data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "If $[x]$ is the greatest integer less than or equal to $x$, then $\\sum_{N=1}^{1024}\\left[\\log _{2} N\\right]$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8204",
"style": "rule"
} | {
"index": "32300",
"split": "train"
} |
math | [
{
"content": "Find the largest integer \\( k \\) such that for at least one natural number \\( n > 1000 \\), the number \\( n! = 1 \\cdot 2 \\cdot \\ldots \\cdot n \\) is divisible by \\( 2^{n+k+2} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "32301",
"split": "train"
} |
math | [
{
"content": "What is half of the absolute value of the difference of the squares of 18 and 16? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "32302",
"split": "train"
} |
math | [
{
"content": "A full container holds 150 watermelons and melons with a total value of 24,000 rubles. The total value of all watermelons is equal to the total value of all melons. How much does one watermelon cost in rubles, given that the container can hold 120 melons (without watermelons) or 160 watermelons (w... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "32303",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$, its first term is $7$, and $a_n= \\frac{1}{2}a_{n-1}+3(n\\geqslant 2)$, find the value of $a_6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{193}{32}",
"style": "rule"
} | {
"index": "32304",
"split": "train"
} |
math | [
{
"content": "On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$ , $IJ=3$ , $JK=4$ , $KH=5$ . Find the value of $13(BD)^2$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "32305",
"split": "train"
} |
math | [
{
"content": "If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{216}",
"style": "rule"
} | {
"index": "32306",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha = \\frac{\\sqrt{5}}{5}$ and $\\sin (\\alpha - \\beta) = -\\frac{\\sqrt{10}}{10}$, where both $\\alpha$ and $\\beta$ are acute angles, find the value of $\\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "32307",
"split": "train"
} |
math | [
{
"content": "What is the maximum value of $\\frac{(3^t - 5t)t}{9^t}$ for real values of $t$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{20}",
"style": "rule"
} | {
"index": "32308",
"split": "train"
} |
math | [
{
"content": "In how many distinct ways can I arrange six keys on a keychain, if I want to keep my house key adjacent to my car key and my office key adjacent to my mailbox key? Two arrangements are not considered different if the keys can be rotated or reflected into the same order. Please output the final ans... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32309",
"split": "train"
} |
math | [
{
"content": "From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95\\%",
"style": "rule"
} | {
"index": "32310",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a(4-2 \\sqrt {7}\\cos B)=b(2 \\sqrt {7}\\cos A-5)$, find the minimum value of $\\cos C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "32311",
"split": "train"
} |
math | [
{
"content": "16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32312",
"split": "train"
} |
math | [
{
"content": "Given six students – three boys and three girls – standing in a row, where girls A, B, and C are not allowed to stand next to each other and girls are not allowed to stand at the ends of the row, calculate the number of different arrangements. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "32313",
"split": "train"
} |
math | [
{
"content": "Find the sum of $432_7$, $54_7$, and $6_7$ in base $7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "525_7",
"style": "rule"
} | {
"index": "32314",
"split": "train"
} |
math | [
{
"content": "Let $A, B, C$ be unique collinear points $ AB = BC =\\frac13$ . Let $P$ be a point that lies on the circle centered at $B$ with radius $\\frac13$ and the circle centered at $C$ with radius $\\frac13$ . Find the measure of angle $\\angle PAC$ in degrees. Please output the final answer... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32315",
"split": "train"
} |
math | [
{
"content": "From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property? ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "32316",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^{2}=2x$ and a point $P(m,2)$ on it, find the value of $m$ and the distance between point $P$ and the focus $F$ of the parabola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "32317",
"split": "train"
} |
math | [
{
