data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given an ellipse with the equation \\\\(\\\\dfrac{x^{2}}{a^{2}}+\\\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\\\) and an eccentricity of \\\\(\\\\dfrac{\\\\sqrt{3}}{2}\\\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ ... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "5400",
"split": "train"
} |
math | [
{
"content": "Define a new operation: \\( x \\odot y = 18 + x - a \\times y \\), where \\( a \\) is a constant. For example:\n\\[ 1 \\odot 2 = 18 + 1 - a \\times 2. \\]\n\nIf \\( 2 \\odot 3 = 8 \\), then what is \\( 3 \\odot 5 \\) and \\( 5 \\odot 3 \\)? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5401",
"split": "train"
} |
math | [
{
"content": "Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1001 + 1000i",
"style": "rule"
} | {
"index": "5402",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110.8333",
"style": "rule"
} | {
"index": "5403",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}$, $\\angle ABC=120^\\circ,AB=3$ and $BC=4$. If perpendiculars constructed to $\\overline{AB}$ at $A$ and to $\\overline{BC}$ at $C$ meet at $D$, then $CD=$\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{8}{\\sqrt{3}}\\quad \\text{(C) } 5\\quad \\text{(D) } \\frac{11}{2}\\quad \\tex... | {
"ground_truth": "\\frac{10}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "5404",
"split": "train"
} |
math | [
{
"content": "Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4944",
"style": "rule"
} | {
"index": "5405",
"split": "train"
} |
math | [
{
"content": "The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "11.54",
"style": "rule"
} | {
"index": "5406",
"split": "train"
} |
math | [
{
"content": "Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten? Please output the ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5407",
"split": "train"
} |
math | [
{
"content": "The product of two positive three-digit palindromes is 436,995. What is their sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1332",
"style": "rule"
} | {
"index": "5408",
"split": "train"
} |
math | [
{
"content": "Consider an equilateral triangle with a side length of 10 cm, where one side of the triangle is also the diameter of a semicircle. Find the sum of the areas of two small shaded regions formed inside the semicircle but outside the triangle. Express your answer in the form \\(a\\pi - b\\sqrt{c}\\) a... | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "5409",
"split": "train"
} |
math | [
{
"content": "A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is formed by revolving... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "5410",
"split": "train"
} |
math | [
{
"content": "Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "5411",
"split": "train"
} |
math | [
{
"content": "Given \\(a > 0\\), \\(b > 0\\), \\(c > 1\\), and \\(a + b = 1\\). Find the minimum value of \\(\\left(\\frac{2a + b}{ab} - 3\\right)c + \\frac{\\sqrt{2}}{c - 1}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "5412",
"split": "train"
} |
math | [
{
"content": "Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \\leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ . Please output the final answer wit... | {
"ground_truth": "\\frac{1}{4e^2}",
"style": "rule"
} | {
"index": "5413",
"split": "train"
} |
math | [
{
"content": "Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.\n\nAnton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of... | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "5414",
"split": "train"
} |
math | [
{
"content": "Given \\(\\lg 2 = 0.30103\\), calculate the number of digits in \\( M = 1 + 10^4 + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + \\frac{10^4 (10^4-1)(10^4-2)}{1 \\cdot 2 \\cdot 3} + \\cdots + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + 10^4 + 1\\). Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "3011",
"style": "rule"
} | {
"index": "5415",
"split": "train"
} |
math | [
{
"content": "There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed t... | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "5416",
"split": "train"
} |
math | [
{
"content": "According to the table below, how many dollars are in the median value of the 59 salaries paid to this company's employees?\n\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Position Title}&\\textbf{\\# with Title}&\\textbf{Salary}\\\\\\hline\nPresident&1&$\\$130{,}000$\\\\\\hline\nVice-President&5&... | {
"ground_truth": "\\$23{,}000",
"style": "rule"
} | {
"index": "5417",
"split": "train"
} |
math | [
{
"content": "What is the sum of all integer solutions to $|n| < |n-3| < 9$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "5418",
"split": "train"
} |
math | [
{
