data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let \\( b = \\frac{\\pi}{2010} \\). Find the smallest positive integer \\( m \\) such that\n\\[\n2\\left[\\cos(b) \\sin(b) + \\cos(4b) \\sin(2b) + \\cos(9b) \\sin(3b) + \\cdots + \\cos(m^2b) \\sin(mb)\\right]\n\\]\nis an integer. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "34400",
"split": "train"
} |
math | [
{
"content": "Given the sets $A=\\{x|x^2 - mx + m^2 - 19 = 0\\}$, $B=\\{x|x^2 - 5x + 6 = 0\\}$, and $C=\\{2, -4\\}$. If $A \\cap B \\neq \\emptyset$ and $A \\cap C = \\emptyset$, find the value of the real number $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "34401",
"split": "train"
} |
math | [
{
"content": "An electronic flea lands on a point $k$ on the number line. In the first step, it jumps 1 unit to the left to $k_1$, in the second step it jumps 2 units to the right to $k_2$, in the third step it jumps 3 units to the left to $k_3$, in the fourth step it jumps 4 units to the right to $k_4$, and so... | {
"ground_truth": "-30.06",
"style": "rule"
} | {
"index": "34402",
"split": "train"
} |
math | [
{
"content": "Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2n)f(2n+1)=9f(n)^{2}+3f(n)$ for all $n \\in \\mathbb{N}$. Compute $f(137)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2215",
"style": "rule"
} | {
"index": "34403",
"split": "train"
} |
math | [
{
"content": "The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a \"lucky coin.\" Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is \"Yes,\" she needs to pay a consultation fee of 20 ... | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "34404",
"split": "train"
} |
math | [
{
"content": "Given the real number \\( x \\), \\([x] \\) denotes the integer part that does not exceed \\( x \\). Find the positive integer \\( n \\) that satisfies:\n\n\\[\n\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2} n\\right] = 1994\n\... | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "34405",
"split": "train"
} |
math | [
{
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 3, \\frac{5 \\pi}{12}, 0 \\right)$ in spherical coordinates to rectangular coordinates. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0,0,3)",
"style": "rule"
} | {
"index": "34406",
"split": "train"
} |
math | [
{
"content": "Suppose \\( a_{n} \\) denotes the last two digits of \\( 7^{n} \\). For example, \\( a_{2} = 49 \\), \\( a_{3} = 43 \\). Find the value of \\( a_{1} + a_{2} + a_{3} + \\cdots + a_{2007} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50199",
"style": "rule"
} | {
"index": "34407",
"split": "train"
} |
math | [
{
"content": "Evaluate \\begin{align*} (5a^2 - 13a + 4)(2a - 3) \\end{align*} for $a = 1\\frac12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "34408",
"split": "train"
} |
math | [
{
"content": "Consider the function $f(x) = \\cos(2x + \\frac{\\pi}{3}) + \\sqrt{3}\\sin 2x + 2m$, where $x \\in \\mathbb{R}$ and $m \\in \\mathbb{R}$.\n(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.\n(II) If $f(x)$ has a minimum value of 0 when $0 \\leq x \\leq \\... | {
"ground_truth": "-\\frac{1}{4}",
"style": "rule"
} | {
"index": "34409",
"split": "train"
} |
math | [
{
"content": "Simplify: $x(3x^2-2)-5(x^2-2x+7)$. Express your answer in the form $Ax^3+Bx^2+Cx+D.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3x^3-5x^2+8x-35",
"style": "rule"
} | {
"index": "34410",
"split": "train"
} |
math | [
{
"content": "There are 30 students in Mrs. Taylor's kindergarten class. If there are twice as many students with blond hair as with blue eyes, 6 students with blond hair and blue eyes, and 3 students with neither blond hair nor blue eyes, how many students have blue eyes? Please output the final answer within... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "34411",
"split": "train"
} |
math | [
{
"content": "Find the largest value of the expression $\\frac{p}{R}\\left( 1- \\frac{r}{3R}\\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{5\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "34412",
"split": "train"
} |
math | [
{
"content": "A $5 \\times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \\times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one co... | {
"ground_truth": "\\boxed{60}",
"style": "rule"
} | {
"index": "34413",
"split": "train"
} |
math | [
{