"content": "The number $0.324375$ can be written as a fraction $\\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2119",
"style": "rule"
} | {
"index": "32318",
"split": "train"
} |
math | [
{
"content": "Given triangle $\\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $\\overrightarrow{BC} \\cdot \\overrightarrow{BA} + 2\\overrightarrow{AC} \\cdot \\overrightarrow{AB} = \\overrightarrow{CA} \\cdot \\overrightarrow{CB}$. <br/>$(1)$ Find the value of $\\frac... | {
"ground_truth": "\\frac{3\\sqrt{2} - \\sqrt{10}}{8}",
"style": "rule"
} | {
"index": "32319",
"split": "train"
} |
math | [
{
"content": "If $2x + 6 = 16$, what is the value of $x + 4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "32320",
"split": "train"
} |
math | [
{
"content": "In a new shooting competition, ten clay targets are set up in four hanging columns with four targets in column $A$, three in column $B$, two in column $C$, and one in column $D$. A shooter must continue following the sequence:\n1) The shooter selects one of the columns.\n2) The shooter must then h... | {
"ground_truth": "12600",
"style": "rule"
} | {
"index": "32321",
"split": "train"
} |
math | [
{
"content": "Let \\( m = \\min \\left\\{ x + 2y + 3z \\mid x^{3} y^{2} z = 1 \\right\\} \\). What is the value of \\( m^{3} \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "32322",
"split": "train"
} |
math | [
{
"content": "In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game a... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "32323",
"split": "train"
} |
math | [
{
"content": "(1) Convert the following angles between degrees and radians: ① -15°; ② $\\frac {7\\pi }{12}$ \n(2) Given that the terminal side of angle $\\alpha$ passes through point P(2sin30°, -2cos30°), find the sine, cosine, and tangent values of angle $\\alpha$. Please output the final answer within \\boxe... | {
"ground_truth": "-\\sqrt {3}",
"style": "rule"
} | {
"index": "32324",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = -x^2 + ax + 3$.\n\n1. When $a=2$, find the interval over which $f(x)$ is monotonically increasing.\n2. If $f(x)$ is an even function, find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "32325",
"split": "train"
} |
math | [
{
"content": "Given the vectors $\\overrightarrow{m}=(x,y)$ and $\\overrightarrow{n}=(x-y)$, let $P$ be a moving point on the curve $\\overrightarrow{m}\\cdot \\overrightarrow{n}=1 (x > 0)$. If the distance from point $P$ to the line $x-y+1=0$ is always greater than $\\lambda$, find the maximum value of the rea... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "32326",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the expression \\((\\sqrt{2(1+\\cos 2x)} - \\sqrt{3-\\sqrt{2}} \\sin x + 1) \\cdot (3 + 2\\sqrt{7-\\sqrt{2}} \\cos y - \\cos 2y)\\). If the answer is not an integer, round it to the nearest whole number. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "32327",
"split": "train"
} |
math | [
{
"content": "Let \\\\(x,y\\\\) satisfy the constraint conditions \\\\(\\begin{cases} & x+y-2\\geqslant 0 \\\\\\\\ & x-y+1\\geqslant 0 \\\\\\\\ & x\\leqslant 3 \\end{cases}\\\\). If the minimum value of \\\\(z=mx+y\\\\) is \\\\(-3\\\\), then the value of \\\\(m\\\\) is . Please output the final answer with... | {
"ground_truth": "- \\dfrac{2}{3}",
"style": "rule"
} | {
"index": "32328",
"split": "train"
} |
math | [
{
"content": "Let $a\\oplus b=3a+4b$ for all real numbers $a$ and $b$. Find $3\\oplus 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "32329",
"split": "train"
} |
math | [
{
"content": "Two people, A and B, visit the \"2011 Xi'an World Horticultural Expo\" together. They agree to independently choose 4 attractions from numbered attractions 1 to 6 to visit, spending 1 hour at each attraction. Calculate the probability that they will be at the same attraction during their last hour... | {
"ground_truth": "\\dfrac{1}{6}",
"style": "rule"
} | {
"index": "32330",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, vertex $B$ is connected to the point that divides side $AC$ into thirds, closer to $A$. The intersection of this segment and the median from $A$ is denoted as $P$, and the centroid of the triangle as $S$. In what ratio does $P$ divide the segment $AS$? Please output the final an... | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": "32331",
"split": "train"
} |
math | [
{
"content": "A circle of radius 1 is internally tangent to two circles of radius 2 at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the figure, that is outside the smaller circle and inside each of the two larger circles? Express your answer in... | {
