data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "**p1.** Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly\na) Three other points?\nb) Four other points?**p2.** Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it possible to change a... | {
"ground_truth": "1376",
"style": "rule"
} | {
"index": "3400",
"split": "train"
} |
math | [
{
"content": "When simplified, $(-\frac{1}{125})^{-2/3}$ becomes: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "3401",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $(a+b)(\\sin A-\\sin B)=c(\\sin C-\\sin B)$. \n$(1)$ Find $A$. \n$(2)$ If $a=4$, find the maximum value of the area $S$ of $\\triangle ABC$. Please output the final ans... | {
"ground_truth": "4 \\sqrt {3}",
"style": "rule"
} | {
"index": "3402",
"split": "train"
} |
math | [
{
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{55}",
"style": "rule"
} | {
"index": "3403",
"split": "train"
} |
math | [
{
"content": "If the arithmetic sequence $\\{a_n\\}$ satisfies $a_{17} + a_{18} + a_{19} > 0$ and $a_{17} + a_{20} < 0$, then the sum of the first $n$ terms of $\\{a_n\\}$ is maximized when $n =$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3404",
"split": "train"
} |
math | [
{
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i = -8; i <= 8; ++i)... | {
"ground_truth": "f(4 - x)",
"style": "rule"
} | {
"index": "3405",
"split": "train"
} |
math | [
{
"content": "Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beauti... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3406",
"split": "train"
} |
math | [
{
"content": "The equation of the directrix of the parabola $y^{2}=6x$ is $x=\\frac{3}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{3}{2}",
"style": "rule"
} | {
"index": "3407",
"split": "train"
} |
math | [
{
"content": "Find all functions $f: \\mathbb R \\to \\mathbb R$ such that \\[ f( xf(x) + f(y) ) = f^2(x) + y \\] for all $x,y\\in \\mathbb R$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x \\text{ or } f(x) = -x",
"style": "rule"
} | {
"index": "3408",
"split": "train"
} |
math | [
{
"content": "Betty goes to the store to get flour and sugar. The amount of flour she buys, in pounds, is at least 6 pounds more than half the amount of sugar, and is no more than twice the amount of sugar. Find the least number of pounds of sugar that Betty could buy. Please output the final answer within \\... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "3409",
"split": "train"
} |
math | [
{
"content": "For a positive constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\rho = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\text{(D)}",
"style": "rule"
} | {
"index": "3410",
"split": "train"
} |
math | [
{
"content": "Simplify: $i^{0}+i^{1}+\\cdots+i^{2009}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+i",
"style": "rule"
} | {
"index": "3411",
"split": "train"
} |
math | [
{
"content": "Given $f(x)$ is an even function defined on $\\mathbb{R}$ and satisfies $f(x+2)=-\\frac{1}{f(x)}$. If $f(x)=x$ for $2\\leq x \\leq 3$, find the value of $f\\left(-\\frac{11}{2}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "3412",
"split": "train"
} |
math | [
{
"content": "On the board, there are three two-digit numbers. One starts with 5, another starts with 6, and the third one starts with 7. The teacher asked three students to each choose any two of these numbers and add them together. The first student got 147, and the results of the second and third students we... | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "3413",
"split": "train"
} |
math | [
{
"content": "A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\\bigtriangleup ADF$. Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "3414",
"split": "train"
} |
math | [
{
"content": "If $x - y = 6$ and $x + y = 12$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "3415",
"split": "train"
} |
math | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.230",
"style": "rule"
} | {
"index": "3416",
"split": "train"
} |
math | [
{
"content": "Given two groups of numerical sequences, each group consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1,2,3, \\ldots, 15$ and their differences are $2,4,6, \\ldots, 30$, respectively. The second group of progressions has ... | {
"ground_truth": "160/151",
"style": "rule"
} | {
"index": "3417",
"split": "train"
} |
math | [
{
"content": "In the tetrahedron P-ABC, $PC \\perpendicular$ plane ABC, $\\angle CAB=90^\\circ$, $PC=3$, $AC=4$, $AB=5$, then the surface area of the circumscribed sphere of this tetrahedron is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50\\pi",