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -5 \\mathbf{v}\\]for all vectors $\\mathbf{v}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -5 & 0 \\\\ 0 & -5 \\end{pmatrix}",
"style": "rule"
} | {
"index": "34414",
"split": "train"
} |
math | [
{
"content": "In a positive geometric sequence $\\{a_n\\}$, $a_1=1$, and the sum of the first $n$ terms is $S_n$. If $-a_3$, $a_2$, and $a_4$ form an arithmetic sequence, then find the value of $S_7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "34415",
"split": "train"
} |
math | [
{
"content": "Given that $\\sum_{k=1}^{40}\\sin 4k=\\tan \\frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\\frac{p}{q}<90,$ find $p+q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "34416",
"split": "train"
} |
math | [
{
"content": "Sale prices at the Ajax Outlet Store are $50\\%$ below original prices. On Saturdays an additional discount of $20\\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\\$ 180$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$72",
"style": "rule"
} | {
"index": "34417",
"split": "train"
} |
math | [
{
"content": "Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the base... | {
"ground_truth": "28812",
"style": "rule"
} | {
"index": "34418",
"split": "train"
} |
math | [
{
"content": "If $4:x^2 = x:16$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34419",
"split": "train"
} |
math | [
{
"content": "What two digits should be added to the right of the number 2013, so that the resulting six-digit number is divisible by 101? Find all possible solutions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "34420",
"split": "train"
} |
math | [
{
"content": "There are 19 weights with values $1, 2, 3, \\ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all the iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. Please output the final answer within \\b... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34421",
"split": "train"
} |
math | [
{
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $\\triangle D'E'F'$. F... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "34422",
"split": "train"
} |
math | [
{
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "34423",
"split": "train"
} |
math | [
{
"content": "An error of $.05''$ is made in the measurement of a line $25''$ long, and an error of $.3''$ is made in the measurement of a line $150''$ long. In comparison with the relative error of the first measurement, find the relative error of the second measurement. Please output the final answer within \... | {
"ground_truth": "0.2\\%",
"style": "rule"
} | {
"index": "34424",
"split": "train"
} |
math | [
{
"content": "An assembly line produces an average of $85\\%$ first-grade products. How many products need to be selected so that with a probability of 0.997, the deviation of the frequency of first-grade products from the probability $p=0.85$ does not exceed $0.01$ in absolute value? Please output the final an... | {
"ground_truth": "11171",
"style": "rule"
} | {
"index": "34425",
"split": "train"
} |
math | [
{
"content": "Given \\( x \\in [0, 2\\pi] \\), determine the maximum value of the function \n\\[ \nf(x) = \\sqrt{4 \\cos^2 x + 4 \\sqrt{6} \\cos x + 6} + \\sqrt{4 \\cos^2 x - 8 \\sqrt{6} \\cos x + 4 \\sqrt{2} \\sin x + 22}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2(\\sqrt{6} + \\sqrt{2})",
"style": "rule"
} | {
"index": "34426",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 6.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{55}{7}",
"style": "rule"
} | {
"index": "34427",
"split": "train"
} |
math | [
{
"content": "The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Express your answer as a ... | {
"ground_truth": "\\frac{60}{7}",
"style": "rule"
} | {
"index": "34428",
"split": "train"
} |
math | [
{
"content": "Solve the following equations:\n1. $4x=20$\n2. $x-18=40$\n3. $x\\div7=12$\n4. $8n\\div2=15$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{15}{4}",
"style": "rule"
} | {
"index": "34429",
"split": "train"
} |
math | [
{
"content": "Given a triangle with side lengths \\( l, m, n \\), where \\( l > m > n \\) are integers, and it is known that:\n\\[\n\\left\\{\\frac{3^{l}}{10^{4}}\\right\\} = \\left\\{\\frac{3^{m}}{10^{4}}\\right\\} = \\left\\{\\frac{3^{n}}{10^{4}}\\right\\}\n\\]\nwhere \\( \\{x\\} = x - \\lfloor x \\rfloor \\)... | {
"ground_truth": "3003",
"style": "rule"
} | {
"index": "34430",
"split": "train"
