data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$ . Find the square of the area of triangle $ADC$... | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "5600",
"split": "train"
} |
math | [
{
"content": "Ellie and Sam run on the same circular track but in opposite directions, with Ellie running counterclockwise and Sam running clockwise. Ellie completes a lap every 120 seconds, while Sam completes a lap every 75 seconds. They both start from the same starting line simultaneously. Ten to eleven min... | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} | {
"index": "5601",
"split": "train"
} |
math | [
{
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect square factor other than one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "5602",
"split": "train"
} |
math | [
{
"content": "Inside the cube \\( ABCD A_1B_1C_1D_1 \\), there is a center \\( O \\) of a sphere with a radius of 10. The sphere intersects the face \\( AA_1D_1D \\) along a circle of radius 1, the face \\( A_1B_1C_1D_1 \\) along a circle of radius 1, and the face \\( CDD_1C_1 \\) along a circle of radius 3. Fi... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5603",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=\\sin \\left( 2x+ \\frac{π}{6} \\right)+ \\frac{3}{2}$, $x\\in R$.\n\n(1) Find the minimum positive period of the function $f(x)$;\n\n(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;\n\n(3) Find the maximum value of the function and the corresponding $x$ ... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "5604",
"split": "train"
} |
math | [
{
"content": "Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pie... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "5605",
"split": "train"
} |
math | [
{
"content": "Crisp All, a basketball player, is dropping dimes and nickels on a number line. Crisp drops a dime on every positive multiple of 10 , and a nickel on every multiple of 5 that is not a multiple of 10. Crisp then starts at 0 . Every second, he has a $\\frac{2}{3}$ chance of jumping from his current ... | {
"ground_truth": "\\frac{20}{31}",
"style": "rule"
} | {
"index": "5606",
"split": "train"
} |
math | [
{
"content": "In a certain country, there are 100 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that for each airline, it must be possible to travel from any city to any other city (possibly with transfers). What is... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "5607",
"split": "train"
} |
math | [
{
"content": "Compute the positive real number $x$ satisfying $x^{\\left(2 x^{6}\\right)}=3$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[6]{3}",
"style": "rule"
} | {
"index": "5608",
"split": "train"
} |
math | [
{
"content": "Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the numbers without proof.\... | {
"ground_truth": "{6,7,7.5,8,8.5,9,10}",
"style": "rule"
} | {
"index": "5609",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region in space defined by\n\\[ |x + 2y + z| + |x + 2y - z| \\le 12 \\]\nand $x, y \\geq 0$, $z \\geq -2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "5610",
"split": "train"
} |
math | [
{
"content": "A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dollar amount that this p... | {
"ground_truth": "865",
"style": "rule"
} | {
"index": "5611",
"split": "train"
} |
math | [
{
"content": "Given that the scores X of 10000 people approximately follow a normal distribution N(100,13^2), it is given that P(61 < X < 139)=0.997, find the number of people scoring no less than 139 points in this exam. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5612",
"split": "train"
} |
math | [
{
"content": "Given that out of 8 teams, there are 3 weak teams, these 8 teams are divided into two groups $A$ and $B$ with 4 teams in each group by drawing lots.\n\n1. The probability that one of the groups $A$ or $B$ has exactly two weak teams.\n2. The probability that group $A$ has at least two weak teams. P... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5613",
"split": "train"
} |
math | [
{
"content": "The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 13\\qquad \\textbf{(C)}\\ 18\\qquad \\textbf{(D)}\\ 27\\qquad \\textbf{(E)}\\ 36$ Please output the final... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5614",
"split": "train"
} |
math | [
{
"content": "For a positive integer $n$ , define $n?=1^n\\cdot2^{n-1}\\cdot3^{n-2}\\cdots\\left(n-1\\right)^2\\cdot n^1$ . Find the positive integer $k$ for which $7?9?=5?k?$ .\n\n*Proposed by Tristan Shin* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5615",
"split": "train"
} |
math | [
{
"content": "On the island of Mumble, the Mumblian alphabet has only $5$ letters, and every word in the Mumblian language has no more than $3$ letters in it. How many words are possible? (A word can use a letter more than once, but $0$ letters does not count as a word.) Please output the final answer within \\... | {
"ground_truth": "155",
"style": "rule"
} | {
"index": "5616",
"split": "train"
} |
math | [
{
"content": "Given $\\cos \\alpha= \\frac {1}{7}$, $\\cos (\\alpha+\\beta)=- \\frac {11}{14}$, and $\\alpha\\in(0, \\frac {\\pi}{2})$, $\\alpha+\\beta\\in( \\frac {\\pi}{2},\\pi)$, find the value of $\\cos \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "5617",