"style": "rule"
} | {
"index": "3418",
"split": "train"
} |
math | [
{
"content": "In an isosceles right triangle $ABC$, with $AB = AC$, a circle is inscribed touching $AB$ at point $D$, $AC$ at point $E$, and $BC$ at point $F$. If line $DF$ is extended to meet $AC$ at point $G$ and $\\angle BAC = 90^\\circ$, and the length of segment $DF$ equals $6$ cm, calculate the length of ... | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "3419",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, $\\angle{BAD}\\cong\\angle{ADC}$ and $\\angle{ABD}\\cong\\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Please output the final answer with... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "3420",
"split": "train"
} |
math | [
{
"content": "Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \\in\\{1,2, \\ldots, N\\}^{4}$ (not necessarily distinct) such that for every integer $n, A n^{3}+B n^{2}+2 C n+D$ is divisible by $N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3421",
"split": "train"
} |
math | [
{
"content": "Simplify: $i^0+i^1+\\cdots+i^{2009}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 + i",
"style": "rule"
} | {
"index": "3422",
"split": "train"
} |
math | [
{
"content": "A point $P$ is chosen in the interior of $\\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the area of $\\triangle ABC... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "3423",
"split": "train"
} |
math | [
{
"content": "After traveling 50 miles by taxi, Ann is charged a fare of $\\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be charged (in dollars) if she had traveled 70 miles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "3424",
"split": "train"
} |
math | [
{
"content": "If\n\\[x + \\sqrt{x^2 - 1} + \\frac{1}{x - \\sqrt{x^2 - 1}} = 20,\\]then find\n\\[x^2 + \\sqrt{x^4 - 1} + \\frac{1}{x^2 + \\sqrt{x^4 - 1}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10201}{200}",
"style": "rule"
} | {
"index": "3425",
"split": "train"
} |
math | [
{
"content": "From the point $(x, y)$, a legal move is a move to $\\left(\\frac{x}{3}+u, \\frac{y}{3}+v\\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \\leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves? Please output the final ... | {
"ground_truth": "\\frac{9 \\pi}{4}",
"style": "rule"
} | {
"index": "3426",
"split": "train"
} |
math | [
{
"content": "$\\frac{1}{1+\\frac{1}{2+\\frac{1}{3}}}=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{10}",
"style": "rule"
} | {
"index": "3427",
"split": "train"
} |
math | [
{
"content": "Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?\n$\\textbf{(A) }129\\qquad \\textbf{(B) }2... | {
"ground_truth": "391",
"style": "rule"
} | {
"index": "3428",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\sqrt {x^{2}-4x+4}-|x-1|$:\n\n1. Solve the inequality $f(x) > \\frac {1}{2}$;\n2. If positive numbers $a$, $b$, $c$ satisfy $a+2b+4c=f(\\frac {1}{2})+2$, find the minimum value of $\\sqrt { \\frac {1}{a}+ \\frac {2}{b}+ \\frac {4}{c}}$. Please output the final answer wit... | {
"ground_truth": "\\frac {7}{3} \\sqrt {3}",
"style": "rule"
} | {
"index": "3429",
"split": "train"
} |
math | [
{
"content": "A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit.\n\nHow many possibilities are there for the rock's age? P... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "3430",
"split": "train"
} |
math | [
{
"content": "Let \\( A, B, C, \\) and \\( D \\) be positive real numbers such that\n\\[\n\\log_{10} (AB) + \\log_{10} (AC) = 3, \\\\\n\\log_{10} (CD) + \\log_{10} (CB) = 4, \\\\\n\\log_{10} (DA) + \\log_{10} (DB) = 5.\n\\]\nCompute the value of the product \\( ABCD \\). Please output the final answer within \\... | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "3431",
"split": "train"
} |
math | [
{
"content": "Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.35",
"style": "rule"
} | {
"index": "3432",
"split": "train"
} |
math | [
{
"content": "Eight students from Adams school worked for $4$ days, six students from Bentley school worked for $6$ days, and seven students from Carter school worked for $10$ days. If a total amount of $\\ 1020$ was paid for the students' work, with each student receiving the same amount for a day's work, dete... | {
"ground_truth": "517.39",
"style": "rule"
} | {
"index": "3433",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{x+1}{3}+\\frac{2-3x}{2}$. Express your answer as a single fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8-7x}{6}",
"style": "rule"
} | {