"ground_truth": "\\frac{5}{3}\\pi - 2\\sqrt{3}",
"style": "rule"
} | {
"index": "32332",
"split": "train"
} |
math | [
{
"content": "Let $\\{b_k\\}$ be a sequence of integers such that $b_1 = 2$ and $b_{m+n} = b_m + b_n + mn + 1$, for all positive integers $m$ and $n$. Find $b_{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "32333",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, where $A > B > C$, if $2 \\cos 2B - 8 \\cos B + 5 = 0$, $\\tan A + \\tan C = 3 + \\sqrt{3}$, and the height $CD$ from $C$ to $AB$ is $2\\sqrt{3}$, then find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12 - 4\\sqrt{3}",
"style": "rule"
} | {
"index": "32334",
"split": "train"
} |
math | [
{
"content": "Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers needed at intersection... | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "32335",
"split": "train"
} |
math | [
{
"content": "Billy wants to determine the average number of cookies in a package of cookies. He purchases 7 packages of cookies, opens them, and counts the number of cookies in each one. He finds that the packages have 8, 10, 12, 15, 16, 17, and 20 cookies. Based on these observations, what is the average numb... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "32336",
"split": "train"
} |
math | [
{
"content": "A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\\ge1,$ the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ are in geometric prog... | {
"ground_truth": "973",
"style": "rule"
} | {
"index": "32337",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, triangle $\\triangle ABC$ has vertices $A(-6, 0)$ and $C(6, 0)$. Vertex $B$ lies on the left branch of the hyperbola $\\frac{x^2}{25} - \\frac{y^2}{11} = 1$. Find the value of $\\frac{\\sin A - \\sin C}{\\sin B}$. Please output the final answer within \\bo... | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "32338",
"split": "train"
} |
math | [
{
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{125}{3}",
"style": "rule"
} | {
"index": "32339",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) is the midpoint of the hypotenuse \\( AC \\) of right triangle \\( ABC \\). Points \\( P \\) and \\( Q \\) on lines \\( AB \\) and \\( BC \\) respectively are such that \\( AP = PM \\) and \\( CQ = QM \\). Find the measure of angle \\( \\angle PQM \\) if \\( \\angle BAC = 17^{\\cir... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "32340",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "32341",
"split": "train"
} |
math | [
{
"content": "When purchasing goods with a total amount of at least 1000 rubles, the store provides a 50% discount on subsequent purchases. With 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 rubles per kg, and sugar ... | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": "32342",
"split": "train"
} |
math | [
{
"content": "A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\frac{3136}{81} \\pi",
"style": "rule"
} | {
"index": "32343",
"split": "train"
} |
math | [
{
"content": "Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6,-12",
"style": "rule"
} | {
"index": "32344",
"split": "train"
} |
math | [
{
"content": "There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n ($1$) for each $k,$ the mathematician who was se... | {
"ground_truth": "332",
"style": "rule"
} | {
"index": "32345",
"split": "train"
} |
math | [
{
"content": "Michael picks a random subset of the complex numbers \\(\\left\\{1, \\omega, \\omega^{2}, \\ldots, \\omega^{2017}\\right\\}\\) where \\(\\omega\\) is a primitive \\(2018^{\\text {th }}\\) root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is \\(... | {
"ground_truth": "\\frac{1009}{2}",
"style": "rule"
} | {
"index": "32346",
"split": "train"
} |
math | [
{
"content": "Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$ . There are two distinct points, $P$ and $Q$ , that are each equidistant from $F$ , from line $L_1$ , and from line $L_2$ . Find the area of ... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "32347",
"split": "train"
} |
math | [
{
"content": "The average of 12 numbers is 90. If the numbers 80, 85, and 92 are removed from the set of numbers, what is the average of the remaining numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{823}{9}",
"style": "rule"
} | {
"index": "32348",
"split": "train"
} |
math | [
{
