
{
"content": "Given the approximate values $\\lg 2 = 0.301$ and $\\lg 3 = 0.477$, find the best approximation for $\\log_{5} 10$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{10}{7}$",
"style": "rule"
} | {
"index": "38721",
"split": "train"
} |
math | [
{
"content": "Find the number of 2's in the factorization of the number $2011 \\cdot 2012 \\cdot 2013 \\cdot \\ldots \\cdot 4020$. Provide the answer in the given field. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "38722",
"split": "train"
} |
math | [
{
"content": "Every card in a deck displays one of three symbols - star, circle, or square, each filled with one of three colors - red, yellow, or blue, and each color is shaded in one of three intensities - light, normal, or dark. The deck contains 27 unique cards, each representing a different symbol-color-in... | {
"ground_truth": "702",
"style": "rule"
} | {
"index": "38723",
"split": "train"
} |
math | [
{
"content": "The bases \\(AB\\) and \\(CD\\) of trapezoid \\(ABCD\\) are 155 and 13 respectively, and its lateral sides are mutually perpendicular. Find the scalar product of the vectors \\(\\overrightarrow{AC}\\) and \\(\\overrightarrow{BD}\\). Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "-2015",
"style": "rule"
} | {
"index": "38724",
"split": "train"
} |
math | [
{
"content": "Given that the coordinates of a direction vector of line $l$ are $({-1,\\sqrt{3}})$, the inclination angle of line $l$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "38725",
"split": "train"
} |
math | [
{
"content": "Is it possible to divide an equilateral triangle into 1,000,000 convex polygons such that any straight line intersects no more than 40 of them? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1707",
"style": "rule"
} | {
"index": "38726",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=\\sin (2017x+\\frac{\\pi }{6})+\\cos (2017x-\\frac{\\pi }{3})$, find the minimum value of $A|x_{1}-x_{2}|$ where $x_{1}$ and $x_{2}$ are real numbers such that $f(x_{1})\\leq f(x)\\leq f(x_{2})$ for all real $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{2\\pi}{2017}",
"style": "rule"
} | {
"index": "38727",
"split": "train"
} |
math | [
{
"content": "Given circle $C$: $(x-1)^{2}+(y-2)^{2}=25$, and line $l$: $(2m+1)x+(m+1)y-7m-4=0$, if the length of the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is _____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "38728",
"split": "train"
} |
math | [
{
"content": "Given the polynomial $f(x)=3x^{9}+3x^{6}+5x^{4}+x^{3}+7x^{2}+3x+1$, calculate the value of $v_{5}$ when $x=3$ using Horner's method. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "761",
"style": "rule"
} | {
"index": "38729",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \\( M \\) to the lines containing the other sides of the hexagon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30\\sqrt{3}",
"style": "rule"
} | {
"index": "38730",
"split": "train"
} |
math | [
{
"content": "There are a theory part and an experimental part in the assessment of a certain course. Each part of the assessment is graded only as \"pass\" or \"fail\", and the whole course assessment is considered \"pass\" if both parts are passed. The probabilities of passing the theory assessment for A, B, ... | {
"ground_truth": "0.254",
"style": "rule"
} | {
"index": "38731",
"split": "train"
} |
math | [
{
"content": "Given 6 digits: \\(0, 1, 2, 3, 4, 5\\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1769580",
"style": "rule"
} | {
"index": "38732",
"split": "train"
} |
math | [
{
"content": "If \\(3 \\times 3 \\times 5 \\times 5 \\times 7 \\times 9 = 3 \\times 3 \\times 7 \\times n \\times n\\), what is a possible value of \\(n\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38733",
"split": "train"
} |
math | [
{
"content": "Given the parametric equations of line $l$ as $\\begin{cases} x=t\\cos α \\\\ y=1+t\\sin α \\end{cases}\\left(t \\text{ is a parameter, } \\frac{π}{2}\\leqslant α < π\\right)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the ... | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "38734",
"split": "train"
} |
math | [
{
"content": "On her previous five attempts Sarah had achieved times, in seconds, of 86, 94, 97, 88 and 96, for swimming 50 meters. After her sixth try she brought her median time down to 92 seconds. What was her time, in seconds, for her sixth attempt? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "38735",
"split": "train"
} |
math | [
{
"content": "The asymptotes of a hyperbola are \\(y = 2x + 3\\) and \\(y = -2x + 1\\). The hyperbola also passes through the point \\((2, 1)\\). Find the distance between the foci of the hyperbola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{30}",
"style": "rule"
} | {
"index": "38736",
"split": "train"
} |
math | [
{
"content": "A man and a dog are walking. The dog waits for the man to start walking along a path and then runs to the end of the path and back to the man a total of four times, always moving at a constant speed. The last time the dog runs back to the man, it covers the remaining distance of 81 meters. The dis... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38737",
"split": "train"
