data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find a constant \\( A > 0 \\) such that for all real \\( x \\geqslant A \\), we have\n\n\\[ x^{3} + x + 1 \\leq \\frac{x^{4}}{1000000} \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000000",
"style": "rule"
} | {
"index": "2600",
"split": "train"
} |
math | [
{
"content": "Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "2601",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid, the larger base is equal to the sum of the smaller base and the length of the altitude, and every diagonal is equal to the length of the smaller base plus half the altitude. Find the ratio of the smaller base to the larger base.\nA) $\\frac{1}{2}$\nB) $\\frac{\\sqrt{5}}{... | {
"ground_truth": "\\frac{2-\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "2602",
"split": "train"
} |
math | [
{
"content": "For real numbers $s,$ the intersection points of the lines $2x + 3y = 8s + 5$ and $x + 2y = 3s + 2$ are plotted. Determine the slope of the line on which all these points lie. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7}{2}",
"style": "rule"
} | {
"index": "2603",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n\\[\\frac{x^2 + 7}{\\sqrt{x^2 + 3}}\\]over all real numbers $x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "2604",
"split": "train"
} |
math | [
{
"content": "Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (list... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "2605",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sqrt{3}\\sin x \\cos x - \\sin^2 x$: \n(1) Find the smallest positive period of $f(x)$ and the intervals where the function is increasing; \n(2) When $x \\in [0, \\frac{\\pi}{2}]$, find the maximum and minimum values of $f(x)$. Please output the final answer within \... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2606",
"split": "train"
} |
math | [
{
"content": "Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new trailer homes were ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2607",
"split": "train"
} |
math | [
{
"content": "John ate a total of 120 peanuts over four consecutive nights. Each night he ate 6 more peanuts than the night before. How many peanuts did he eat on the fourth night? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "2608",
"split": "train"
} |
math | [
{
"content": "3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000. Otherwise, it will be $25-27 \\frac{|A-C|}{\\min (A, C)-1000}$. Ple... | {
"ground_truth": "1019",
"style": "rule"
} | {
"index": "2609",
"split": "train"
} |
math | [
{
"content": "Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round. Please output the fi... | {
"ground_truth": "\\frac{35}{486}",
"style": "rule"
} | {
"index": "2610",
"split": "train"
} |
math | [
{
"content": "Find the fraction that equals $0.\\overline{73}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{73}{99}",
"style": "rule"
} | {
"index": "2611",
"split": "train"
} |
math | [
{
"content": "At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$\n\\frac{1}{4} \\qquad \\frac{1}{5} \\qquad \\frac{1}{6} \\qquad \\frac{1}{7} \\qquad \\frac{1}{8}\n$$ Please output the final answer wit... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "2612",
"split": "train"
} |
math | [
{
"content": "What is the degree of the polynomial $(3x^2 +11)^{12}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2613",
"split": "train"
} |
math | [
{
"content": "Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\\sin \\angle ADB$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 \\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "2614",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the p... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "2615",
"split": "train"
} |
math | [
{
"content": "Given the equation of the Monge circle of the ellipse $\\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\\Gamma$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{{\\sqrt{2}}}{2}",
"style": "rule"
} | {
"index": "2616",
"split": "train"
} |
math | [
{
"content": "A cyclist initially traveled at a speed of 20 km/h. After covering one-third of the distance, the cyclist looked at the clock and decided to increase the speed by 20%. With the new speed, the cyclist traveled the remaining part of the distance. What is the average speed of the cyclist? Please outp... | {