"content": "What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.7));\n\nxaxis(0,8,Ticks(1.0,NoZero));\nyaxis(0,8,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);... | {
"ground_truth": "67\\%",
"style": "rule"
} | {
"index": "5419",
"split": "train"
} |
math | [
{
"content": "Find the distance between the foci of the ellipse\n\\[\\frac{x^2}{36} + \\frac{y^2}{9} = 5.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{5.4}",
"style": "rule"
} | {
"index": "5420",
"split": "train"
} |
math | [
{
"content": "Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitel... | {
"ground_truth": "5/11",
"style": "rule"
} | {
"index": "5421",
"split": "train"
} |
math | [
{
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "c = a",
"style": "rule"
} | {
"index": "5422",
"split": "train"
} |
math | [
{
"content": "The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means \"less than\"; \nthe symbol $>$ means \"greater than.\" \nThe set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: Please output th... | {
"ground_truth": "$-1<x<7$",
"style": "rule"
} | {
"index": "5423",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "5424",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\cos(2x - \\frac{\\pi}{3}) + 2\\sin^2x$,\n(I) Find the period of the function $f(x)$ and the intervals where it is monotonically increasing;\n(II) When $x \\in [0, \\frac{\\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. Please output the final ans... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5425",
"split": "train"
} |
math | [
{
"content": "Find all values of $x$ so that $\\arccos x > \\arcsin x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left[ -1, \\frac{1}{\\sqrt{2}} \\right)",
"style": "rule"
} | {
"index": "5426",
"split": "train"
} |
math | [
{
"content": "Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "5427",
"split": "train"
} |
math | [
{
"content": "Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle Sc... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5428",
"split": "train"
} |
math | [
{
"content": "Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "5429",
"split": "train"
} |
math | [
{
"content": "If the real number sequence: -1, $a_1$, $a_2$, $a_3$, -81 forms a geometric sequence, determine the eccentricity of the conic section $x^2+ \\frac{y^2}{a_2}=1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "5430",
"split": "train"
} |
math | [
{
"content": "For which natural number \\( K \\) does the expression \\(\\frac{n^{2}}{1.001^{n}}\\) reach its maximum value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2001",
"style": "rule"
} | {
"index": "5431",
"split": "train"
} |
math | [
{
"content": "A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "5432",
"split": "train"
} |
math | [
{
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. Find the area of \\(\\triangle ABC\... | {
"ground_truth": "2535",
"style": "rule"
} | {
"index": "5433",
"split": "train"
} |
math | [
{
"content": "The Fibonacci sequence $1,1,2,3,5,8,13,21,\\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5434",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C$: $\\begin{cases}x=2\\cos θ \\\\\\\\ y=\\sqrt{3}\\sin θ\\end{cases}$, find the value of $\\frac{1}{m}+\\frac{1}{n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "5435",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\\angle BAF = 18^\\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that the value of $\\ang... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "5436",
"split": "train"
} |
math | [
{
"content": "A right triangle has legs of lengths 126 and 168 units. What is the perimeter of the triangle formed by the points where the angle bisectors intersect the opposite sides? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "230.61",
"style": "rule"
} | {
"index": "5437",
"split": "train"
} |
math | [
{
"content": "What is the largest integer \\( k \\) whose square \\( k^2 \\) is a factor of \\( 10! \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "5438",
"split": "train"
} |
math | [
{
"content": "On the first day, Barry Sotter used his magic wand to make an object's length increase by $\\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \\frac{1}{3} x.$ On the second day, he increased the object's new length from the previous day by $... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "5439",
"split": "train"
} |
math | [
{
"content": "Find, as a function of $\\, n, \\,$ the sum of the digits of \\[9 \\times 99 \\times 9999 \\times \\cdots \\times \\left( 10^{2^n} - 1 \\right),\\] where each factor has twice as many digits as the previous one. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ 9 \\cdot 2^n \\]",
"style": "rule"
} | {
"index": "5440",