} |
math | [
{
"content": "Given that in the polar coordinate system, the equation of the curve $\\Omega$ is $\\rho=6\\cos \\theta$. With the pole as the origin of the rectangular coordinate system, the polar axis as the positive semi-axis of the $x$-axis, and the same length unit in both coordinate systems, establish a rec... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34431",
"split": "train"
} |
math | [
{
"content": "How many days have passed from March 19, 1990, to March 23, 1996, inclusive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2197",
"style": "rule"
} | {
"index": "34432",
"split": "train"
} |
math | [
{
"content": "Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\\$27.30$ each. If Sandy can cut... | {
"ground_truth": " \\$ 54.60",
"style": "rule"
} | {
"index": "34433",
"split": "train"
} |
math | [
{
"content": "Let us call a number \\( \\mathrm{X} \\) \"50-podpyirayushchim\" if for any 50 real numbers \\( a_{1}, \\ldots, a_{50} \\) whose sum is an integer, there exists at least one \\( a_i \\) such that \\( \\left|a_{i}-\\frac{1}{2}\\right| \\geq X \\).\n\nFind the greatest 50-podpyirayushchee \\( X \\),... | {
"ground_truth": "0.01",
"style": "rule"
} | {
"index": "34434",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C:\\frac{{{x}^{2}}}{{{a}^{2}}}+\\frac{{{y}^{2}}}{{{b}^{2}}}=1\\left( a > b > 0 \\right)$ with left and right focal points ${F}_{1},{F}_{2}$, and a point $P\\left( 1,\\frac{\\sqrt{2}}{2} \\right)$ on the ellipse such that $\\vert P{F}_{1}\\vert+\\vert P{F}_{2}\\vert=2 \\sqrt{2}$.\... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "34435",
"split": "train"
} |
math | [
{
"content": "Calculate: $-{1^{2022}}+{({\\frac{1}{3}})^{-2}}+|{\\sqrt{3}-2}|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10-\\sqrt{3}",
"style": "rule"
} | {
"index": "34436",
"split": "train"
} |
math | [
{
"content": "In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "34437",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^3 + 3(a-1)x^2 - 12ax + b$ has a local maximum $M$ at $x=x_1$ and a local minimum $N$ at $x=x_2$,\n\n(1) If the tangent line of the graph of $f(x)$ at its intersection with the y-axis is $24x - y - 10 = 0$, find the values of $x_1$, $x_2$, $M$, and $N$.\n\n(2) If $f(1) ... | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "34438",
"split": "train"
} |
math | [
{
"content": "A certain school has a total of 1,000 students, including 380 first-year students and 180 male second-year students. It is known that the probability of drawing a female second-year student from all the students is 0.19. If stratified sampling (by grade) is used to draw 100 students from the entir... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "34439",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\\frac{|MO|}{|MF|}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "34440",
"split": "train"
} |
math | [
{
"content": "Given the digits 0, 1, 2, 3, 4, 5, calculate the number of six-digit numbers that can be formed without repetition and alternating between odd and even digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34441",
"split": "train"
} |
math | [
{
"content": "Suppose that the number $\\sqrt{7200} - 61$ can be expressed in the form $(\\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34442",
"split": "train"
} |
math | [
{
"content": "How many unordered pairs of positive integers that are relatively prime to each other have a sum of $285? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "34443",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be an odd function defined on $(-\\infty, +\\infty)$, and $f(x+2) = -f(x)$. Given that $f(x) = x$ for $0 \\leq x \\leq 1$, find $f(3\\pi)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 - 3\\pi",
"style": "rule"
} | {
"index": "34444",
"split": "train"
} |
math | [
{
"content": "Compute $2011 \\cdot 2012 \\cdot 2013 \\cdot 2014$ modulo 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34445",
"split": "train"
} |
math | [
{
"content": "We call a four-digit number with the following property a \"centered four-digit number\": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit number is exactly in the... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "34446",
"split": "train"
} |
math | [
{