} |
math | [
{
"content": "Simplify the expression:\n\n$$\n\\frac{\\frac{1}{\\sqrt{3+x} \\cdot \\sqrt{x+2}}+\\frac{1}{\\sqrt{3-x} \\cdot \\sqrt{x-2}}}{\\frac{1}{\\sqrt{3+x} \\cdot \\sqrt{x+2}}-\\frac{1}{\\sqrt{3-x} \\cdot \\sqrt{x-2}}} ; \\quad x=\\sqrt{6} \\text {. }\n$$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "38738",
"split": "train"
} |
math | [
{
"content": "Let \\[P(x) = 24x^{24} + \\sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\\] Let $z_{1},z_{2},\\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let\n$\\sum_{k = 1}^{r}|b_{k}| = m + n\\sqrt {... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38739",
"split": "train"
} |
math | [
{
"content": "For all $m, n$ satisfying $1 \\leqslant n \\leqslant m \\leqslant 5$, the number of distinct hyperbolas represented by the polar equation $\\rho = \\frac{1}{1 - c_{m}^{n} \\cos \\theta}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38740",
"split": "train"
} |
math | [
{
"content": "Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white. Please output the final... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "38741",
"split": "train"
} |
math | [
{
"content": "Let's call a number \"remarkable\" if it has exactly 4 different natural divisors, among which there are two such that neither is a multiple of the other. How many \"remarkable\" two-digit numbers exist? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38742",
"split": "train"
} |
math | [
{
"content": "Given Harry has 4 sisters and 6 brothers, and his sister Harriet has S sisters and B brothers, calculate the product of S and B. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38743",
"split": "train"
} |
math | [
{
"content": "Simplify $2w+4w+6w+8w+10w+12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30w+12",
"style": "rule"
} | {
"index": "38744",
"split": "train"
} |
math | [
{
"content": "While waiting at the post office, Lena moved 40 feet closer to the counter over a period of 20 minutes. At this rate, how many minutes will it take her to move the remaining 100 meters to the counter? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "164.042",
"style": "rule"
} | {
"index": "38745",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB$ is congruent to $AC$, the measure of angle $ABC$ is $72^{\\circ}$ and segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. If point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "38746",
"split": "train"
} |
math | [
{
"content": "A square and a regular nonagon are coplanar and share a common side $\\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{130}{1}",
"style": "rule"
} | {
"index": "38747",
"split": "train"
} |
math | [
{
"content": "An eight-sided die (numbered 1 through 8) is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "960",
"style": "rule"
} | {
"index": "38748",
"split": "train"
} |
math | [
{
"content": "A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38749",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\) with sides \\(BC=7\\), \\(AC=5\\), and \\(AB=3\\), an angle bisector \\(AD\\) is drawn. A circle is circumscribed around triangle \\(ABD\\), and a circle is inscribed in triangle \\(ACD\\). Find the product of their radii. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "35/32",
"style": "rule"
} | {
"index": "38750",
"split": "train"
} |
math | [
{
"content": "What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98752",
"style": "rule"
} | {
"index": "38751",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides corresponding to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $b = a \\sin C + c \\cos A$,\n\n(1) Find the value of $A + B$;\n\n(2) If $c = \\sqrt{2}$, find the maximum area of $\\triangle ABC$. Please output the final answer ... | {
"ground_truth": "\\frac{1 + \\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38752",
"split": "train"
} |
math | [
{
"content": "Carina has three pins, labeled $A, B$ , and $C$ , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?\n... | {
"ground_truth": "\\[ 128 \\]",
"style": "rule"
} | {
"index": "38753",
"split": "train"
} |
math | [
{
"content": "If the real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-2)^{2}+(y-1)^{2}=1\\), then the minimum value of \\(x^{2}+y^{2}\\) is \\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6-2\\sqrt{5}",
"style": "rule"
} | {
"index": "38754",
"split": "train"
} |
math | [
{
"content": "Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "298",
"style": "rule"
} | {
"index": "38755",
"split": "train"
} |
math | [
{
"content": "A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\pi",
"style": "rule"
} | {
"index": "38756",
"split": "train"
} |
math | [
{
"content": "How many positive rational numbers less than \\(\\pi\\) have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "38757",
"split": "train"
} |
math | [
{
"content": "Alice buys three burgers and two sodas for $\\$3.20$, and Bill buys two burgers and a soda for $\\$2.00$. How many cents does a burger cost? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "38758",
"split": "train"
} |
math | [
{
"content": "With all angles measured in degrees, the product $\\prod_{k=1}^{22} \\sec^2(4k)^\\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "38759",
