data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Determine the smallest positive integer $m$ such that $11m-3$ and $8m + 5$ have a common factor greater than $1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "34800",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), side \\(AB\\) is 21, the bisector \\(BD\\) is \\(8 \\sqrt{7}\\), and \\(DC\\) is 8. Find the perimeter of the triangle \\(ABC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34801",
"split": "train"
} |
math | [
{
"content": "Given $cosθ+cos(θ+\\frac{π}{3})=\\frac{\\sqrt{3}}{3},θ∈(0,\\frac{π}{2})$, find $\\sin \\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-1 + 2\\sqrt{6}}{6}",
"style": "rule"
} | {
"index": "34802",
"split": "train"
} |
math | [
{
"content": "In a cafeteria line, the number of people ahead of Kaukab is equal to two times the number of people behind her. There are $n$ people in the line. What is a possible value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "34803",
"split": "train"
} |
math | [
{
"content": "If the surface area of a cone is $3\\pi$, and its lateral surface unfolds into a semicircle, then the diameter of the base of the cone is ___. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "34804",
"split": "train"
} |
math | [
{
"content": "If $23=x^4+\\frac{1}{x^4}$, then what is the value of $x^2+\\frac{1}{x^2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34805",
"split": "train"
} |
math | [
{
"content": "Given the planar vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $\\overrightarrow{a} \\cdot (\\overrightarrow{a} + \\overrightarrow{b}) = 3$, and $|\\overrightarrow{a}| = 2$, $|\\overrightarrow{b}| = 1$, find the angle between the vectors $\\overrightarrow{a}$ and $\\overrigh... | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "34806",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = x - \\frac{x^3}{2} + \\frac{x^5}{2 \\cdot 4} - \\frac{x^7}{2 \\cdot 4 \\cdot 6} + \\cdots \\), and \\( g(x) = 1 + \\frac{x^2}{2^2} + \\frac{x^4}{2^2 \\cdot 4^2} + \\frac{x^6}{2^2 \\cdot 4^2 \\cdot 6^2} + \\cdots \\). Find \\( \\int_{0}^{\\infty} f(x) g(x) \\, dx \\). Please output t... | {
"ground_truth": "\\sqrt{e}",
"style": "rule"
} | {
"index": "34807",
"split": "train"
} |
math | [
{
"content": "For any interval $\\mathcal{A}$ in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form $\\frac 1x$ where $x$ is an element in $\\mathcal{A}$ . Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the l... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34808",
"split": "train"
} |
math | [
{
"content": "In Morse code, each symbol is represented by a sequence of dashes and dots. How many distinct symbols can be represented using sequences of 1, 2, 3, or 4 total dots and/or dashes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34809",
"split": "train"
} |
math | [
{
"content": "A box contains one hundred multicolored balls: 28 red, 20 green, 13 yellow, 19 blue, 11 white, and 9 black. What is the minimum number of balls that must be drawn from the box, without looking, to ensure that at least 15 balls of one color are among them? Please output the final answer within \\bo... | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "34810",
"split": "train"
} |
math | [
{
"content": "How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "34811",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \\cdots - 98 + 102$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-52",
"style": "rule"
} | {
"index": "34812",
"split": "train"
} |
math | [
{
"content": "Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34813",
"split": "train"
} |
math | [
{
"content": "Oleg drew an empty 50×50 table and wrote a number above each column and next to each row.\n\nIt turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "34814",
"split": "train"
} |
math | [
{
"content": "If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34815",
"split": "train"
} |
math | [
{
"content": "What is GCF(LCM(16, 21), LCM(14, 18))? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "34816",
"split": "train"
} |
math | [
{
"content": "To meet the shopping needs of customers during the \"May Day\" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "34817",
"split": "train"
} |
math | [
{
"content": "A rigid board with a mass \\( m \\) and a length \\( l = 20 \\) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \\( 2m \\) was placed at its very edge. How far from the stone can a ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34818",
"split": "train"
} |
math | [
{
