data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \\in \\mathbb{Z}$, she ascends to heaven, but if she lan... | {
"ground_truth": "\\frac{13}{22}",
"style": "rule"
} | {
"index": "34600",
"split": "train"
} |
math | [
{
"content": "Given that $A$, $B$, $C$ are the three internal angles of $\\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\\cos ^{2} \\frac {A}{2}+\\cos A=0$.\n(1) Find the value of angle $A$;\n(2) If $a=2 \\sqrt {3},b+c=4$, find the area of $\\triangle ABC$. Please output the final... | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "34601",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| > 0$ and\n\\[az^2 + bz + c = 0.\\]Find the largest possible value of $|z|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "34602",
"split": "train"
} |
math | [
{
"content": "Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "34603",
"split": "train"
} |
math | [
{
"content": "In right triangle $XYZ$ with $\\angle Z = 90^\\circ$, the length $XY = 15$ and the length $XZ = 8$. Find $\\sin Y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{161}}{15}",
"style": "rule"
} | {
"index": "34604",
"split": "train"
} |
math | [
{
"content": "Let $T_{L}=\\sum_{n=1}^{L}\\left\\lfloor n^{3} / 9\\right\\rfloor$ for positive integers $L$. Determine all $L$ for which $T_{L}$ is a square number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "L=1 \\text{ or } L=2",
"style": "rule"
} | {
"index": "34605",
"split": "train"
} |
math | [
{
"content": "Compute the remainder when $$\\sum_{k=1}^{30303} k^{k}$$ is divided by 101. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "34606",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are non-zero numbers such that $x=1+\\frac{1}{y}$ and $y=1+\\frac{1}{x}$, then $y$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x",
"style": "rule"
} | {
"index": "34607",
"split": "train"
} |
math | [
{
"content": "There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a ci... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "34608",
"split": "train"
} |
math | [
{
"content": "Suppose that for some $a,b,c$ we have $a+b+c = 6$, $ab+ac+bc = 5$ and $abc = -12$. What is $a^3+b^3+c^3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "34609",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = \\begin{cases} x^{2}+1 & (x>0) \\\\ 2f(x+1) & (x\\leq 0) \\end{cases}$, find $f(2)$ and $f(-2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34610",
"split": "train"
} |
math | [
{
"content": "In how many ways can 7 people sit around a round table, considering that two seatings are the same if one is a rotation of the other, and additionally, one specific person must sit between two particular individuals? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "34611",
"split": "train"
} |
math | [
{
"content": "Billy Bones has two coins - one gold and one silver. One of them is symmetrical, and the other is not. It is unknown which coin is asymmetrical, but it is known that the asymmetrical coin lands heads with a probability of $p=0.6$.\n\nBilly Bones tossed the gold coin, and it landed heads immediatel... | {
"ground_truth": "5/9",
"style": "rule"
} | {
"index": "34612",
"split": "train"
} |
math | [
{
"content": "Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . Find the maximum value of $A\\cdot M\\cdot C+A\\cdot M+M\\cdot C+C\\cdot A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "34613",
"split": "train"
} |
math | [
{
"content": "Given the function $f\\left( x \\right)=\\sqrt{3}\\sin\\left( \\omega x-\\frac{\\pi }{6} \\right)(\\omega > 0)$, the distance between two adjacent highest points on the graph is $\\pi$.\n\n(1) Find the value of $\\omega$ and the equation of the axis of symmetry for the function $f\\left( x \\right... | {
"ground_truth": "\\frac{3\\sqrt{5}-1}{8}",
"style": "rule"
} | {
"index": "34614",
"split": "train"
} |
math | [
{
"content": "The numbers $5,6,10,17$, and 21 are rearranged so that the sum of the first three numbers is equal to the sum of the last three numbers. Which number is in the middle of this rearrangement? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34615",
"split": "train"
} |
math | [
{
"content": "Find the coefficient of \\(x^9\\) in the polynomial expansion of \\((1+3x-2x^2)^5\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "34616",
"split": "train"
} |
math | [
{
"content": "Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible. Please output the final an... | {
"ground_truth": "15\\sqrt{2}",
"style": "rule"
} | {
"index": "34617",
"split": "train"
} |
math | [
{
"content": "Which of the following integers cannot be written as a product of two integers, each greater than 1: 6, 27, 53, 39, 77? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "34618",
"split": "train"
} |
math | [
{
"content": "Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the ... | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "34619",
"split": "train"
} |
math | [
{