"content": "The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.\n\n[asy]\ndraw((-2,0)--(7,0),linewidth(1),Arrows);\ndraw((0,-1)--(0,4),linewidth(1),Arrows);\n\ndraw((1,.25)--(1,-.25)... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "34447",
"split": "train"
} |
math | [
{
"content": "If $x+\\sqrt{25}=\\sqrt{36}$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "34448",
"split": "train"
} |
math | [
{
"content": "Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted. Please output the final answer within \\... | {
"ground_truth": "\\frac{1}{256}",
"style": "rule"
} | {
"index": "34449",
"split": "train"
} |
math | [
{
"content": "From 1000 to 9999, a four-digit number is randomly chosen. The probability that all the digits in this number are different and the units digit is an odd number is ( ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{56}{225}$",
"style": "rule"
} | {
"index": "34450",
"split": "train"
} |
math | [
{
"content": "Tanks has a pile of 5 blue cards and 5 red cards. Every morning, he takes a card and throws it down a well. What is the probability that the first card he throws down and the last card he throws down are the same color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "34451",
"split": "train"
} |
math | [
{
"content": "A tetrahedron \\(ABCD\\) has edge lengths 7, 13, 18, 27, 36, 41, with \\(AB = 41\\). Determine the length of \\(CD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "34452",
"split": "train"
} |
math | [
{
"content": "What is the smallest prime divisor of $5^{23} + 7^{17}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34453",
"split": "train"
} |
math | [
{
"content": "Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9 \\sqrt{3}",
"style": "rule"
} | {
"index": "34454",
"split": "train"
} |
math | [
{
"content": "The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \\sqrt{a^{2} + b^{2}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "34455",
"split": "train"
} |
math | [
{
"content": "A proper divisor of a number is one that is not equal to the number itself. What is the sum of the proper divisors of \\( 450 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "759",
"style": "rule"
} | {
"index": "34456",
"split": "train"
} |
math | [
{
"content": "What is the least common multiple of 3, 4, 6 and 15? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34457",
"split": "train"
} |
math | [
{
"content": "Given that socks cost $4 per pair and a T-shirt costs $5 more than a pair of socks, and each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games, with a total cost of $2366, determine the number of members in the League. Please output the ... | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "34458",
"split": "train"
} |
math | [
{
"content": "\\( BL \\) is the angle bisector of triangle \\( ABC \\). Find its area if it is known that \\( |AL| = 3 \\), \\( |BL| = 6\\sqrt{5} \\), and \\( |CL| = 4 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{21 \\sqrt{55}}{4}",
"style": "rule"
} | {
"index": "34459",
"split": "train"
} |
math | [
{
"content": "Compute $\\sqrt{125}\\cdot\\sqrt{45}\\cdot \\sqrt{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75\\sqrt{10}",
"style": "rule"
} | {
"index": "34460",
"split": "train"
} |
math | [
{
"content": "Given \\\\(\\cos \\left(\\alpha+ \\frac{\\pi}{6}\\right)-\\sin \\alpha= \\frac{2 \\sqrt{3}}{3} \\\\), determine the value of \\\\(\\sin \\left(\\alpha- \\frac{7\\pi}{6}\\right) \\\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "34461",
"split": "train"
} |
math | [
{
"content": "Find the smallest solution to the equation \\[\\frac{2x}{x-2} + \\frac{2x^2-24}{x} = 11.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1-\\sqrt{65}}{4}",
"style": "rule"
} | {
"index": "34462",
"split": "train"
} |
math | [
{
"content": "Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize h... | {
"ground_truth": "\\frac{7}{36}",
"style": "rule"
} | {
"index": "34463",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), \\( AB = AC \\), \\( \\angle A = 80^\\circ \\), and point \\( D \\) is inside the triangle such that \\( \\angle DAB = \\angle DBA = 10^\\circ \\). Find the measure of \\( \\angle ACD \\). Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34464",
"split": "train"
} |
math | [
{