"content": "In the diagram, $\\triangle ABC$ is right-angled at $A$ with $AB = 3$ and $AC = 3\\sqrt{3}$. Altitude $AD$ intersects median $BE$ at point $G$. Find the length of $AG$. Assume the diagram has this configuration: \n\n[Contextual ASY Diagram]\n```\ndraw((0,0)--(9,0)--(0,10*sqrt(3))--cycle);\ndraw((0... | {
"ground_truth": "-\\frac{1.5\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "34819",
"split": "train"
} |
math | [
{
"content": "Integers $0 \\leq a, b, c, d \\leq 9$ satisfy $$\\begin{gathered} 6 a+9 b+3 c+d=88 \\\\ a-b+c-d=-6 \\\\ a-9 b+3 c-d=-46 \\end{gathered}$$ Find $1000 a+100 b+10 c+d$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6507",
"style": "rule"
} | {
"index": "34820",
"split": "train"
} |
math | [
{
"content": "Dani wrote the integers from 1 to \\( N \\). She used the digit 1 fifteen times. She used the digit 2 fourteen times.\nWhat is \\( N \\) ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "34821",
"split": "train"
} |
math | [
{
"content": "A triangle with angles \\( A, B, C \\) satisfies the following conditions:\n\\[\n\\frac{\\sin A + \\sin B + \\sin C}{\\cos A + \\cos B + \\cos C} = \\frac{12}{7},\n\\]\nand\n\\[\n\\sin A \\sin B \\sin C = \\frac{12}{25}.\n\\]\n\nGiven that \\( \\sin C \\) takes on three possible values \\( s_1, s_... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "34822",
"split": "train"
} |
math | [
{
"content": "Let $M$ be the number of $8$-digit positive integers such that the digits are in both increasing order and even. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "34823",
"split": "train"
} |
math | [
{
"content": "Define a sequence of integers by $T_1 = 2$ and for $n\\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \\cdots + T_{256}$ is divided by 255.\n\n*Ray Li.* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "34824",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer $n$ such that\n$\\frac 1{\\sin 45^\\circ\\sin 46^\\circ}+\\frac 1{\\sin 47^\\circ\\sin 48^\\circ}+\\cdots+\\frac 1{\\sin 133^\\circ\\sin 134^\\circ}=\\frac 1{\\sin n^\\circ}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "34825",
"split": "train"
} |
math | [
{
"content": "The quadratic polynomial $P(x),$ with real coefficients, satisfies\n\\[P(x^3 + x) \\ge P(x^2 + 1)\\]for all real numbers $x.$ Find the sum of the roots of $P(x).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34826",
"split": "train"
} |
math | [
{
"content": "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Pleas... | {
"ground_truth": "409",
"style": "rule"
} | {
"index": "34827",
"split": "train"
} |
math | [
{
"content": "Calculate the number of minutes in a week. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "34828",
"split": "train"
} |
math | [
{
"content": "In unit square $ABCD,$ the inscribed circle $\\omega$ intersects $\\overline{CD}$ at $M,$ and $\\overline{AM}$ intersects $\\omega$ at a point $P$ different from $M.$ What is $AP?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{10}",
"style": "rule"
} | {
"index": "34829",
"split": "train"
} |
math | [
{
"content": "Each of the first eight prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac17",
"style": "rule"
} | {
"index": "34830",
"split": "train"
} |
math | [
{
"content": "Square $PQRS$ has sides of length 1. Points $T$ and $U$ are on $\\overline{QR}$ and $\\overline{RS}$, respectively, so that $\\triangle PTU$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\\overline{PT}.$ The length of a side of this smalle... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34831",
"split": "train"
} |
math | [
{
"content": "Evaluate $3x^y + 4y^x$ when $x=2$ and $y=3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34832",
"split": "train"
} |
math | [
{
"content": "Given a regular tetrahedron with an edge length of \\(2 \\sqrt{6}\\), a sphere is centered at the centroid \\(O\\) of the tetrahedron. The total length of the curves where the sphere intersects with the four faces of the tetrahedron is \\(4 \\pi\\). Find the radius of the sphere centered at \\(O\\... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "34833",
"split": "train"
} |
math | [
{
"content": "A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in ... | {
"ground_truth": "258",
"style": "rule"
} | {
"index": "34834",
"split": "train"
} |
math | [
{
"content": "Let $P(x) = (x-1)(x-4)(x-5)$. Determine how many polynomials $Q(x)$ there exist such that there exists a polynomial $R(x)$ of degree 3 with $P(Q(x)) = P(x) \\cdot R(x)$, and the coefficient of $x$ in $Q(x)$ is 6. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "34835",
"split": "train"
} |
math | [
{
"content": "A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "34836",
"split": "train"
} |
math | [
{