"ground_truth": "22.5",
"style": "rule"
} | {
"index": "2617",
"split": "train"
} |
math | [
{
"content": "In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete th... | {
"ground_truth": "1803",
"style": "rule"
} | {
"index": "2618",
"split": "train"
} |
math | [
{
"content": "How many odd divisors does $10!$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "2619",
"split": "train"
} |
math | [
{
"content": "How many two-digit numbers have digits whose sum is a prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "2620",
"split": "train"
} |
math | [
{
"content": "A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? Please output the... | {
"ground_truth": "385",
"style": "rule"
} | {
"index": "2621",
"split": "train"
} |
math | [
{
"content": "An \"$n$-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\\cdots ,k,\\cdots,n,\\text{ }n\\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong th... | {
"ground_truth": "180(n-4)",
"style": "rule"
} | {
"index": "2622",
"split": "train"
} |
math | [
{
"content": "In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\\ell$ at the same point $A,$ but they may be on either side of $\\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? Please output t... | {
"ground_truth": "65\\pi",
"style": "rule"
} | {
"index": "2623",
"split": "train"
} |
math | [
{
"content": "A cylinder has both its front and left views as rectangles with length 4 and height 3. Calculate the surface area of this cylinder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20\\pi",
"style": "rule"
} | {
"index": "2624",
"split": "train"
} |
math | [
{
"content": "What is the least positive multiple of 21 that is greater than 380? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "399",
"style": "rule"
} | {
"index": "2625",
"split": "train"
} |
math | [
{
"content": "The solutions to the equations $z^2=4+4\\sqrt{15}i$ and $z^2=2+2\\sqrt 3i,$ where $i=\\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\\sqrt q-r\\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neith... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2626",
"split": "train"
} |
math | [
{
"content": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 points. In a game betwee... | {
"ground_truth": "0.3072",
"style": "rule"
} | {
"index": "2627",
"split": "train"
} |
math | [
{
"content": "Let $S = \\{r_1, r_2, \\ldots, r_n\\} \\subseteq \\{1, 2, 3, \\ldots, 50\\}$, and the sum of any two numbers in $S$ cannot be divisible by 7. The maximum value of $n$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "2628",
"split": "train"
} |
math | [
{
"content": "A truck travels due west at $\\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\\frac{1}{2}\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truck. Determine the ave... | {
"ground_truth": "208",
"style": "rule"
} | {
"index": "2629",
"split": "train"
} |
math | [
{
"content": "In quadrilateral \\(ABCD\\), \\(AB = BC = 9\\) cm, \\(AD = DC = 8\\) cm, \\(AB\\) is perpendicular to \\(BC\\), and \\(AD\\) is perpendicular to \\(DC\\). What is the area of quadrilateral \\(ABCD\\) in square centimeters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "82.5",
"style": "rule"
} | {
"index": "2630",
"split": "train"
} |
math | [
{
"content": "Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2631",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15.5",
"style": "rule"
} | {
"index": "2632",
"split": "train"
} |
math | [
{
"content": "Find the $3 \\times 3$ matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -4 \\mathbf{v}\\]for all three-dimensional vectors $\\mathbf{v}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -4 & 0 & 0 \\\\ 0 & -4 & 0 \\\\ 0 & 0 & -4 \\end{pmatrix}",
"style": "rule"
} | {
"index": "2633",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive real numbers with $\\frac{1}{x+y}=\\frac{1}{x}-\\frac{1}{y}$, what is the value of $\\left(\\frac{x}{y}+\\frac{y}{x}\\right)^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "2634",
"split": "train"
} |
math | [
{
"content": "Given the sets $M=\\{2, 4, 6, 8\\}$, $N=\\{1, 2\\}$, $P=\\left\\{x \\mid x= \\frac{a}{b}, a \\in M, b \\in N\\right\\}$, determine the number of proper subsets of set $P$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "2635",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 0^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "2636",