"index": "3434",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\\left\\{\\begin{array}{l}{x=2+2\\cos\\varphi}\\\\{y=2\\sin\\varphi}\\end{array}\\right.$ ($\\varphi$ is the parameter). Taking the origin $O$ as the pole and the positive half of the $x$-axis as the polar axi... | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "3435",
"split": "train"
} |
math | [
{
"content": "A malfunctioning digital clock shows the time $9: 57 \\mathrm{AM}$; however, the correct time is $10: 10 \\mathrm{AM}$. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of butt... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3436",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the angles A, B, and C are represented by vectors AB and BC with an angle θ between them. Given that the dot product of AB and BC is 6, and that $6(2-\\sqrt{3})\\leq|\\overrightarrow{AB}||\\overrightarrow{BC}|\\sin(\\pi-\\theta)\\leq6\\sqrt{3}$.\n\n(I) Find the value of $\\tan 15^... | {
"ground_truth": "\\sqrt{3}-1",
"style": "rule"
} | {
"index": "3437",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\\left\\{{\\begin{array}{l}{x=1+t\\cos\\alpha}\\\\{y=t\\sin\\alpha}\\end{array}}\\right.$ ($t$ is the parameter, $0\\leqslant \\alpha\\ \\ \\lt \\pi$). Taking the origin $O$ as the pole and the non-negative $x... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "3438",
"split": "train"
} |
math | [
{
"content": "Find the positive value of $x$ that satisfies $cd = x-3i$ given $|c|=3$ and $|d|=5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{6}",
"style": "rule"
} | {
"index": "3439",
"split": "train"
} |
math | [
{
"content": "Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inches, and Dan's cookie... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "3440",
"split": "train"
} |
math | [
{
"content": "A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "3441",
"split": "train"
} |
math | [
{
"content": "A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbit... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "3442",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1(-4,0)$ and $F\\_2(4,0)$ are the two foci of the ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, and $P$ is a point on the ellipse such that the area of $\\triangle PF\\_1F\\_2$ is $3\\sqrt{3}$, find the value of $\\cos\\angle{F\\_1PF\\_2}$. Please output the final answer within \... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3443",
"split": "train"
} |
math | [
{
"content": "Donna has $n$ boxes of doughnuts. Each box contains $13$ doughnuts.\n\nAfter eating one doughnut, Donna is able to rearrange the remaining doughnuts into bags so that each bag contains $9$ doughnuts, and none are left over. What is the smallest possible value of $n$? Please output the final answer... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "3444",
"split": "train"
} |
math | [
{
"content": "The integer $x$ has 12 positive factors. The numbers 12 and 15 are factors of $x$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "3445",
"split": "train"
} |
math | [
{
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given diagram). In each s... | {
"ground_truth": "640",
"style": "rule"
} | {
"index": "3446",
"split": "train"
} |
math | [
{
"content": "When 542 is multiplied by 3, what is the ones (units) digit of the result? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3447",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\frac{3}{5} + \\frac{2}{3} + 1\\frac{1}{15}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\frac{1}{3}",
"style": "rule"
} | {
"index": "3448",
"split": "train"
} |
math | [
{
"content": "Find the maximum constant $k$ such that $\\frac{k a b c}{a+b+c} \\leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3449",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AC}\\perp\\overline{CD}$. If $CD = 15$, $\\tan D = 2$, and $\\tan B = 2.5$, then what is $BC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{261}",
"style": "rule"
} | {
"index": "3450",
"split": "train"
} |
math | [
{
"content": "What is the value of $99^3 + 3(99^2) + 3(99) + 1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1,\\!000,\\!000",
"style": "rule"
} | {
"index": "3451",
"split": "train"
} |
math | [
{
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 + 61 = 145x - 7,$ what is the other value of $x$ that solves the equation? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{68}{81}",
"style": "rule"
} | {
"index": "3452",
"split": "train"
} |
math | [
{