"split": "train"
} |
math | [
{
"content": "Given the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ on the interval $(0, +\\infty)$, determine the value of $m$ that makes it a decreasing function. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5618",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\frac{c}{2} = b - a\\cos C$,\n(1) Determine the measure of angle $A$.\n(2) If $a=\\sqrt{15}$ and $b=4$, find the length of side $c$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2 - \\sqrt{3}",
"style": "rule"
} | {
"index": "5619",
"split": "train"
} |
math | [
{
"content": "Compute the smallest base-10 positive integer greater than 15 that is a palindrome both in base 2 and base 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5620",
"split": "train"
} |
math | [
{
"content": "A deck consists of 32 cards divided into 4 suits, each containing 8 cards. In how many ways can we choose 6 cards such that all four suits are represented among them? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "415744",
"style": "rule"
} | {
"index": "5621",
"split": "train"
} |
math | [
{
"content": "In parallelogram $ABCD$, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the... | {
"ground_truth": "$\\frac{9k}{8}$",
"style": "rule"
} | {
"index": "5622",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be nonzero real numbers such that $\\tfrac{1}{3a}+\\tfrac{1}{b}=2011$ and $\\tfrac{1}{a}+\\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1509",
"style": "rule"
} | {
"index": "5623",
"split": "train"
} |
math | [
{
"content": "Find the largest integer $n$ for which $12^n$ evenly divides $20!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "5624",
"split": "train"
} |
math | [
{
"content": "Given a trapezoid \\( ABCD \\) with \\( AD \\parallel BC \\). It turns out that \\( \\angle ABD = \\angle BCD \\). Find the length of segment \\( BD \\) if \\( BC = 36 \\) and \\( AD = 64 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "5625",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ such that the four-digit number $\\underline{x15x}$ is divisible by 18. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5626",
"split": "train"
} |
math | [
{
"content": "Given the probability of failing on the first attempt and succeeding on the second attempt to guess the last digit of a phone number is requested, calculate the probability of this event. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "5627",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ true for all real $t$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "5628",
"split": "train"
} |
math | [
{
"content": "Fill in the blanks with appropriate numbers. \n6.8 + 4.1 + __ = 12 __ + 6.2 + 7.6 = 20 19.9 - __ - 5.6 = 10 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.3",
"style": "rule"
} | {
"index": "5629",
"split": "train"
} |
math | [
{
"content": "A two-gallon container had all of its dimensions tripled. How many gallons does the new container hold? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "5630",
"split": "train"
} |
math | [
{
"content": "Triangle $A B C$ has $A B=1, B C=\\sqrt{7}$, and $C A=\\sqrt{3}$. Let $\\ell_{1}$ be the line through $A$ perpendicular to $A B, \\ell_{2}$ the line through $B$ perpendicular to $A C$, and $P$ the point of intersection of $\\ell_{1}$ and $\\ell_{2}$. Find $P C$. Please output the final answer with... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "5631",
"split": "train"
} |
math | [
{
"content": "Four vertices of a cube are given as \\(A=(1, 2, 3)\\), \\(B=(1, 8, 3)\\), \\(C=(5, 2, 3)\\), and \\(D=(5, 8, 3)\\). Calculate the surface area of the cube. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "5632",
"split": "train"
} |
math | [
{
"content": "Let \\(ABC\\) be a triangle such that \\(\\frac{|BC|}{|AB| - |BC|} = \\frac{|AB| + |BC|}{|AC|}\\). Determine the ratio \\(\\angle A : \\angle C\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 : 2",
"style": "rule"
} | {
"index": "5633",
"split": "train"
} |
math | [
{
"content": "Two circles, each with a radius of 12 cm, overlap such that each circle passes through the center of the other. Determine the length, in cm, of the common chord formed by these circles. Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "12\\sqrt{3}",
"style": "rule"
} | {
"index": "5634",
"split": "train"
} |
math | [
{
"content": "What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5635",
"split": "train"
} |
math | [
{
"content": "A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \\mathrm{~km}$ on the rear wheel or $25000 \\mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the front and rear pairs o... | {
"ground_truth": "18750",
"style": "rule"
} | {
"index": "5636",
"split": "train"
} |
math | [
{
"content": "If A and B can only undertake the first three tasks, while the other three can undertake all four tasks, calculate the total number of different selection schemes for the team leader group to select four people from five volunteers to undertake four different tasks. Please output the final answer ... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "5637",