"split": "train"
} |
math | [
{
"content": "In the figure, $ABCD$ is a rectangle, $AZ=WC=6$ units, $AB=12$ units and the area of trapezoid $ZWCD$ is 120 square units. What is the area of triangle $BQW$? [asy]\ndraw((0,0)--(12,0)--(12,20)--(0,20)--(0,0)--(12,20));\ndraw((0,14)--(12,6));\nlabel(\"$A$\",(0,20),W);\nlabel(\"$Z$\",(0,14),W);\nla... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "2637",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $E$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\\_1$ perpendicular to the $x$-axis and $\\sin \\angle MF\\_2F\\_1 = \\frac{1}{3}$. Find the eccentricity of $E$. Please output the final ans... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "2638",
"split": "train"
} |
math | [
{
"content": "For any real numbers \\( a \\) and \\( b \\), the inequality \\( \\max \\{|a+b|,|a-b|,|2006-b|\\} \\geq C \\) always holds. Find the maximum value of the constant \\( C \\). (Note: \\( \\max \\{x, y, z\\} \\) denotes the largest among \\( x, y, \\) and \\( z \\).) Please output the final answer wi... | {
"ground_truth": "1003",
"style": "rule"
} | {
"index": "2639",
"split": "train"
} |
math | [
{
"content": "In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initially, the organisati... | {
"ground_truth": "10895",
"style": "rule"
} | {
"index": "2640",
"split": "train"
} |
math | [
{
"content": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?\n\n[asy]\nunitsize(0.7cm);\npair A,B,C;\nA=(0,8);\nB=(-2.8,0);\nC=(2.... | {
"ground_truth": "16\\sqrt{2}",
"style": "rule"
} | {
"index": "2641",
"split": "train"
} |
math | [
{
"content": "Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "2642",
"split": "train"
} |
math | [
{
"content": "Three lathes \\( A, B, C \\) each process the same type of standard parts at a certain work efficiency. Lathe \\( A \\) starts 10 minutes earlier than lathe \\( C \\), and lathe \\( C \\) starts 5 minutes earlier than lathe \\( B \\). After lathe \\( B \\) has been working for 10 minutes, the numb... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2643",
"split": "train"
} |
math | [
{
"content": "Given the parametric equation of curve $C\\_1$ as $\\begin{cases} x=a\\cos \\theta \\\\ y=b\\sin \\theta \\end{cases}$ $(a > b > 0, \\theta$ is the parameter$)$, and the point $M(1, \\frac{ \\sqrt{3}}{2})$ on curve $C\\_1$ corresponds to the parameter $\\theta= \\frac{ \\pi}{3}$. Establish a polar... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "2644",
"split": "train"
} |
math | [
{
"content": "What are the rightmost three digits of $7^{1984}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "2645",
"split": "train"
} |
math | [
{
"content": "A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation? Please output the final answer within \... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "2646",
"split": "train"
} |
math | [
{
"content": "Compute the ordered pair of positive integers $(x,y)$ such that\n\n\\begin{align*}\nx^y+1&=y^x,\\\\\n2x^y&=y^x+7.\n\\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,3)",
"style": "rule"
} | {
"index": "2647",
"split": "train"
} |
math | [
{
"content": "In tetrahedron \\(ABCD\\), \\(AB = 1\\), \\(BC = 5\\), \\(CD = 7\\), \\(DA = 5\\), \\(AC = 5\\), \\(BD = 2\\sqrt{6}\\). Find the distance between skew lines \\(AC\\) and \\(BD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{11}}{10}",
"style": "rule"
} | {
"index": "2648",
"split": "train"
} |
math | [
{
"content": "Let $S = \\{2^0,2^1,2^2,\\ldots,2^{10}\\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all of these differences. Find $N.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16398",
"style": "rule"
} | {
"index": "2649",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$ .\n\n*Proposed by Kyle Lee* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "2650",
"split": "train"
} |
math | [
{
"content": "A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green be... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "2651",
"split": "train"
} |
math | [
{
"content": "Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.\n\nDetermine the maximum possible number of positive integers among the 2018 integers. Please output the final an... | {
"ground_truth": "1009",
"style": "rule"
} | {
"index": "2652",
"split": "train"