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled triangles with $\\angle AEB=\\angle BEC = \\angle CED = 45^\\circ$ and $AE=32$. Find the length of $CE.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3453",
"split": "train"
} |
math | [
{
"content": "Let $r$ and $s$ be positive integers such that\\[\\frac{5}{11} < \\frac{r}{s} < \\frac{4}{9}\\]and $s$ is as small as possible. What is $s - r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "3454",
"split": "train"
} |
math | [
{
"content": "Integers a, b, c, d, and e satisfy the following three properties: \n(i) $2 \\le a < b <c <d <e <100$ \n(ii) $ \\gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence. \nWhat is the value of c? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3455",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c$ be positive real numbers such that $a \\leq b \\leq c \\leq 2 a$. Find the maximum possible value of $$\\frac{b}{a}+\\frac{c}{b}+\\frac{a}{c}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{2}",
"style": "rule"
} | {
"index": "3456",
"split": "train"
} |
math | [
{
"content": "According to the Shannon formula $C=W\\log_{2}(1+\\frac{S}{N})$, if the bandwidth $W$ is not changed, but the signal-to-noise ratio $\\frac{S}{N}$ is increased from $1000$ to $12000$, then find the approximate percentage increase in the value of $C$. Please output the final answer within \\boxed{}... | {
"ground_truth": "36\\%",
"style": "rule"
} | {
"index": "3457",
"split": "train"
} |
math | [
{
"content": "A pyramid with a square base has a base edge of 20 cm and a height of 40 cm. Two smaller similar pyramids are cut away from the original pyramid: one has an altitude that is one-third the original, and another that is one-fifth the original altitude, stacked atop the first smaller pyramid. What is... | {
"ground_truth": "\\frac{3223}{3375}",
"style": "rule"
} | {
"index": "3458",
"split": "train"
} |
math | [
{
"content": "Given $π < α < 2π$, $\\cos (α-9π)=- \\dfrac {3}{5}$, find the value of $\\cos (α- \\dfrac {11π}{2})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{4}{5}",
"style": "rule"
} | {
"index": "3459",
"split": "train"
} |
math | [
{
"content": "From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible? Please... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "3460",
"split": "train"
} |
math | [
{
"content": "Four players stand at distinct vertices of a square. They each independently choose a vertex of the square (which might be the vertex they are standing on). Then, they each, at the same time, begin running in a straight line to their chosen vertex at 10 mph, stopping when they reach the vertex. If... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "3461",
"split": "train"
} |
math | [
{
"content": "The lengths of the sides of a non-degenerate triangle are $x$, 13 and 37 units. How many integer values of $x$ are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "3462",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the centroid of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. The area of triangle $ABC$ is 24. Find the area of triangle $G_1 G_2 G_3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "3463",
"split": "train"
} |
math | [
{
"content": "Calculate:<br/>$(1)-4^2÷(-32)×(\\frac{2}{3})^2$;<br/>$(2)(-1)^{10}÷2+(-\\frac{1}{2})^3×16$;<br/>$(3)\\frac{12}{7}×(\\frac{1}{2}-\\frac{2}{3})÷\\frac{5}{14}×1\\frac{1}{4}$;<br/>$(4)1\\frac{1}{3}×[1-(-4)^2]-(-2)^3÷\\frac{4}{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "3464",
"split": "train"
} |
math | [
{
"content": "Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gi... | {
"ground_truth": "\\frac{5}{192}",
"style": "rule"
} | {
"index": "3465",
"split": "train"
} |
math | [
{
"content": "One hundred bricks, each measuring $3''\\times12''\\times20''$, are to be stacked to form a tower 100 bricks tall. Each brick can contribute $3''$, $12''$, or $20''$ to the total height of the tower. However, each orientation must be used at least once. How many different tower heights can be achi... | {
"ground_truth": "187",
"style": "rule"
} | {
"index": "3466",
"split": "train"
} |
math | [
{
"content": "Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "3467",
"split": "train"
} |
math | [
{
"content": "If $\\sin x,$ $\\cos x,$ $\\tan x$ form a geometric sequence, in this order, then find $\\cot^6 x - \\cot^2 x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "3468",
"split": "train"
} |
math | [
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{221}",
"style": "rule"
} | {
"index": "3469",
"split": "train"
} |
math | [
{