"content": "Calculate the sum $\\frac{3}{50} + \\frac{5}{500} + \\frac{7}{5000}$.\nA) $0.0714$\nB) $0.00714$\nC) $0.714$\nD) $0.0357$\nE) $0.00143$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.0714",
"style": "rule"
} | {
"index": "34620",
"split": "train"
} |
math | [
{
"content": "An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only n... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34621",
"split": "train"
} |
math | [
{
"content": "Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "34622",
"split": "train"
} |
math | [
{
"content": "Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than they disassemble one. When they arrived at the workshop in the morning, several grinders were already assembled.\n\nAt 7:00 AM, Hubert started assembling and Robert started di... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34623",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the expression for \\( a, b > 0 \\):\n\n$$\n\\frac{|4a - 10b| + |2(a - b\\sqrt{3}) - 5(a\\sqrt{3} + b)|}{\\sqrt{a^2 + b^2}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{87}",
"style": "rule"
} | {
"index": "34624",
"split": "train"
} |
math | [
{
"content": "Let the three-digit number \\( n = abc \\). If \\( a, b, \\) and \\( c \\) as the lengths of the sides can form an isosceles (including equilateral) triangle, then how many such three-digit numbers \\( n \\) are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "34625",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word \"SUCCESS\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "34626",
"split": "train"
} |
math | [
{
"content": "Given Karl's rectangular garden measures \\(30\\) feet by \\(50\\) feet with a \\(2\\)-feet wide uniformly distributed pathway and Makenna's garden measures \\(35\\) feet by \\(55\\) feet with a \\(3\\)-feet wide pathway, compare the areas of their gardens, assuming the pathways take up gardening ... | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "34627",
"split": "train"
} |
math | [
{
"content": "What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "272",
"style": "rule"
} | {
"index": "34628",
"split": "train"
} |
math | [
{
"content": "An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right. How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region below that line? Ple... | {
"ground_truth": "637",
"style": "rule"
} | {
"index": "34629",
"split": "train"
} |
math | [
{
"content": "If Sarah is leading a class of 35 students, and each time Sarah waves her hands a prime number of students sit down, determine the greatest possible number of students that could have been standing before her third wave. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "34630",
"split": "train"
} |
math | [
{
"content": "Add $-45.367$, $108.2$, and $23.7654$, then round your answer to the nearest tenth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86.6",
"style": "rule"
} | {
"index": "34631",
"split": "train"
} |
math | [
{
"content": "An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite... | {
"ground_truth": "\\dfrac{79}{90}",
"style": "rule"
} | {
"index": "34632",
"split": "train"
} |
math | [
{
"content": "Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{126}",
"style": "rule"
} | {
"index": "34633",
"split": "train"
} |
math | [
{
"content": "Compute $\\arccos (\\sin 2).$ All functions are in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - \\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "34634",
"split": "train"
} |
math | [
{
"content": "Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "231",
"style": "rule"
} | {
"index": "34635",
"split": "train"
} |
math | [
{
"content": "A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{500}{3} \\pi",
"style": "rule"
} | {
"index": "34636",
"split": "train"
} |
math | [
{
"content": "Given the ellipse C₁: $$\\frac {x^{2}}{a^{2}}$$+ $$\\frac {y^{2}}{b^{2}}$$\\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y<sup>2</sup>\\=4 $$\\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\\frac { \\sqrt {6}}{3}$$.\n(I) Find the equation of C₁.\n(II) A moving... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "34637",
"split": "train"
} |
math | [
{
"content": "A motorcycle travels due west at $\\frac{5}{8}$ mile per minute on a long straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\\frac{1}{2}$ mile per minute. At time $t=0$, the center of the storm is $100$ miles due north of the motorcycle. At time $t=... | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "34638",
"split": "train"
} |
math | [
{
"content": "Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{a+b}{2}",
"style": "rule"
} | {
"index": "34639",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{2}",
"style": "rule"
} | {
"index": "34640",
"split": "train"
} |
math | [
{