} |
math | [
{
"content": "The number $0.428125$ can be written as a fraction $\\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "457",
"style": "rule"
} | {
"index": "2653",
"split": "train"
} |
math | [
{
"content": "Suppose $\\sin N = \\frac{2}{3}$ in the diagram below. What is $LN$?\n\n\n\n[asy]\npair L,M,N;\nM = (0,0);\nN = (17.89,0);\nL = (0,16);\ndraw(L--M--N--L);\ndraw(rightanglemark(L,M,N,18));\nlabel(\"$M$\",M,SW);\nlabel(\"$N$\",N,SE);\nlabel(\"$L$\",L,NE);\nlabel(\"$16$\",L/2,W);\n[/asy] Please outp... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2654",
"split": "train"
} |
math | [
{
"content": "If point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{2}",
"style": "rule"
} | {
"index": "2655",
"split": "train"
} |
math | [
{
"content": "Let $a$ be the proportion of teams that correctly answered problem 1 on the Guts round. Estimate $A=\\lfloor 10000a\\rfloor$. An estimate of $E$ earns $\\max (0,\\lfloor 20-|A-E| / 20\\rfloor)$ points. If you have forgotten, question 1 was the following: Two hexagons are attached to form a new pol... | {
"ground_truth": "2539",
"style": "rule"
} | {
"index": "2656",
"split": "train"
} |
math | [
{
"content": "Find the ratio of $AE:EC$ in $\\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\\overline{AC}$ with $BE=6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{18}{7}",
"style": "rule"
} | {
"index": "2657",
"split": "train"
} |
math | [
{
"content": "Suppose $A B C$ is a triangle such that $A B=13, B C=15$, and $C A=14$. Say $D$ is the midpoint of $\\overline{B C}, E$ is the midpoint of $\\overline{A D}, F$ is the midpoint of $\\overline{B E}$, and $G$ is the midpoint of $\\overline{D F}$. Compute the area of triangle $E F G$. Please output th... | {
"ground_truth": "\\frac{21}{4}",
"style": "rule"
} | {
"index": "2658",
"split": "train"
} |
math | [
{
"content": "Reading from left to right, a sequence consists of 6 X's, followed by 24 Y's, followed by 96 X's. After the first \\(n\\) letters, reading from left to right, one letter has occurred twice as many times as the other letter. What is the sum of the four possible values of \\(n\\)? Please output the ... | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "2659",
"split": "train"
} |
math | [
{
"content": "If $\\lceil{\\sqrt{x}}\\rceil=12$, how many possible integer values of $x$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "2660",
"split": "train"
} |
math | [
{
"content": "A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2661",
"split": "train"
} |
math | [
{
"content": "In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.\n\nWhat is the probability that we will reach the finish line in more than two rolls? Please... | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "2662",
"split": "train"
} |
math | [
{
"content": "10 times 10,000 is ; 10 times is 10 million; times 10 million is 100 million. There are 10,000s in 100 million. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "2663",
"split": "train"
} |
math | [
{
"content": "The coefficient of the $x^2$ term in the expansion of $\\sum\\limits_{k=1}^{n}{(x+1)}^{k}$ is equal to the coefficient of the $x^{10}$ term. Determine the positive integer value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "2664",
"split": "train"
} |
math | [
{
"content": "Our school's girls volleyball team has 14 players, including a set of 3 triplets: Missy, Lauren, and Liz. In how many ways can we choose 6 starters if the only restriction is that not all 3 triplets can be in the starting lineup? Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "2838",
"style": "rule"
} | {
"index": "2665",
"split": "train"
} |
math | [
{
"content": "The area of the floor in a square room is 225 square feet. The homeowners plan to cover the floor with rows of 6-inch by 6-inch tiles. How many tiles will be in each row? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "2666",
"split": "train"
} |
math | [
{
"content": "Compute \\[\\sum_{n=1}^{1000} \\frac{1}{n^2 + n}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1000}{1001}",
"style": "rule"
} | {
"index": "2667",
"split": "train"
} |
math | [
{
"content": "The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "2668",
"split": "train"
} |
math | [
{