"content": "(1) The definite integral $\\int_{-1}^{1}(x^{2}+\\sin x)dx=$ ______.\n\n(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.\n\n(3) Given the function $f(x)=\\begin{cases}1-\\log_{a}(x+2), & x\\geqslant 0 \\\\... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "32349",
"split": "train"
} |
math | [
{
"content": "For $x \\ge 0$ the smallest value of $\\frac {4x^2 + 8x + 13}{6(1 + x)}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32350",
"split": "train"
} |
math | [
{
"content": "The kindergarten teacher evenly distributed 270 apples, 180 pears, and 235 oranges to the larger group of children. The remaining quantities of apples, pears, and oranges are in the ratio $3:2:1$. How many children are there in the larger group? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "32351",
"split": "train"
} |
math | [
{
"content": "Given that 3 people are to be selected from 5 girls and 2 boys, if girl A is selected, determine the probability that at least one boy is selected. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "32352",
"split": "train"
} |
math | [
{
"content": "Given $x+m≤{e}^{\\frac{2x}{m}+n}$ holds for any $x\\in \\left(-m,+\\infty \\right)$, then the minimum value of $m\\cdot n$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{e^2}",
"style": "rule"
} | {
"index": "32353",
"split": "train"
} |
math | [
{
"content": "A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of student... | {
"ground_truth": "-13.5",
"style": "rule"
} | {
"index": "32354",
"split": "train"
} |
math | [
{
"content": "Let \\( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \\) be nonnegative real numbers whose sum is 300. Let \\( M \\) be the maximum of the four numbers \\( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \\) and \\( x_{4} + x_{5} \\). Find the least possible value of \\( M \\). Please output the final answer wi... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "32355",
"split": "train"
} |
math | [
{
"content": "In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: $42=\\left(-8053873881207597 \\_\\right)^{3}+(80435758145817515)^{3}+(12602123297335631)^{3}$. One of the digits, labeled by an underscore, is missing. What is that digit? Please output th... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "32356",
"split": "train"
} |
math | [
{
"content": "Find the sum of the squares of the solutions to the equation\n\\[\\left| x^2 - x + \\frac{1}{2010} \\right| = \\frac{1}{2010}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2008}{1005}",
"style": "rule"
} | {
"index": "32357",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$ with diagonals $AC$ and $BD$, intersecting at $O$, suppose $BO=5$, $OD = 7$, $AO=9$, $OC=4$, and $AB=7$. The length of $AD$ is:\n**A)** $15$\n**B)** $\\sqrt{210}$\n**C)** $14$\n**D)** $\\sqrt{220}$\n**E)** $13$ Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\sqrt{210}",
"style": "rule"
} | {
"index": "32358",
"split": "train"
} |
math | [
{
"content": "In the non-convex quadrilateral $ABCD$ shown below, $\\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "32359",
"split": "train"
} |
math | [
{
"content": "How many pairs of positive integers $(x, y)$ have the property that the ratio $x: 4$ equals the ratio $9: y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "32360",
"split": "train"
} |
math | [
{
"content": "From the 200 natural numbers from 1 to 200, how many numbers must be selected to ensure that there are at least 2 numbers whose sum is a multiple of 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "82",
"style": "rule"
} | {
"index": "32361",
"split": "train"
} |
math | [
{
"content": "Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32362",
"split": "train"
} |
math | [
{
"content": "A hexagon has its vertices alternately on two concentric circles with radii 3 units and 5 units and centered at the origin on a Cartesian plane. Each alternate vertex starting from the origin extends radially outward to the larger circle. Calculate the area of this hexagon. Please output the final... | {
"ground_truth": "\\frac{51\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "32363",
"split": "train"
} |
math | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.298",
"style": "rule"
} | {
"index": "32364",
"split": "train"
} |
math | [
{
"content": "If the perimeter of a square is 28, what is the side length of the square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "32365",
"split": "train"
} |
math | [
{
"content": "Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "32366",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF = 60^{\\circ}$. The area of triangle $DEF$ is $14\\sqrt{3}$. The two possible values of the length of side $AB$ are... | {