"content": "A triangle has a base of 20 inches. Two lines are drawn parallel to the base, intersecting the other two sides and dividing the triangle into four regions of equal area. Determine the length of the parallel line closer to the base. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34641",
"split": "train"
} |
math | [
{
"content": "At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product ... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34642",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\(A B C D\\), the bases \\(A D\\) and \\(B C\\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \\(A B D\\) is tangent to lines \\(B C\\) and \\(C D\\). Find the perimeter of the trapezoid. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "34643",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ that has center $I$ . If $IA = 5, IB = 7, IC = 4, ID = 9$ , find the value of $\\frac{AB}{CD}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35/36",
"style": "rule"
} | {
"index": "34644",
"split": "train"
} |
math | [
{
"content": "The solution to the equation \\(\\arcsin x + \\arcsin 2x = \\arccos x + \\arccos 2x\\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "34645",
"split": "train"
} |
math | [
{
"content": "Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in 250 grams of her le... | {
"ground_truth": "665",
"style": "rule"
} | {
"index": "34646",
"split": "train"
} |
math | [
{
"content": "Simplify $\\dfrac{30}{45} \\cdot \\dfrac{75}{128} \\cdot \\dfrac{256}{150}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "34647",
"split": "train"
} |
math | [
{
"content": "There are constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\\alpha+\\beta$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "137",
"style": "rule"
} | {
"index": "34648",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of bishops that can be placed on an $8 \\times 8$ chessboard such that at most three bishops lie on any diagonal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "34649",
"split": "train"
} |
math | [
{
"content": "Let the function $f(x)= \\sqrt{3}\\cos^2\\omega x+\\sin \\omega x\\cos \\omega x+a$ where $\\omega > 0$, $a\\in\\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\\dfrac{\\pi}{6}$.\n\n(Ⅰ) Find the smallest positive period of $f(x)... | {
"ground_truth": "\\dfrac{ \\sqrt{3}+1}{2}",
"style": "rule"
} | {
"index": "34650",
"split": "train"
} |
math | [
{
"content": "A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$, $S$, and the upper ... | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "34651",
"split": "train"
} |
math | [
{
"content": "A student passes through three intersections with traffic lights, labeled \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\), on the way to school. It is known that the probabilities of encountering a red light at intersections \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) are \\\\( \\dfrac {1}{3} \\\\), \\\\(... | {
"ground_truth": "\\dfrac {235}{3}",
"style": "rule"
} | {
"index": "34652",
"split": "train"
} |
math | [
{
"content": "There are two straight lines, each of which passes through four points of the form $(1,0,a), (b,1,0), (0,c,1),$ and $(6d,6d,-d),$ where $a,b,c,$ and $d$ are real numbers, not necessarily in that order. Enter all possible values of $d,$ separated by commas. Please output the final answer within \\... | {
"ground_truth": "\\frac{1}{3}, \\frac{1}{8}",
"style": "rule"
} | {
"index": "34653",
"split": "train"
} |
math | [
{
"content": "Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "217",
"style": "rule"
} | {
"index": "34654",
"split": "train"
} |
math | [
{
"content": "The triangle $\\triangle ABC$ is an isosceles triangle where $AC = 6$ and $\\angle A$ is a right angle. If $I$ is the incenter of $\\triangle ABC,$ then what is $BI$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{2} - 6",
"style": "rule"
} | {
"index": "34655",
"split": "train"
} |
math | [
{
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "799",
"style": "rule"
} | {
"index": "34656",
"split": "train"
} |
math | [
{
"content": "Let $O$ be an interior point of triangle $ABC$, and let $s_1=OA+OB+OC$. If $s_2=AB+BC+CA$, then Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$s_2\\ge 2s_1,s_1 \\le s_2$",
"style": "rule"
} | {
"index": "34657",
"split": "train"
} |
math | [
{
"content": "Compute the radius of the inscribed circle of a triangle with sides 15,16 , and 17 . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{21}",
"style": "rule"
} | {
"index": "34658",
"split": "train"
} |
math | [
{
"content": "Let $a_0=29$ , $b_0=1$ and $$ a_{n+1} = a_n+a_{n-1}\\cdot b_n^{2019}, \\qquad b_{n+1}=b_nb_{n-1} $$ for $n\\geq 1$ . Determine the smallest positive integer $k$ for which $29$ divides $\\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$ . P... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "34659",
"split": "train"
} |
math | [
{
"content": "Given that the digits of the integers must be from the set $\\{1,3,4,5,6,8\\}$ and the integers must be between $300$ and $800$, determine how many even integers have all distinct digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34660",