"split": "train"
} |
math | [
{
"content": "For how many integers $n$ between $1$ and $50$, inclusive, is $\\frac{(n^2-1)!}{(n!)^n}$ an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "5638",
"split": "train"
} |
math | [
{
"content": "The rank of a rational number $q$ is the unique $k$ for which $q=\\frac{1}{a_{1}}+\\cdots+\\frac{1}{a_{k}}$, where each $a_{i}$ is the smallest positive integer such that $q \\geq \\frac{1}{a_{1}}+\\cdots+\\frac{1}{a_{i}}$. Let $q$ be the largest rational number less than \\frac{1}{4}$ with rank 3... | {
"ground_truth": "(5,21,421)",
"style": "rule"
} | {
"index": "5639",
"split": "train"
} |
math | [
{
"content": "An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for ... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "5640",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 3 is inscribed in a right trapezoid, where the shorter base is 4. Find the length of the longer base of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5641",
"split": "train"
} |
math | [
{
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(3000,0),(3000,2000),$ and $(0,2000)$. What is the probability that $x > 5y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{20}",
"style": "rule"
} | {
"index": "5642",
"split": "train"
} |
math | [
{
"content": "Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\\in \\{-2,-1,0,1,2\\}$ and $b\\in \\{-3,-2,-1,1,2,3\\}$. No three of these parabolas have a common point. How man... | {
"ground_truth": "810",
"style": "rule"
} | {
"index": "5643",
"split": "train"
} |
math | [
{
"content": "The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14238",
"style": "rule"
} | {
"index": "5644",
"split": "train"
} |
math | [
{
"content": "There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "5645",
"split": "train"
} |
math | [
{
"content": "Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "5646",
"split": "train"
} |
math | [
{
"content": "Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\\angle G_{3} G_{1} G... | {
"ground_truth": "\\frac{169}{23}",
"style": "rule"
} | {
"index": "5647",
"split": "train"
} |
math | [
{
"content": "What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "5648",
"split": "train"
} |
math | [
{
"content": "Given the progression $10^{\\frac{1}{11}}, 10^{\\frac{2}{11}}, 10^{\\frac{3}{11}}, 10^{\\frac{4}{11}},\\dots , 10^{\\frac{n}{11}}$. The least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is Please output the final answer within \\boxed{}.",... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5649",
"split": "train"
} |
math | [
{
"content": "Write $4.3+3.88$ as a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.18",
"style": "rule"
} | {
"index": "5650",
"split": "train"
} |
math | [
{
"content": "Jane's quiz scores were 98, 97, 92, 85 and 93. What was her mean score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "5651",
"split": "train"
} |
math | [
{
"content": "The operation $\\&$ is defined as $a \\& b = (a+b)(a-b)$. What is the value of $6 \\& 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "5652",
"split": "train"
} |
math | [
{
"content": "A four-digit number \\(\\overline{abcd} (1 \\leqslant a \\leqslant 9, 0 \\leqslant b, c, d \\leqslant 9)\\) is called a \\(P\\) type number if \\(a > b, b < c, c > d\\). It is called a \\(Q\\) type number if \\(a < b, b > c, c < d\\). Let \\(N(P)\\) and \\(N(Q)\\) be the number of \\(P\\) type and... | {
"ground_truth": "285",
"style": "rule"
} | {
"index": "5653",
"split": "train"
} |
math | [
{
"content": "Find the value of the first term in the geometric sequence $a,b,c,32,64$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5654",
"split": "train"
} |
math | [
{
"content": "There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5655",
"split": "train"
} |
math | [
{
"content": "If \"$x^2 > 1$\" is a necessary but not sufficient condition for \"$x < a$\", then the maximum value of $a$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5656",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 3:2.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA = (0,0);\nB = (5,1);\nP = ... | {
"ground_truth": "\\left( \\frac{2}{5}, \\frac{3}{5} \\right)",
"style": "rule"
} | {
"index": "5657",
"split": "train"
} |
math | [
{
"content": "Dan is walking down the left side of a street in New York City and must cross to the right side at one of 10 crosswalks he will pass. Each time he arrives at a crosswalk, however, he must wait $t$ seconds, where $t$ is selected uniformly at random from the real interval $[0,60](t$ can be different... | {
"ground_truth": "60\\left(1-\\left(\\frac{1}{10}\\right)^{\\frac{1}{9}}\\right)",
"style": "rule"
} | {
"index": "5658",
"split": "train"
} |
math | [
{
"content": "A solid triangular prism is made up of 27 identical smaller solid triangular prisms. The length of every edge of each of the smaller prisms is 1. If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted? Please output ... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "5659",
"split": "train"
} |
math | [
{