"ground_truth": "989",
"style": "rule"
} | {
"index": "32367",
"split": "train"
} |
math | [
{
"content": "Let $g$ be a function defined on the positive integers, such that\n\\[g(xy) = g(x) + g(y)\\] for all positive integers $x$ and $y.$ Given $g(12) = 18$ and $g(48) = 26,$ find $g(600).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "32368",
"split": "train"
} |
math | [
{
"content": "Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32369",
"split": "train"
} |
math | [
{
"content": "Using the digits 1, 2, 3, 4, 5, how many even four-digit numbers less than 4000 can be formed if each digit can be used more than once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "32370",
"split": "train"
} |
math | [
{
"content": "The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \\leqslant x \\leqslant 4, 1 \\leqslant y \\leqslant 4$, and where $x$ and $y$ are integers is $\\qquad$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "516",
"style": "rule"
} | {
"index": "32371",
"split": "train"
} |
math | [
{
"content": "The mean (average), the median, and the mode of the five numbers \\( 12, 9, 11, 16, x \\) are all equal. What is the value of \\( x \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "32372",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $(\\sqrt{3}b-c)\\cos A=a\\cos C$, find the value of $\\cos A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "32373",
"split": "train"
} |
math | [
{
"content": "How many paths are there from point $A$ to point $B$ in a $7 \\times 6$ grid, if every step must be up or to the right, and you must not pass through the cell at position $(3,3)$?\n[asy]size(4cm,4cm);int w=7;int h=6;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w;... | {
"ground_truth": "262",
"style": "rule"
} | {
"index": "32374",
"split": "train"
} |
math | [
{
"content": "Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32375",
"split": "train"
} |
math | [
{
"content": "Circles with radii $1$, $2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{5}",
"style": "rule"
} | {
"index": "32376",
"split": "train"
} |
math | [
{
"content": "Sector $OAB$ is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form. [asy]\nimport olympiad; import geometry; size(100); d... | {
"ground_truth": "3\\sqrt{2}-3",
"style": "rule"
} | {
"index": "32377",
"split": "train"
} |
math | [
{
"content": "A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$? Ple... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32378",
"split": "train"
} |
math | [
{
"content": "Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \\times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns.\nDetermine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following c... | {
"ground_truth": "n(k-1)^2",
"style": "rule"
} | {
"index": "32379",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b\\sin^{2}\\frac{A}{2}+a\\sin^{2}\\frac{B}{2}=\\frac{C}{2}$.\n1. If $c=2$, find the perimeter of $\\triangle ABC$.\n2. If $C=\\frac{\\pi}{3}$ and the area of $\\triangle ABC$ is $2\\... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "32380",
"split": "train"
} |
math | [
{
"content": "The hyperbola C: $\\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) has one focus that is also the focus of the parabola $y^{2} = 4x$. Line $l$ is an asymptote of $C$ and intersects the circle $(x-1)^{2} + y^{2} = a^{2}$ at points $A$ and $B$. If $|AB| = b$, determine the eccentr... | {
"ground_truth": "\\dfrac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "32381",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a_n\\}$, it is given that $a_{13}=1$ and $a_{12} > a_{13}$. Find the largest integer $n$ for which $(a_1-\\frac{1}{a_1})+(a_2-\\frac{1}{a_2})+(a_3-\\frac{1}{a_3})+\\cdots+(a_n-\\frac{1}{a_n}) > 0$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32382",
"split": "train"
} |
math | [
{
"content": "The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 o... | {
"ground_truth": "875",
"style": "rule"
} | {
"index": "32383",
"split": "train"
} |
math | [
{
"content": "In a city, a newspaper stand buys the \"Evening News\" from the publisher at a price of $0.20 per copy and sells it at $0.30 per copy. Unsold newspapers can be returned to the publisher at $0.05 per copy. In a month (considered as 30 days), there are 20 days when 400 copies can be sold each day, a... | {
"ground_truth": "825",
"style": "rule"
} | {
"index": "32384",
"split": "train"
} |
math | [
{