"content": "Let $f: \\mathbb{Z}^{2} \\rightarrow \\mathbb{Z}$ be a function such that, for all positive integers $a$ and $b$, $$f(a, b)= \\begin{cases}b & \\text { if } a>b \\\\ f(2 a, b) & \\text { if } a \\leq b \\text { and } f(2 a, b)<a \\\\ f(2 a, b)-a & \\text { otherwise }\\end{cases}$$ Compute $f\\lef... | {
"ground_truth": "203",
"style": "rule"
} | {
"index": "3470",
"split": "train"
} |
math | [
{
"content": "Starting with the display \"1,\" calculate the fewest number of keystrokes needed to reach \"400\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3471",
"split": "train"
} |
math | [
{
"content": "There exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ unique integers $a_k$ ($1\\le k\\le r$) with each $a_k$ either $1$ or $- 1$ such that\\[a_13^{n_1} + a_23^{n_2} + \\cdots + a_r3^{n_r} = 2008.\\]Find $n_1 + n_2 + \\cdots + n_r$. Please output the final answer within \... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "3472",
"split": "train"
} |
math | [
{
"content": "A biologist sequentially placed 150 beetles into ten jars. Each subsequent jar contains more beetles than the previous one. The number of beetles in the first jar is at least half the number of beetles in the tenth jar. How many beetles are in the sixth jar? Please output the final answer within \... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3473",
"split": "train"
} |
math | [
{
"content": "The product of three consecutive integers is 210. What is their sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3474",
"split": "train"
} |
math | [
{
"content": "Given complex numbers \\( x \\) and \\( y \\), find the maximum value of \\(\\frac{|3x+4y|}{\\sqrt{|x|^{2} + |y|^{2} + \\left|x^{2}+y^{2}\\right|}}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "3475",
"split": "train"
} |
math | [
{
"content": "In the regular hexagon \\( A B C D E F \\), two of the diagonals, \\( F C \\) and \\( B D \\), intersect at \\( G \\). The ratio of the area of quadrilateral FEDG to the area of \\( \\triangle B C G \\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5: 1",
"style": "rule"
} | {
"index": "3476",
"split": "train"
} |
math | [
{
"content": "Find all functions $f:\\mathbb{R} \\rightarrow \\mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x",
"style": "rule"
} | {
"index": "3477",
"split": "train"
} |
math | [
{
"content": "Find the smallest integer $n$ such that an $n\\times n$ square can be partitioned into $40\\times 40$ and $49\\times 49$ squares, with both types of squares present in the partition, if\r\na) $40|n$ ; b) $49|n$ ; c) $n\\in \\mathbb N$ . Please output the final answer within \\boxed{}.",... | {
"ground_truth": "1960",
"style": "rule"
} | {
"index": "3478",
"split": "train"
} |
math | [
{
"content": "Given \\( x_{1}, x_{2}, \\cdots, x_{1993} \\) satisfy:\n\\[\n\\left|x_{1}-x_{2}\\right|+\\left|x_{2}-x_{3}\\right|+\\cdots+\\left|x_{1992}-x_{1993}\\right|=1993,\n\\]\nand\n\\[\ny_{k}=\\frac{x_{1}+x_{2}+\\cdots+x_{k}}{k} \\quad (k=1,2,\\cdots,1993),\n\\]\nwhat is the maximum possible value of \\( ... | {
"ground_truth": "1992",
"style": "rule"
} | {
"index": "3479",
"split": "train"
} |
math | [
{
"content": "In $\\Delta ABC$, $\\overline{DE} \\parallel \\overline{AB}, CD = 4$ cm, $DA = 10$ cm, and $CE = 6$ cm. What is the number of centimeters in the length of $\\overline{CB}$?\n\n[asy]pair A,B,C,D,E;\nA = (-2,-4); B = (4,-4); C = (0,0); D = A/3; E = B/3;\ndraw(E--D--C--B--A--D);\nlabel(\"A\",A,W); la... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "3480",
"split": "train"
} |
math | [
{
"content": "A sequence of natural numbers $\\left\\{x_{n}\\right\\}$ is constructed according to the following rules:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\text{ for } n \\geq 1.\n$$\n\nIt is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$? Please output th... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3481",
"split": "train"
} |
math | [
{
"content": "Given a nonnegative real number $x$, let $\\langle x\\rangle$ denote the fractional part of $x$; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$, where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$. Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^... | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "3482",
"split": "train"
} |
math | [
{
"content": "How many of the integers between 1 and 1500, inclusive, can be expressed as the difference of the squares of two positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1125",
"style": "rule"
} | {
"index": "3483",
"split": "train"
} |
math | [
{