"split": "train"
} |
math | [
{
"content": "In the parallelepiped $ABCD-{A'}{B'}{C'}{D'}$, the base $ABCD$ is a square with side length $2$, the length of the side edge $AA'$ is $3$, and $\\angle {A'}AB=\\angle {A'}AD=60^{\\circ}$. Find the length of $AC'$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{29}",
"style": "rule"
} | {
"index": "38760",
"split": "train"
} |
math | [
{
"content": "Justine has two fair dice, one with sides labeled $1,2,\\ldots, m$ and one with sides labeled $1,2,\\ldots, n.$ She rolls both dice once. If $\\tfrac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$ .\n\n... | {
"ground_truth": "996",
"style": "rule"
} | {
"index": "38761",
"split": "train"
} |
math | [
{
"content": "Given a moving point \\( P \\) on the \\( x \\)-axis, \\( M \\) and \\( N \\) lie on the circles \\((x-1)^{2}+(y-2)^{2}=1\\) and \\((x-3)^{2}+(y-4)^{2}=3\\) respectively. Find the minimum value of \\( |PM| + |PN| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{10} - \\sqrt{3} - 1",
"style": "rule"
} | {
"index": "38762",
"split": "train"
} |
math | [
{
"content": "The letters $\\text{A}$, $\\text{J}$, $\\text{H}$, $\\text{S}$, $\\text{M}$, $\\text{E}$ and the digits $1$, $9$, $8$, $9$ are \"cycled\" separately as follows and put together in a numbered list:\n\\[\\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ &... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38763",
"split": "train"
} |
math | [
{
"content": "In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\\frac{AP}{AB}=\\frac{61}{2022}$ and a point $Q$ on the segment $AD$ is taken such that $\\frac{AQ}{AD}=\\frac{61}{2065}$ .If $PQ$ intersects $AC$ at $T$ , find $\\frac{AC}{AT}$ to the nearest integer... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "38764",
"split": "train"
} |
math | [
{
"content": "The values of $x$ and $y$ are always positive, and $x^2$ and $y$ vary inversely. If $y$ is 10 when $x$ is 2, then find $x$ when $y$ is 4000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "38765",
"split": "train"
} |
math | [
{
"content": "If the six digits 1, 2, 3, 5, 5 and 8 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 15? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38766",
"split": "train"
} |
math | [
{
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$. If $a_5=3$ and $S_{10}=40$, then the minimum value of $nS_n$ is . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-32",
"style": "rule"
} | {
"index": "38767",
"split": "train"
} |
math | [
{
"content": "On the game show $\\text{\\emph{Wheel of Fortune Redux}}$, you see the following spinner. Given that each region is of the same area, what is the probability that you will earn exactly $\\$3200$ in your first three spins? The spinner includes the following sections: $\"\\$2000\"$, $\"\\$300\"$, $\... | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "38768",
"split": "train"
} |
math | [
{
"content": "The number 121 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 100 and 500? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "38769",
"split": "train"
} |
math | [
{
"content": "When $x^5$, $x+\\frac{1}{x}$ and $1+\\frac{2}{x} + \\frac{3}{x^2}$ are multiplied, the product is a polynomial of degree. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38770",
"split": "train"
} |
math | [
{
"content": "Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "38771",
"split": "train"
} |
math | [
{
"content": "The point $P$ $(4,5)$ is reflected over the $y$-axis to $Q$. Then $Q$ is reflected over the line $y=-x$ to $R$. What is the area of triangle $PQR$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38772",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\\begin{cases}x=2+t \\\\ y=1+t\\end{cases} (t \\text{ i... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "38773",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^{2}=4x$, a line with a slope of $\\frac{\\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "38774",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $1.000 + 0.101 + 0.011 + 0.001$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.113",
"style": "rule"
} | {
"index": "38775",
"split": "train"
} |
math | [
{
"content": "$\\triangle KWU$ is an equilateral triangle with side length $12$ . Point $P$ lies on minor arc $\\overarc{WU}$ of the circumcircle of $\\triangle KWU$ . If $\\overline{KP} = 13$ , find the length of the altitude from $P$ onto $\\overline{WU}$ .\n\n*Proposed by Bradley Guo* Please outpu... | {
"ground_truth": "\\frac{25\\sqrt{3}}{24}",
"style": "rule"
} | {
"index": "38776",
"split": "train"
} |
math | [
{
"content": "Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a constant speed, but the... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "38777",
"split": "train"
} |
math | [
{
"content": "Person A and Person B start walking towards each other from points $A$ and $B$ respectively, which are 10 kilometers apart. If they start at the same time, they will meet at a point 1 kilometer away from the midpoint of $A$ and $B$. If Person A starts 5 minutes later than Person B, they will meet ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38778",