"content": "Find the increments of the argument and the function for \\( y = 2x^2 + 1 \\) when the argument \\( x \\) changes from 1 to 1.02. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.0808",
"style": "rule"
} | {
"index": "5660",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 8.$ Find the minimum value of\n\\[(3a + b)(a + 3c)(2bc + 4).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "5661",
"split": "train"
} |
math | [
{
"content": "If $4x + 14 = 8x - 48$, what is the value of $2x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "5662",
"split": "train"
} |
math | [
{
"content": "Determine the integer \\( m \\), \\( -180 \\leq m \\leq 180 \\), such that \\(\\sin m^\\circ = \\sin 945^\\circ.\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-135",
"style": "rule"
} | {
"index": "5663",
"split": "train"
} |
math | [
{
"content": "The product, $\\log_a b \\cdot \\log_b a$ is equal to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5664",
"split": "train"
} |
math | [
{
"content": "My friend June likes numbers that have an interesting property: they are divisible by 4. How many different pairings of last two digits are possible in numbers that June likes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "5665",
"split": "train"
} |
math | [
{
"content": "Within the range of 0° to 360°, find the angle(s) with the same terminal side as -120°. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "5666",
"split": "train"
} |
math | [
{
"content": "When f(t)=0.1, solve for t in the function model: $f(t)=\\frac{1}{{1+{e^{-0.22(3t-40)}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5667",
"split": "train"
} |
math | [
{
"content": "The largest prime factor of $999999999999$ is greater than $2006$ . Determine the remainder obtained when this prime factor is divided by $2006$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1877",
"style": "rule"
} | {
"index": "5668",
"split": "train"
} |
math | [
{
"content": "In a certain country, there are 200 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with l... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "5669",
"split": "train"
} |
math | [
{
"content": "In Mrs. Warner's class, there are 30 students. Strangely, 15 of the students have a height of 1.60 m and 15 of the students have a height of 1.22 m. Mrs. Warner lines up \\(n\\) students so that the average height of any four consecutive students is greater than 1.50 m and the average height of an... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "5670",
"split": "train"
} |
math | [
{
"content": "The line $y = 3x - 11$ is parameterized by the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} r \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 4 \\\\ k \\end{pmatrix}.\\]Enter the ordered pair $(r,k).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(4,12)",
"style": "rule"
} | {
"index": "5671",
"split": "train"
} |
math | [
{
"content": "The points $Q(1,-1), R(-1,0)$ and $S(0,1)$ are three vertices of a parallelogram. What could be the coordinates of the fourth vertex of the parallelogram? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-2,2)",
"style": "rule"
} | {
"index": "5672",
"split": "train"
} |
math | [
{
"content": "Determine the number of decreasing sequences of positive integers \\(b_1 \\geq b_2 \\geq b_3 \\geq \\cdots \\geq b_7 \\leq 1500\\) such that \\(b_i - i\\) is divisible by 3 for \\(1 \\leq i \\le 7\\). Express the number of such sequences as \\({m \\choose n}\\) for some integers \\(m\\) and \\(n\\... | {
"ground_truth": "506",
"style": "rule"
} | {
"index": "5673",
"split": "train"
} |
math | [
{
"content": "An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "615",
"style": "rule"
} | {
"index": "5674",
"split": "train"
} |
math | [
{
"content": "What is the sum of the proper divisors of $600$? Additionally, determine if $600$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "5675",
"split": "train"
} |
math | [
{
"content": "To ensure the safety of property during the Spring Festival holiday, an office needs to arrange for one person to be on duty each day for seven days. Given that there are 4 people in the office, and each person needs to work for either one or two days, the number of different duty arrangements is ... | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": "5676",
"split": "train"
} |
math | [
{
"content": "Consider parallelogram $A B C D$ with $A B>B C$. Point $E$ on $\\overline{A B}$ and point $F$ on $\\overline{C D}$ are marked such that there exists a circle $\\omega_{1}$ passing through $A, D, E, F$ and a circle $\\omega_{2}$ passing through $B, C, E, F$. If $\\omega_{1}, \\omega_{2}$ partition ... | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "5677",
"split": "train"
} |
math | [
{
"content": "A certain senior high school has a total of 3200 students, with 1000 students each in the second and third grades. A stratified sampling method is used to draw a sample of size 160. The number of first-grade students that should be drawn is ______ . Please output the final answer within \\boxed{}.... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "5678",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = 4\\cos x \\cos\\left(x - \\frac{\\pi}{3}\\right) - 2$.\n(I) Find the smallest positive period of the function $f(x)$.\n(II) Find the maximum and minimum values of the function $f(x)$ in the interval $\\left[-\\frac{\\pi}{6}, \\frac{\\pi}{4}\\right]$. Please output the fi... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5679",