"content": "Two teams, Team A and Team B, participate in a table tennis group match. The ratio of their strength is $3:2$. Assuming both teams play to their normal levels, find the probability that Team A wins after playing 4 games in a best-of-5 series (3 wins out of 5 games). Please output the final answer ... | {
"ground_truth": "\\frac{162}{625}",
"style": "rule"
} | {
"index": "2669",
"split": "train"
} |
math | [
{
"content": "Calculate the force with which water presses on a dam, whose cross-section has the shape of an isosceles trapezoid. The density of water is $\\rho=1000 \\, \\text{kg} / \\text{m}^{3}$, and the acceleration due to gravity $g$ is $10 \\, \\text{m} / \\text{s}^{2}$.\n\nHint: The pressure at depth $x$... | {
"ground_truth": "640000",
"style": "rule"
} | {
"index": "2670",
"split": "train"
} |
math | [
{
"content": "Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified a... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2671",
"split": "train"
} |
math | [
{
"content": "Teams A and B are playing a volleyball match. Currently, Team A needs to win one more game to become the champion, while Team B needs to win two more games to become the champion. If the probability of each team winning each game is 0.5, then the probability of Team A becoming the champion is $\\b... | {
"ground_truth": "0.75",
"style": "rule"
} | {
"index": "2672",
"split": "train"
} |
math | [
{
"content": "Let's consider the sum of five consecutive odd numbers starting from 997 up to 1005. If the sum $997 + 999 + 1001 + 1003 + 1005 = 5100 - M$, find $M$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "2673",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$. Please output the final an... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "2674",
"split": "train"
} |
math | [
{
"content": "Let \\( P \\) be any point inside a regular tetrahedron \\( ABCD \\) with side length \\( \\sqrt{2} \\). The distances from point \\( P \\) to the four faces are \\( d_1, d_2, d_3, d_4 \\) respectively. What is the minimum value of \\( d_1^2 + d_2^2 + d_3^2 + d_4^2 \\)? Please output the final ans... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2675",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the maximum of \\( |x^2 - 2xy| \\) over \\( 0 \\leq x \\leq 1 \\) for \\( y \\) in \\( \\mathbb{R} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "2676",
"split": "train"
} |
math | [
{
"content": "Julia invested a certain amount of money in two types of assets: real estate and mutual funds. The total amount she invested was $\\$200,000$. If she invested 6 times as much in real estate as she did in mutual funds, what was her total investment in real estate? Please output the final answer wit... | {
"ground_truth": "171,428.58",
"style": "rule"
} | {
"index": "2677",
"split": "train"
} |
math | [
{
"content": "On a \\(6 \\times 6\\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40/119",
"style": "rule"
} | {
"index": "2678",
"split": "train"
} |
math | [
{
"content": "In the sequence $\\{a_n\\}$, $a_1= \\sqrt{2}$, $a_n= \\sqrt{a_{n-1}^2 + 2}\\ (n\\geqslant 2,\\ n\\in\\mathbb{N}^*)$, let $b_n= \\frac{n+1}{a_n^4(n+2)^2}$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\\{b_n\\}$. The value of $16S_n+ \\frac{1}{(n+1)^2}+ \\frac{1}{(n+2)^2}$ is __... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "2679",
"split": "train"
} |
math | [
{
"content": "Let $p = 101$ and let $S$ be the set of $p$ -tuples $(a_1, a_2, \\dots, a_p) \\in \\mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \\to \\{0, 1, \\dots, p-1\\}$ such that\n\n\t\n- $f(a + b) + f(a - b) \\equiv 2\\big(f(a) + f(b)\\big) \\pmod{p}$ for all $a, b \\... | {
"ground_truth": "5152",
"style": "rule"
} | {
"index": "2680",
"split": "train"
} |
math | [
{
"content": "Given that the quiz consists of 4 multiple-choice questions, each with 3 choices, calculate the probability that the contestant wins the quiz. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "2681",
"split": "train"
} |
math | [
{
"content": "A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{8}",
"style": "rule"
} | {
"index": "2682",
"split": "train"
} |
math | [
{
"content": "Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:\n\n- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.\n\nAnd Dan remarks to Jack:\n\n- If I give you 14 sheep for one horse, your herd will have three times as many heads... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "2683",