"content": "Given two non-zero vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ that satisfy $| \\overrightarrow {a} + \\overrightarrow {b} | = | \\overrightarrow {a} - \\overrightarrow {b} | = \\sqrt {2} | \\overrightarrow {a} |$, find the angle between vector $\\overrightarrow {b}$ and $\\overright... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "34465",
"split": "train"
} |
math | [
{
"content": "The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height \\( a \\) to a mountain of height \\( b \\) takes \\( (b-a)^{2} \\) time. Suppose that you start on the mountain of height 1 and that you ca... | {
"ground_truth": "212",
"style": "rule"
} | {
"index": "34466",
"split": "train"
} |
math | [
{
"content": "What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "34467",
"split": "train"
} |
math | [
{
"content": "Circle $O$ is located on the coordinate plane with center at $(2,3)$. One endpoint of a diameter is at $(-1,-1)$. What are the coordinates of the other endpoint of this diameter? Express your answer as an ordered pair. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(5,7)",
"style": "rule"
} | {
"index": "34468",
"split": "train"
} |
math | [
{
"content": "Let $A,$ $B,$ $C$ be the angles of a non-right triangle. Compute\n\\[\\begin{vmatrix} \\tan A & 1 & 1 \\\\ 1 & \\tan B & 1 \\\\ 1 & 1 & \\tan C \\end{vmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34469",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2x^{3}+ax^{2}+bx+1$, it reaches an extreme value of $-6$ at $x=1$.\n\n(1) Find the values of the real numbers $a$ and $b$.\n(2) Find the maximum and minimum values of the function $f(x)$ on the interval $[−2,2]$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "34470",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. Claudio has $n$ cards, each labeled with a different number from 1 to n. He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive integer $m$, it is possible to select some subset of the cards so that the... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "34471",
"split": "train"
} |
math | [
{
"content": "The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34472",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region in space defined by\n\\[|x - y + z| + |x - y - z| \\le 10\\]and $x, y, z \\ge 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "34473",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin(α + \\frac{π}{6}) - \\cos α = \\frac{1}{3}$, find the value of $2 \\sin α \\cos(α + \\frac{π}{6})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{18}",
"style": "rule"
} | {
"index": "34474",
"split": "train"
} |
math | [
{
"content": "The knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knight... | {
"ground_truth": "\\frac{7}{27}",
"style": "rule"
} | {
"index": "34475",
"split": "train"
} |
math | [
{
"content": "Mary and James each sit in a row of 10 chairs. They choose their seats at random. Two of the chairs (chair number 4 and chair number 7) are broken and cannot be chosen. What is the probability that they do not sit next to each other? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "34476",
"split": "train"
} |
math | [
{
"content": "A basket contains 12 apples and 15 bananas. If 3 more bananas are added to the basket, what fraction of the fruit in the basket will be bananas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "34477",
"split": "train"
} |
math | [
{
"content": "The function $f: \\mathbb{Z}^{2} \\rightarrow \\mathbb{Z}$ satisfies - $f(x, 0)=f(0, y)=0$, and - $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "77500",
"style": "rule"
} | {
"index": "34478",
"split": "train"
} |
math | [
{
"content": "Given $\\frac{x}{y}=\\frac{10}{4}.$ If $y = 18$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "34479",
"split": "train"
} |
math | [
{
"content": "Find the value of $$\\sum_{1 \\leq a<b<c} \\frac{1}{2^{a} 3^{b} 5^{c}}$$ (i.e. the sum of $\\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1624}",
"style": "rule"
} | {
"index": "34480",
"split": "train"
} |
math | [
{
"content": "What is the sum of $x + y$ if the sequence $3, ~8, ~13, \\ldots, ~x, ~y, ~33$ forms an arithmetic sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "34481",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $3a^{2}+\\left[a^{2}+\\left(5a^{2}-2a\\right)-3\\left(a^{2}-3a\\right)\\right]$, where $a=-2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34482",
"split": "train"
} |
math | [
{