"split": "train"
} |
math | [
{
"content": "For some complex number $\\omega$ with $|\\omega| = 2,$ there is some real $\\lambda > 1$ such that $\\omega,$ $\\omega^2,$ and $\\lambda \\omega$ form an equilateral triangle in the complex plane. Find $\\lambda.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{13}}{2}",
"style": "rule"
} | {
"index": "5680",
"split": "train"
} |
math | [
{
"content": "What is the remainder of $19^{1999}$ divided by 25? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5681",
"split": "train"
} |
math | [
{
"content": "The range of the function $g(x) = \\frac{2}{2+4x^2}$ can be written as an interval $(a,b]$. What is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5682",
"split": "train"
} |
math | [
{
"content": "Each of the twelve letters in ``MATHEMATICS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``CALM''? Express your answer as a common fraction. Please output the final answer wi... | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} | {
"index": "5683",
"split": "train"
} |
math | [
{
"content": "Given an ellipse E: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}$$\\=1 (a>b>0) passing through point P ($$\\sqrt {3}$$, $$\\frac {1}{2}$$) with its left focus at F ($$-\\sqrt {3}$$, 0).\n\n1. Find the equation of ellipse E.\n2. If A is the right vertex of ellipse E, and the line passing through ... | {
"ground_truth": "$\\frac {2 \\sqrt {5}+ \\sqrt {15}}{4}$",
"style": "rule"
} | {
"index": "5684",
"split": "train"
} |
math | [
{
"content": "Joe has exactly enough paint to paint the surface of a cube whose side length is 2. It turns out that this is also exactly enough paint to paint the surface of a sphere. If the volume of this sphere is $\\frac{K \\sqrt{6}}{\\sqrt{\\pi}}$, then what is $K$? Please output the final answer within \... | {
"ground_truth": "K=8",
"style": "rule"
} | {
"index": "5685",
"split": "train"
} |
math | [
{
"content": "When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5686",
"split": "train"
} |
math | [
{
"content": "What is the sum of the last three digits of each term in the following part of the Fibonacci Factorial Series: $1!+2!+3!+5!+8!+13!+21!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "249",
"style": "rule"
} | {
"index": "5687",
"split": "train"
} |
math | [
{
"content": "Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of t... | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "5688",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 16 positive divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "5689",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a_n\\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5150",
"style": "rule"
} | {
"index": "5690",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{5}{\\sqrt{125}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "5691",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1660",
"style": "rule"
} | {
"index": "5692",
"split": "train"
} |
math | [
{
"content": "Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5693",
"split": "train"
} |
math | [
{
"content": "Evaluate\n\\[\\log_{10}(\\tan 1^{\\circ})+\\log_{10}(\\tan 2^{\\circ})+\\log_{10}(\\tan 3^{\\circ})+\\cdots+\\log_{10}(\\tan 88^{\\circ})+\\log_{10}(\\tan 89^{\\circ}).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "5694",
"split": "train"
} |
math | [
{
"content": "Find the projection of $\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}$ onto the plane $3x - y + 4z = 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -1/2 \\\\ 5/2 \\\\ 1 \\end{pmatrix}",
"style": "rule"
} | {
"index": "5695",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered integer tuples \\((k_1, k_2, k_3, k_4)\\) that satisfy \\(0 \\leq k_i \\leq 20\\) for \\(i = 1, 2, 3, 4\\), and \\(k_1 + k_3 = k_2 + k_4\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6181",
"style": "rule"
} | {
"index": "5696",
"split": "train"
} |
math | [
{
"content": "A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \\geq 0$. Find $\\operatorname{gcd}(a_{999}, a_{2004})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "677",
"style": "rule"
} | {
"index": "5697",
"split": "train"
} |
math | [
{
"content": "The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \\begin{align*} f(x,x) &=x, \\\\ f(x,y) &=f(y,x), \\quad \\text{and} \\\\ (x + y) f(x,y) &= yf(x,x + y). \\end{align*}Calculate $f(14,52)$. Please output the final answer within \\box... | {
"ground_truth": "364",
"style": "rule"
} | {
"index": "5698",
"split": "train"
} |
math | [
{
"content": "Given that $\\text{1 mile} = \\text{8 furlongs}$ and $\\text{1 furlong} = \\text{40 rods}$, the number of rods in one mile is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "320",
"style": "rule"
} | {
"index": "5699",
"split": "train"
} |
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