"split": "train"
} |
math | [
{
"content": "Consider a five-digit integer in the form $AB,BCA$, where $A$, $B$, and $C$ are distinct digits. What is the largest possible value of $AB,BCA$ that is divisible by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98,879",
"style": "rule"
} | {
"index": "34661",
"split": "train"
} |
math | [
{
"content": "The largest and smallest possible three-digit numbers that can be formed using the digits 5, 1, and 9 are found by maximizing and minimizing the order of the digits, respectively. Calculate the difference between these two numbers. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "792",
"style": "rule"
} | {
"index": "34662",
"split": "train"
} |
math | [
{
"content": "A club has 10 members, 5 boys and 5 girls. Two of the members are chosen at random. What is the probability that they are both girls? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2}{9}",
"style": "rule"
} | {
"index": "34663",
"split": "train"
} |
math | [
{
"content": "In the sum shown, $P, Q$ and $R$ represent three different single digits. What is the value of $P+Q+R$?\n\n\\begin{tabular}{r}\n$P 7 R$ \\\\\n$+\\quad 39 R$ \\\\\n\\hline$R Q 0$\n\\end{tabular} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "34664",
"split": "train"
} |
math | [
{
"content": "A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "34665",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "34666",
"split": "train"
} |
math | [
{
"content": "In the trapezium \\(ABCD\\), the lines \\(AB\\) and \\(DC\\) are parallel, \\(BC = AD\\), \\(DC = 2 \\times AD\\), and \\(AB = 3 \\times AD\\). The angle bisectors of \\(\\angle DAB\\) and \\(\\angle CBA\\) intersect at the point \\(E\\). What fraction of the area of the trapezium \\(ABCD\\) is th... | {
"ground_truth": "3/5",
"style": "rule"
} | {
"index": "34667",
"split": "train"
} |
math | [
{
"content": "For which natural numbers \\( n \\) is the sum \\( 5^n + n^5 \\) divisible by 13? What is the smallest \\( n \\) that satisfies this condition? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34668",
"split": "train"
} |
math | [
{
"content": "If $a, b$, and $c$ are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "34669",
"split": "train"
} |
math | [
{
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{15})$ be a permutation of $(1,2,3,\\ldots,15)$ for which\n$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$\nFind the number of such permutations. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "3003",
"style": "rule"
} | {
"index": "34670",
"split": "train"
} |
math | [
{
"content": "The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\\sin{\\frac{m\\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m<n.$ Find $n.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "34671",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $\\cos C=-\\frac{1}{4}$.\n1. If $b=3$, find the value of $c$.\n2. If $c=2\\sqrt{6}$, find the value of $\\sin B$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\frac{\\sqrt{10}}{4}",
"style": "rule"
} | {
"index": "34672",
"split": "train"
} |
math | [
{
"content": "The arithmetic progressions $\\{2, 5, 8, 11, \\ldots\\}$ and $\\{3, 10, 17, 24, \\ldots \\}$ have some common values. What is the largest value less than 500 that they have in common? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "479",
"style": "rule"
} | {
"index": "34673",
"split": "train"
} |
math | [
{
"content": "Given the parameter equation of line $l$ and the equation of circle $C$ in the polar coordinate system, find the rectangular coordinate equation of circle $C$ and the minimum value of $\\frac{1}{|PA|} + \\frac{1}{|PB|}$, where $P(1, 2)$ and $A$, $B$ are the intersection points of line $l$ and circ... | {
"ground_truth": "\\frac{2\\sqrt{7}}{7}",
"style": "rule"
} | {
"index": "34674",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13.5",
"style": "rule"
} | {
"index": "34675",
"split": "train"
} |
math | [
{
"content": "It is now between 10:00 and 11:00 o'clock, and six minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10:15",
"style": "rule"
} | {
"index": "34676",
"split": "train"
} |
math | [
{
"content": "Find the number of functions of the form \\( f(x) = ax^3 + bx^2 + cx + d \\) such that\n\\[ f(x)f(-x) = f(x^3). \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34677",
"split": "train"
} |
math | [
{
"content": "Given $a, b, c, d \\in \\mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \\cdot 10^{3} + b \\cdot 10^{2} + c \\cdot 10 + d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1949",
"style": "rule"
} | {
"index": "34678",
"split": "train"
} |
math | [
{
"content": "Let non-negative real numbers \\(a_1, a_2, \\ldots, a_{100}\\) satisfy: \\( a_i + a_{i+1} + a_{i+2} \\leq 1 \\) for \\( 1 \\leq i \\leq 100 \\), where \\(a_{101} = a_1\\) and \\(a_{102} = a_2\\). Find the maximum value of \\(\\sum_{i=1}^{100} a_i a_{i+2}\\). Please output the final answer within \... | {