"content": "Find the value of \\(m + n\\) where \\(m\\) and \\(n\\) are integers defined as follows: The positive difference between the roots of the quadratic equation \\(5x^2 - 9x - 14 = 0\\) can be expressed as \\(\\frac{\\sqrt{m}}{n}\\), with \\(m\\) not divisible by the square of any prime number. Please... | {
"ground_truth": "366",
"style": "rule"
} | {
"index": "34483",
"split": "train"
} |
math | [
{
"content": "During a break between voyages, a sailor turned 20 years old. All six crew members gathered in the cabin to celebrate. \"I am twice the age of the cabin boy and 6 years older than the engineer,\" said the helmsman. \"And I am as much older than the cabin boy as I am younger than the engineer,\" no... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34484",
"split": "train"
} |
math | [
{
"content": "Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at constant but different sp... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "34485",
"split": "train"
} |
math | [
{
"content": "Given that $\\angle C=90^{\\circ}$, \\(A D=D B\\), \\(D E \\perp A B\\), \\(A B=20\\), and \\(A C=12\\), find the area of quadrilateral \\(AD EC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58 \\frac{1}{2}",
"style": "rule"
} | {
"index": "34486",
"split": "train"
} |
math | [
{
"content": "Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? \n\nA) 16\nB) 20\nC) 24\nD) 30\nE) 45 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "34487",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD,$ $P$ is a point on side $\\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\\tan \\angle APD = 3,$ then find $AB.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34488",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = x^2 + 12x + 30 \\). Find the largest real root of the equation \\( f(f(f(x))) = 0 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6 + \\sqrt[8]{6}",
"style": "rule"
} | {
"index": "34489",
"split": "train"
} |
math | [
{
"content": "Due to the increasing personnel exchanges between Hefei and Nanjing as a result of urban development, there is a plan to build a dedicated railway to alleviate traffic pressure, using a train as a shuttle service. It is known that the daily round-trip frequency $y$ of the train is a linear functio... | {
"ground_truth": "7920",
"style": "rule"
} | {
"index": "34490",
"split": "train"
} |
math | [
{
"content": "Given the equation of a line is $Ax+By=0$, choose two different numbers from the set $\\{1, 2, 3, 4, 5\\}$ to be the values of $A$ and $B$ each time, and find the number of different lines obtained. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "34491",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\\overline{AB}$ with $D$ between $A$ and $E$ such that $\\overline{CD}$ and $\\overline{CE}$ trisect $\\angle C.$ If $\\frac{DE}{BE} = \\frac{8}{15},$ then $\\tan B$ can be written as $\\frac{m \\sqrt{p... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "34492",
"split": "train"
} |
math | [
{
"content": "A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "34493",
"split": "train"
} |
math | [
{
"content": "A rectangle has length 8 cm and width $\\pi$ cm. A semi-circle has the same area as the rectangle. What is its radius? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34494",
"split": "train"
} |
math | [
{
"content": "A circle intersects the $y$ -axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$ . Compute the sum of all possible values of $ab - a - b$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "34495",
"split": "train"
} |
math | [
{
"content": "Given right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<150$, determine the number of possible triangles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34496",
"split": "train"
} |
math | [
{
"content": "A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within... | {
"ground_truth": "4640",
"style": "rule"
} | {
"index": "34497",
"split": "train"
} |
math | [
{
"content": "Mark writes the expression $\\sqrt{d}$ for each positive divisor $d$ of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \\sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a ... | {
"ground_truth": "3480",
"style": "rule"
} | {
"index": "34498",
"split": "train"
} |
math | [
{
"content": "A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34499",
"split": "train"
} |
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