"split": "train"
} |
math | [
{
"content": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime ... | {
"ground_truth": "515",
"style": "rule"
} | {
"index": "2684",
"split": "train"
} |
math | [
{
"content": "The number 21 is the sum of two consecutive integers 10 and 11. What is the largest number of positive, consecutive integers whose sum is 21? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "2685",
"split": "train"
} |
math | [
{
"content": "Given that $| \\overrightarrow{a}|=3 \\sqrt {2}$, $| \\overrightarrow{b}|=4$, $\\overrightarrow{m}= \\overrightarrow{a}+ \\overrightarrow{b}$, $\\overrightarrow{n}= \\overrightarrow{a}+λ \\overrightarrow{b}$, $ < \\overrightarrow{a}, \\overrightarrow{b} > =135^{\\circ}$, if $\\overrightarrow{m} \\... | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "2686",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = \\frac{1}{x^3 + 3x^2 + 2x} \\). Determine the smallest positive integer \\( n \\) such that \n\\[ f(1) + f(2) + f(3) + \\cdots + f(n) > \\frac{503}{2014}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "2687",
"split": "train"
} |
math | [
{
"content": "For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \\in (0, +\\infty)$, then the value of the real number $m$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2688",
"split": "train"
} |
math | [
{
"content": "Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. Th... | {
"ground_truth": "5\\pi",
"style": "rule"
} | {
"index": "2689",
"split": "train"
} |
math | [
{
"content": "In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was ... | {
"ground_truth": "4.8\\%",
"style": "rule"
} | {
"index": "2690",
"split": "train"
} |
math | [
{
"content": "The Great Eighteen Hockey League is divided into two divisions, with nine teams in each division. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled? Please output the final answer within \\boxed{}... | {
"ground_truth": "378",
"style": "rule"
} | {
"index": "2691",
"split": "train"
} |
math | [
{
"content": "Alina and Masha wanted to create an interesting version of the school tour of the Olympiad. Masha proposed several problems and rejected every second problem of Alina (exactly half). Alina also proposed several problems and only accepted every third problem of Masha (exactly one-third). In the end... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2692",
"split": "train"
} |
math | [
{
"content": "A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing i... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "2693",
"split": "train"
} |
math | [
{
"content": "In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\\overline{FB}$ and $\\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ Please output the final... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "2694",
"split": "train"
} |
math | [
{
"content": "It is known that, for all positive integers $k$,\n$1^2+2^2+3^2+\\ldots+k^{2}=\\frac{k(k+1)(2k+1)}6$.\n Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $200$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "2695",
"split": "train"
} |
math | [
{
"content": "In this addition problem, each letter stands for a different digit.\n$\\begin{array}{cccc}&T & W & O\\\\ +&T & W & O\\\\ \\hline F& O & U & R\\end{array}$\nIf T = 7 and the letter O represents an even number, what is the only possible value for W? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "2696",
"split": "train"
} |
math | [
{
"content": "In the figure, the length of side $AB$ of square $ABCD$ is $\\sqrt{50}$ and $BE=1$. What is the area of the inner square $EFGH$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "2697",
"split": "train"
} |
math | [
{
"content": "In the convex quadrilateral $ABCD$ angle $\\angle{BAD}=90$ , $\\angle{BAC}=2\\cdot\\angle{BDC}$ and $\\angle{DBA}+\\angle{DCB}=180$ . Then find the angle $\\angle{DBA}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "2698",
"split": "train"
} |
math | [
{
"content": "Let $\\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\\otimes$ is continuous, commutative $(a \\otimes b=b \\otimes a)$, distributive across multiplication $(a \\otimes(b c)=(a \\otimes b)(a \\otimes c))$, and that $2 \... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "2699",
"split": "train"
} |
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