"split": "train"
} |
math | [
{
"content": "Given that the positive real numbers \\(a_{1}, a_{2}, a_{3}, a_{4}\\) satisfy the conditions \\(a_{1} \\geqslant a_{2} a_{3}^{2}, a_{2} \\geqslant a_{3} a_{4}^{2}, a_{3} \\geqslant a_{4} a_{1}^{2}, a_{4} \\geqslant a_{1} a_{2}^{2}\\), find the maximum value of \\(a_{1} a_{2} a_{3} a_{4}\\left(a_{1... | {
"ground_truth": "1/256",
"style": "rule"
} | {
"index": "5441",
"split": "train"
} |
math | [
{
"content": "Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that\n\\[Q(2) = Q(4) = Q(6) = Q(8) = b\\]and\n\\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\\]\nWhat is the smallest possible value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "315",
"style": "rule"
} | {
"index": "5442",
"split": "train"
} |
math | [
{
"content": "A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged seats? Please output th... | {
"ground_truth": "1/32",
"style": "rule"
} | {
"index": "5443",
"split": "train"
} |
math | [
{
"content": "A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrila... | {
"ground_truth": "314",
"style": "rule"
} | {
"index": "5444",
"split": "train"
} |
math | [
{
"content": "Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \\leq a, c \\leq 59$ and $1 \\leq b, d \\leq 5$. Please outpu... | {
"ground_truth": "366",
"style": "rule"
} | {
"index": "5445",
"split": "train"
} |
math | [
{
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "5446",
"split": "train"
} |
math | [
{
"content": "In the diagram, $ABCD$ and $EFGD$ are squares each of area 16. If $H$ is the midpoint of both $BC$ and $EF$, find the total area of polygon $ABHFGD$.\n\n[asy]\nunitsize(3 cm);\n\npair A, B, C, D, E, F, G, H;\n\nF = (0,0);\nG = (1,0);\nD = (1,1);\nE = (0,1);\nH = (E + F)/2;\nA = reflect(D,H)*(G);\... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "5447",
"split": "train"
} |
math | [
{
"content": "Given the sets \\( M = \\{1, 2, 3\\} \\) and \\( N = \\{1, 2, 3, 4, 5\\} \\), define the function \\( f: M \\rightarrow N \\). Let the points \\( A(1, f(1)), B(2, f(2)), C(3, f(3)) \\) form a triangle \\( \\triangle ABC \\). The circumcenter of \\( \\triangle ABC \\) is \\( D \\), and it is given ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5448",
"split": "train"
} |
math | [
{
"content": "If $f^{-1}(g(x))=x^3-1$ and $g$ has an inverse, find $g^{-1}(f(7))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5449",
"split": "train"
} |
math | [
{
"content": "In the expansion of the polynomial $$(x+ \\frac {1}{ \\sqrt {x}})^{6}( \\sqrt {x}-1)^{10}$$, the constant term is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-495",
"style": "rule"
} | {
"index": "5450",
"split": "train"
} |
math | [
{
"content": "How many distinct, positive factors does $1100$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5451",
"split": "train"
} |
math | [
{
"content": "What is the sum of the 2009 fractions of the form $\\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 1 through 2009? Express your answer as a decimal to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.499",
"style": "rule"
} | {
"index": "5452",
"split": "train"
} |
math | [
{
"content": "Given the real numbers \\( x \\) and \\( y \\) that satisfy \\( xy + 6 = x + 9y \\) and \\( y \\in (-\\infty, 1) \\), find the maximum value of \\((x+3)(y+1)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27 - 12\\sqrt{2}",
"style": "rule"
} | {
"index": "5453",
"split": "train"
} |
math | [
{
"content": "Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. Diagonals $AC$ and $BD$ intersect at $E$, where $AC = 18$, and $\\triangle AED$ and $\\triangle BEC$ have equal perimeters. Calculate the length of $AE$.\n**A)** $6$ \n**B)** $7.2$ \n**C)** $7.5$ \n**D)** $9$ \n**E)** $10$ Please output ... | {
"ground_truth": "7.2",
"style": "rule"
} | {
"index": "5454",
"split": "train"
} |
math | [
{
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "4 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "5455",
"split": "train"
} |
math | [
{
"content": "Point P is located on side AB of triangle ABC. What is the probability that the area of triangle PBC is less than or equal to 1/3 of the area of triangle ABC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "5456",
"split": "train"
} |
math | [
{
"content": "Given two non-zero vectors $\\overrightarrow{m}$ and $\\overrightarrow{n}$ with an angle of $\\frac{\\pi}{3}$ between them, and the magnitude of $\\overrightarrow{n}$ is a positive scalar multiple of the magnitude of $\\overrightarrow{m}$, i.e., $|\\overrightarrow{n}| = λ|\\overrightarrow{m}| (λ >... | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "5457",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5458",
"split": "train"
} |
math | [
{