"split": "train"
} |
math | [
{
"content": "There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done? Please output the final answer within \\boxe... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "38779",
"split": "train"
} |
math | [
{
"content": "If $f(x)=4^x$ then $f(x+1)-f(x)$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3f(x)",
"style": "rule"
} | {
"index": "38780",
"split": "train"
} |
math | [
{
"content": "Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \\angle{A}\\equal{}60^o$, $ BC\\equal{}CD\\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both inte... | {
"ground_truth": "\\frac{38}{7}",
"style": "rule"
} | {
"index": "38781",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be reals. Find the least possible value of $(2 \\cos \\alpha+5 \\sin \\beta-8)^{2}+(2 \\sin \\alpha+5 \\cos \\beta-15)^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38782",
"split": "train"
} |
math | [
{
"content": "There are 12 students in a classroom; 6 of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political align... | {
"ground_truth": "\\frac{341}{55}",
"style": "rule"
} | {
"index": "38783",
"split": "train"
} |
math | [
{
"content": "Carefully observe the arrangement pattern of the following hollow circles ($○$) and solid circles ($●$): $○●○○●○○○●○○○○●○○○○○●○○○○○○●…$. If this pattern continues, a series of $○$ and $●$ will be obtained. The number of $●$ in the first $100$ of $○$ and $●$ is $\\_\\_\\_\\_\\_\\_\\_$. Please outpu... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38784",
"split": "train"
} |
math | [
{
"content": "Find the number of natural numbers \\( k \\), not exceeding 333300, such that \\( k^{2} - 2k \\) is exactly divisible by 303. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4400",
"style": "rule"
} | {
"index": "38785",
"split": "train"
} |
math | [
{
"content": "Given the function $y=\\sin(\\omega x)$ ($\\omega>0$) is increasing in the interval $[0, \\frac{\\pi}{3}]$ and its graph is symmetric about the point $(3\\pi, 0)$, the maximum value of $\\omega$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "38786",
"split": "train"
} |
math | [
{
"content": "We write the following equation: \\((x-1) \\ldots (x-2020) = (x-1) \\ldots (x-2020)\\). What is the minimal number of factors that need to be erased so that there are no real solutions? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "38787",
"split": "train"
} |
math | [
{
"content": "How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38788",
"split": "train"
} |
math | [
{
"content": "In how many ways can you fill a $3 \\times 3$ table with the numbers 1 through 9 (each used once) such that all pairs of adjacent numbers (sharing one side) are relatively prime? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "38789",
"split": "train"
} |
math | [
{
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that $f(0)\\neq 0$ and for all $x,y\\in\\mathbb{R}$,\n \\[ f(x+y)^2 = 2f(x)f(y) + \\max \\left\\{ f(x^2+y^2), f(x^2)+f(y^2) \\right\\}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = -1 \\text{ and } f(x) = x - 1.",
"style": "rule"
} | {
"index": "38790",
"split": "train"
} |
math | [
{
"content": "What is the remainder when the sum $1 + 7 + 13 + 19 + \\cdots + 253 + 259$ is divided by $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "38791",
"split": "train"
} |
math | [
{
"content": "100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who won exactly two mat... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "38792",
"split": "train"
} |
math | [
{
"content": "Rounded to 3 decimal places, what is $\\frac{8}{11}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.727",
"style": "rule"
} | {
"index": "38793",
"split": "train"
} |
math | [
{
"content": "Rectangle \\(ABCD\\) has length 9 and width 5. Diagonal \\(AC\\) is divided into 5 equal parts at \\(W, X, Y\\), and \\(Z\\). Determine the area of the shaded region. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38794",
"split": "train"
} |
math | [
{
"content": "In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38795",
"split": "train"
} |
math | [
{
"content": "Given the three digits 2, 4 and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38796",
"split": "train"
} |
math | [
{
"content": "Sasha and Masha each picked a natural number and communicated them to Vasya. Vasya wrote the sum of these numbers on one piece of paper and their product on another piece, then hid one of the pieces and showed the other (on which the number 2002 was written) to Sasha and Masha. Seeing this number,... | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "38797",
"split": "train"
} |
math | [
{
"content": "What is the largest three-digit integer whose digits are distinct and form a geometric sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "964",
"style": "rule"
} | {
"index": "38798",
"split": "train"
} |
math | [
{
"content": "Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in \"random\" mode, he can listen... | {
"ground_truth": "7/11",
"style": "rule"
} | {
"index": "38799",
"split": "train"
} |
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