"content": "A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.2",
"style": "rule"
} | {
"index": "3484",
"split": "train"
} |
math | [
{
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies \\(a_{1}=1, \\sqrt{\\frac{1}{a_{n}^{2}}+4}=\\frac{1}{a_{n+1}}\\). Let \\(S_{n}=\\sum_{i=1}^{n} a_{i}^{2}\\). If \\(S_{2 n+1}-S_{n} \\leqslant \\frac{t}{30}\\) holds for any \\(n \\in \\mathbf{N}^{*}\\), what is the smallest positive integer \\(... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3485",
"split": "train"
} |
math | [
{
"content": "Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "3486",
"split": "train"
} |
math | [
{
"content": "What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\\le2$ and $|x|+|y|+|z-2|\\le2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "3487",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos (2x+\\varphi)$, where $|\\varphi| \\leqslant \\frac{\\pi}{2}$, if $f\\left( \\frac{8\\pi}{3}-x\\right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\\sin 2x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "3488",
"split": "train"
} |
math | [
{
"content": "Given that Kira needs to store 25 files onto disks, each with 2.0 MB of space, where 5 files take up 0.6 MB each, 10 files take up 1.0 MB each, and the rest take up 0.3 MB each, determine the minimum number of disks needed to store all 25 files. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3489",
"split": "train"
} |
math | [
{
"content": "From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ the population increased by $j\\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\left(i+j+\\frac{ij}{100}\\right)\\%$",
"style": "rule"
} | {
"index": "3490",
"split": "train"
} |
math | [
{
"content": "Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \\{ 1,2,\\ldots, mn\\}$. Suppose that there exists a $2n$-element set $T$ such that\r\n(a) each element of $T$ is an $m$-element subset of $S_{m}$;\r\n(b) each pair of elements of $T$ shares at most one common element;... | {
"ground_truth": "2n - 1",
"style": "rule"
} | {
"index": "3491",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, point $G$ satisfies $\\overrightarrow{GA}+\\overrightarrow{GB}+\\overrightarrow{GC}=\\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\\overrightarrow{AM}=m\\overrightarrow{AB}$ $(m>0)$ and $\\overright... | {
"ground_truth": "\\frac{4}{3}+\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "3492",
"split": "train"
} |
math | [
{
"content": "Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7\\pi - 4\\sqrt{3}\\pi",
"style": "rule"
} | {
"index": "3493",
"split": "train"
} |
math | [
{
"content": "Given that $0 < \\alpha < \\pi$ and $\\cos{\\alpha} = -\\frac{3}{5}$, find the value of $\\tan{\\alpha}$ and the value of $\\cos{2\\alpha} - \\cos{(\\frac{\\pi}{2} + \\alpha)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{25}",
"style": "rule"
} | {
"index": "3494",
"split": "train"
} |
math | [
{
"content": "Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \\geq 1$, if $T_n = \\Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, an... | {
"ground_truth": "\\frac{1509}{128}",
"style": "rule"
} | {
"index": "3495",
"split": "train"
} |
math | [
{
"content": "How many students are there in our city? The number expressing the quantity of students is the largest of all numbers where any two adjacent digits form a number that is divisible by 23. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46923",
"style": "rule"
} | {
"index": "3496",
"split": "train"
} |
math | [
{
"content": "For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ \\cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "3497",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the plane passing through the origin with normal vector $\\begin{pmatrix} 1 \\\\ 1 \\\\ -1 \\end{pmatrix}.$ Find the matrix $\\mathbf{R}$ such that for any vector $\\mathbf{v},$ $\\mathbf{R} \\mathbf{v}$ is the reflection of $\\mathbf{v}$ through plane $P.$ Please output the final answ... | {
"ground_truth": "\\begin{pmatrix} \\frac{1}{3} & -\\frac{2}{3} & \\frac{2}{3} \\\\ -\\frac{2}{3} & \\frac{1}{3} & \\frac{2}{3} \\\\ \\frac{2}{3} & \\frac{2}{3} & \\frac{1}{3} \\end{pmatrix}",
"style": "rule"
} | {
"index": "3498",
"split": "train"
} |
math | [
{
"content": "A population consists of $20$ individuals numbered $01$, $02$, $\\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select two numbers from le... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3499",
"split": "train"
} |
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