"content": "Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "757",
"style": "rule"
} | {
"index": "5459",
"split": "train"
} |
math | [
{
"content": "In the regular quadrangular pyramid \\(P-ABCD\\), \\(M\\) and \\(N\\) are the midpoints of \\(PA\\) and \\(PB\\) respectively. If the tangent of the dihedral angle between a side face and the base is \\(\\sqrt{2}\\), find the cosine of the angle between skew lines \\(DM\\) and \\(AN\\). Please out... | {
"ground_truth": "1/6",
"style": "rule"
} | {
"index": "5460",
"split": "train"
} |
math | [
{
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "236",
"style": "rule"
} | {
"index": "5461",
"split": "train"
} |
math | [
{
"content": "Determine the maximum possible value of \\[\\frac{\\left(x^2+5x+12\\right)\\left(x^2+5x-12\\right)\\left(x^2-5x+12\\right)\\left(-x^2+5x+12\\right)}{x^4}\\] over all non-zero real numbers $x$ .\n\n*2019 CCA Math Bonanza Lightning Round #3.4* Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "5462",
"split": "train"
} |
math | [
{
"content": "How many ways can one color the squares of a $6 \\times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67950",
"style": "rule"
} | {
"index": "5463",
"split": "train"
} |
math | [
{
"content": "Find $\\left(\\frac{1}{2}\\right)^{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "5464",
"split": "train"
} |
math | [
{
"content": "Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \\{1,2,\\dots,2p\\}$ are there, the sum of whose elements is divisible by $ p$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\boxed{2 + \\frac{1}{p} \\left(\\binom{2p}{p} - 2 \\right)}",
"style": "rule"
} | {
"index": "5465",
"split": "train"
} |
math | [
{
"content": "Find the number of 0-1 binary sequences formed by six 0's and six 1's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "357",
"style": "rule"
} | {
"index": "5466",
"split": "train"
} |
math | [
{
"content": "What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16.67\\%",
"style": "rule"
} | {
"index": "5467",
"split": "train"
} |
math | [
{
"content": "Let \\[\\mathbf{N} = \\begin{pmatrix} x & y & z \\\\ y & z & x \\\\ z & x & y \\end{pmatrix}\\] be a matrix with real entries such that $\\mathbf{N}^3 = \\mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5468",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $A=30^{\\circ}$, $2 \\overrightarrow{AB}\\cdot \\overrightarrow{AC}=3 \\overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "5469",
"split": "train"
} |
math | [
{
"content": "Determine the area of a triangle with side lengths 7, 7, and 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{42.75}}{2}",
"style": "rule"
} | {
"index": "5470",
"split": "train"
} |
math | [
{
"content": "On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches. Please output... | {
"ground_truth": ".34",
"style": "rule"
} | {
"index": "5471",
"split": "train"
} |
math | [
{
"content": "A tetrahedron is formed using the vertices of a cube. How many such distinct tetrahedrons can be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "5472",
"split": "train"
} |
math | [
{
"content": "Determine the base seven product of the numbers $321_7$ and $13_7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4503_7",
"style": "rule"
} | {
"index": "5473",
"split": "train"
} |
math | [
{
"content": "At Typico High School, $60\\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\\%$ say that they dislike it, and the rest say that they like it. What fraction o... | {
"ground_truth": "25\\%",
"style": "rule"
} | {
"index": "5474",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \\geq 2$, find the minimum value of $x+ \\sqrt {3}y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3} - 2",
"style": "rule"
} | {
"index": "5475",
"split": "train"
} |
math | [
{
"content": "A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "5476",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x)$ is a rational function such that $3f\\left(\\dfrac{1}{x}\\right)+\\dfrac{2f(x)}{x}=x^2$ for $x\\neq 0$. Find $f(-2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{67}{20}",
"style": "rule"
} | {
"index": "5477",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer $n$ such that $6375$ is a factor of $n!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5478",
"split": "train"
} |
math | [
{
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\omega$, the area of $WXYZ$ can be expressed as the quadratic p... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5479",
"split": "train"
} |
math | [
{
"content": "Someone says that 7 times their birth year divided by 13 gives a remainder of 11, and 13 times their birth year divided by 11 gives a remainder of 7. How old will this person be in the year 1954? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "5480",