"content": "For certain ordered pairs $(a,b)\\,$ of real numbers, the system of equations\n$ax+by=1\\,$\n$x^2+y^2=50\\,$\n has at least one solution, and each solution is an ordered pair $(x,y)\\,$ of integers. How many such ordered pairs $(a,b)\\,$ are there? Please output the final answer within \\boxed{}."... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "32385",
"split": "train"
} |
math | [
{
"content": "Find the value of $\\frac{\\sin^{2}B+\\sin^{2}C-\\sin^{2}A}{\\sin B \\sin C}$ given that $\\frac{\\sin B}{\\sin C}=\\frac{AC}{AB}$, $\\frac{\\sin C}{\\sin B}=\\frac{AB}{AC}$, and $\\frac{\\sin A}{\\sin B \\sin C}=\\frac{BC}{AC \\cdot AB}$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{83}{80}",
"style": "rule"
} | {
"index": "32386",
"split": "train"
} |
math | [
{
"content": "For an arithmetic sequence $\\{a_n\\}$ with a non-zero common difference, some terms $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ... form a geometric sequence $\\{a_{k_n}\\}$, and it is given that $k_1 \\neq 1$, $k_2 \\neq 2$, $k_3 \\neq 6$. Find the value of $k_4$. Please output the final answer within \\bo... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "32387",
"split": "train"
} |
math | [
{
"content": "To implement the \"double reduction\" policy and promote intangible cultural heritage, a certain school plans to organize extracurricular interest classes for students to visit the following projects: $A$ (Xiu Shui pottery art), $B$ (Xiu Shui tea-picking opera), $C$ (Jiujiang mountain song), $D$ (... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "32388",
"split": "train"
} |
math | [
{
"content": "We write on the board the equation\n\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016),\n$$\n\nwhere there are 2016 linear factors on each side. What is the smallest positive value of $k$ such that we can omit exactly $k$ of these 4032 linear factors in such a way that there is at least o... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "32389",
"split": "train"
} |
math | [
{
"content": "In a competition with five participants A, B, C, D, and E, determine the probability that neither B nor C appears adjacent to A. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "32390",
"split": "train"
} |
math | [
{
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "32391",
"split": "train"
} |
math | [
{
"content": "Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\\frac{1}{2}$-mile stretch in a school zone where h... | {
"ground_truth": "3 \\frac{3}{4}",
"style": "rule"
} | {
"index": "32392",
"split": "train"
} |
math | [
{
"content": "What is the least integer whose square is 75 more than its double? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "32393",
"split": "train"
} |
math | [
{
"content": "Find all $a,$ $0^\\circ < a < 360^\\circ,$ such that $\\cos a,$ $\\cos 2a,$ and $\\cos 3a$ form an arithmetic sequence, in that order. Enter the solutions, separated by commas, in degrees. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ",
"style": "rule"
} | {
"index": "32394",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\\circ}$ around these lines maps $\\mathcal{O}$ onto itself? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "32395",
"split": "train"
} |
math | [
{
"content": "Let $\\omega$ be the unit circle centered at the origin of $R^2$ . Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$ . Please output the final answer within \\bo... | {
"ground_truth": "\\frac{\\sqrt{2} - 1}{2}",
"style": "rule"
} | {
"index": "32396",
"split": "train"
} |
math | [
{
"content": "Carl and Dina each arrive at a café at a random time between 1:00 PM and 1:45 PM. Each stays for 30 minutes before leaving. What is the probability that Carl and Dina meet each other at the café? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "32397",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sin (\\\\omega x)$, where $\\\\omega > 0$.\n\n(1) When $ \\\\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\\\dfrac{\\\\mathrm{ }\\\\!\\\\!\\\\pi\\\\!\\\\!{ }}{2})$ and explain the reason.\n\n(2) When $ \\\\omega =2$, the graph of the functi... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32398",
"split": "train"
} |
math | [
{
"content": "There are 60 empty boxes $B_1,\\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.\nIn the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consis... | {
"ground_truth": "960",
"style": "rule"
} | {
"index": "32399",
"split": "train"
} |
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