"ground_truth": "25/2",
"style": "rule"
} | {
"index": "34679",
"split": "train"
} |
math | [
{
"content": "Find the smallest integer $n$ such that $\\sqrt{n+99}-\\sqrt{n}<1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2402",
"style": "rule"
} | {
"index": "34680",
"split": "train"
} |
math | [
{
"content": "What is the greatest prime factor of 221? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "34681",
"split": "train"
} |
math | [
{
"content": "Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Pr... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "34682",
"split": "train"
} |
math | [
{
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + 1 < y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"index": "34683",
"split": "train"
} |
math | [
{
"content": "A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:\n\n- If an adventurer has diamonds, then they have either emeralds or sapphires (but not b... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34684",
"split": "train"
} |
math | [
{
"content": "License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the one pictured. North Dakota, on the other hand, has a different format, ... | {
"ground_truth": "28121600",
"style": "rule"
} | {
"index": "34685",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 300^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "34686",
"split": "train"
} |
math | [
{
"content": "Positive real numbers $x \\neq 1$ and $y \\neq 1$ satisfy $\\log_2{x} = \\log_y{16}$ and $xy = 64$. What is $(\\log_2{\\tfrac{x}{y}})^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "34687",
"split": "train"
} |
math | [
{
"content": "Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\\text{eight}}.$ Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "585",
"style": "rule"
} | {
"index": "34688",
"split": "train"
} |
math | [
{
"content": "The polynomial\n$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34689",
"split": "train"
} |
math | [
{
"content": "Two arithmetic sequences, $C$ and $D$, begin with the terms 25 and 40 respectively, both with common differences of absolute value 15, with sequence $C$ increasing and sequence $D$ decreasing. What is the absolute value of the difference between the 60th terms of sequences $C$ and $D$? Please outp... | {
"ground_truth": "1755",
"style": "rule"
} | {
"index": "34690",
"split": "train"
} |
math | [
{
"content": "Find the curve defined by the equation\n\\[r = 4 \\tan \\theta \\sec \\theta.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(C)}",
"style": "rule"
} | {
"index": "34691",
"split": "train"
} |
math | [
{
"content": "In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\\ell_A$ has slope 1 and passes through $A$. Line $\\ell_B$ is vertical and passes through $B$. Line $\\ell_C$ has slope $-1$ and passes through $C$. The three lines $\\ell_A$, $\\ell_B$, and $\\ell_... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "34692",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$ with right angle at $Z$, $\\angle XYZ < 45^\\circ$ and $XY = 6$. A point $Q$ on $\\overline{XY}$ is chosen such that $\\angle YQZ = 3\\angle XQZ$ and $QZ = 2$. Determine the ratio $\\frac{XQ}{YQ}$ in simplest form. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{7 + 3\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "34693",
"split": "train"
} |
math | [
{
"content": "Given that the perimeter of each of the nine small equilateral triangles is $6 \\mathrm{cm}$, calculate the perimeter of $\\triangle A B C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "34694",
"split": "train"
} |
math | [
{
"content": "Find one-fourth of 6 times 8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34695",
"split": "train"
} |
math | [
{
"content": "Consider the set of points that are inside or within two units of a rectangular parallelepiped that measures 2 by 3 by 6 units. Calculate the total volume of this set, expressing your answer in the form $\\frac{m+n\\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n$ and $p$ being rela... | {
"ground_truth": "\\frac{540 + 164\\pi}{3}",
"style": "rule"
} | {
"index": "34696",
"split": "train"
} |
math | [
{
"content": "A baseball league has nine teams. During the season, each of the nine teams plays exactly three games with each of the other teams. What is the total number of games played? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "34697",
"split": "train"
} |
math | [
{
"content": "The weight of grain in a sample of 256 grains is 18 grains, and the total weight of rice is 1536 dan. Calculate the amount of mixed grain in the total batch of rice. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "34698",
"split": "train"
} |
math | [
{
"content": "Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34699",
"split": "train"
} |
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