"split": "train"
} |
math | [
{
"content": "A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied? Please output the ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5481",
"split": "train"
} |
math | [
{
"content": "There are 10 different mathematics books, 9 different Chinese language books, and 8 different English books. In how many different ways can you take two books of different subjects? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "242",
"style": "rule"
} | {
"index": "5482",
"split": "train"
} |
math | [
{
"content": "Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is either odd or greater tha... | {
"ground_truth": "\\frac{17}{40}",
"style": "rule"
} | {
"index": "5483",
"split": "train"
} |
math | [
{
"content": "Given the set $A = \\{x | x < 1 \\text{ or } x > 5\\}$, and $B = \\{x | a \\leq x \\leq b\\}$, and $A \\cup B = \\mathbb{R}$, $A \\cap B = \\{x | 5 < x \\leq 6\\}$, find the value of $2a - b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "5484",
"split": "train"
} |
math | [
{
"content": "When plotted in the standard rectangular coordinate system, trapezoid $ABCD$ has vertices $A(1, -2)$, $B(1, 1)$, $C(5, 7)$ and $D(5, 1)$. What is the area of trapezoid $ABCD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5485",
"split": "train"
} |
math | [
{
"content": "Given the function $y=\\sin (2x+\\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\\sin 2x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "5486",
"split": "train"
} |
math | [
{
"content": "Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3),$ $(5,5),$ $(7,7),$ and $(9,9),$ determine whether there is a point where the graphs of $y=h(3x)$ and $y=3j(x)$ intersect, and find the sum of the coordinates of that point if it exists. Please output the final answer within \\boxe... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5487",
"split": "train"
} |
math | [
{
"content": "Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digits and all 26 letters... | {
"ground_truth": "1170561600",
"style": "rule"
} | {
"index": "5488",
"split": "train"
} |
math | [
{
"content": "Given that in the geometric sequence $\\{a_n\\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5489",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute $$\\sum_{n=1}^{2020} f(n)^{2}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3431",
"style": "rule"
} | {
"index": "5490",
"split": "train"
} |
math | [
{
"content": "In 1960, there were 450,000 cases of measles reported in the U.S. In 1996, there were 500 cases reported. How many cases of measles would have been reported in 1987 if the number of cases reported from 1960 to 1996 decreased linearly? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "112,\\!875",
"style": "rule"
} | {
"index": "5491",
"split": "train"
} |
math | [
{
"content": "Let\n\\[z = \\frac{(-11 + 13i)^3 \\cdot (24 - 7i)^4}{3 + 4i},\\]and let $w = \\frac{\\overline{z}}{z}.$ Compute $|w|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5492",
"split": "train"
} |
math | [
{
"content": "A triangle has altitudes of lengths 15, 21, and 35. Find its area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "5493",
"split": "train"
} |
math | [
{
"content": "The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "5494",
"split": "train"
} |
math | [
{
"content": "On a road of length $A B = 8 \\text{ km}$, buses travel in both directions at a speed of $12 \\text{ km/h}$. The first bus from each location starts at 6 o'clock, with subsequent buses departing every 10 minutes.\n\nA pedestrian starts walking from $A$ to $B$ at $\\frac{81}{4}$ hours; their speed ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5495",
"split": "train"
} |
math | [
{
"content": "Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). Please output the final answer w... | {
"ground_truth": "75600",
"style": "rule"
} | {
"index": "5496",
"split": "train"
} |
math | [
{
"content": "The greatest common divisor (GCD) of 17 and 51 is , and the least common multiple (LCM) is . The GCD of 6 and 8 is , and the LCM of 8 and 9 is . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "5497",
"split": "train"
} |
math | [
{
"content": "How many natural numbers between 200 and 400 are divisible by 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "5498",
"split": "train"
} |
math | [
{
"content": "Paint both sides of a small wooden board. It takes 1 minute to paint one side, but you must wait 5 minutes for the paint to dry before painting the other side. How many minutes will it take to paint 6 wooden boards in total? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5499",
"split": "train"
} |
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