"content": "Given a number \\\\(x\\\\) randomly selected from the interval \\\\(\\left[-\\frac{\\pi}{4}, \\frac{2\\pi}{3}\\right]\\\\), find the probability that the function \\\\(f(x)=3\\sin\\left(2x- \\frac{\\pi}{6}\\right)\\\\) is not less than \\\\(0\\\\). Please output the final answer within \\boxed{}."... | {
"ground_truth": "\\frac{6}{11}",
"style": "rule"
} | {
"index": "34837",
"split": "train"
} |
math | [
{
"content": "Four concentric circles are drawn with radii of 1, 3, 5 and 7. The inner circle is painted black, the ring around it is white, the next ring is black and the outer ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction. Please output the fina... | {
"ground_truth": "\\frac{17}{32}",
"style": "rule"
} | {
"index": "34838",
"split": "train"
} |
math | [
{
"content": "Given two points A (-2, 0), B (0, 2), and point C is any point on the circle $x^2+y^2-2x=0$, determine the minimum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3- \\sqrt{2}",
"style": "rule"
} | {
"index": "34839",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\sqrt{25\\sqrt{15\\sqrt{9}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{15}",
"style": "rule"
} | {
"index": "34840",
"split": "train"
} |
math | [
{
"content": "Every evening, Laura gathers three socks randomly to pack for her gym session next morning. She has 12 black socks, 10 white socks, and 6 striped socks in her drawer. What is the probability that all three socks she picks are of the same type? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{60}{546}",
"style": "rule"
} | {
"index": "34841",
"split": "train"
} |
math | [
{
"content": "Let $\\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\\an... | {
"ground_truth": "592",
"style": "rule"
} | {
"index": "34842",
"split": "train"
} |
math | [
{
"content": "For a given positive integer \\( k \\), let \\( f_{1}(k) \\) represent the square of the sum of the digits of \\( k \\), and define \\( f_{n+1}(k) = f_{1}\\left(f_{n}(k)\\right) \\) for \\( n \\geq 1 \\). Find the value of \\( f_{2005}\\left(2^{2006}\\right) \\). Please output the final answer wit... | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "34843",
"split": "train"
} |
math | [
{
"content": "We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "34844",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers is 22. Their difference is 4. What is the greater of the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "34845",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "950",
"style": "rule"
} | {
"index": "34846",
"split": "train"
} |
math | [
{
"content": "Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\\plus{}5^q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2, 3), (2, 5), (3, 2), (5, 2), (5, 5), (5, 313), (313, 5)",
"style": "rule"
} | {
"index": "34847",
"split": "train"
} |
math | [
{
"content": "Two subsets of the set $T = \\{w, x, y, z, v\\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order? Please output the final answer within \\boxed... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "34848",
"split": "train"
} |
math | [
{
"content": "Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \\le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "34849",
"split": "train"
} |
math | [
{
"content": "A teacher gave a test to a class in which $10\\%$ of the students are juniors and $90\\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test? Please outp... | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "34850",
"split": "train"
} |
math | [
{
"content": "Given the cyclist encounters red lights at each of 4 intersections with probability $\\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters the first red light a... | {
"ground_truth": "\\frac{4}{27}",
"style": "rule"
} | {
"index": "34851",
"split": "train"
} |
math | [
{
"content": "On a straight segment of a one-way, single-lane highway, cars travel at the same speed and follow a safety rule where the distance from the back of one car to the front of the next is equal to the car’s speed divided by 10 kilometers per hour, rounded up to the nearest whole number (e.g., a car tr... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "34852",
"split": "train"
} |
math | [
{
"content": "If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "34853",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34854",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ for which $11n - 3$ and $8n + 2$ share a common factor greater than $1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "34855",
"split": "train"
} |
math | [
{
"content": "In the year 2023, the International Mathematical Olympiad will be hosted by a country. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \\cdot B \\cdot C = 2023$. What is the largest possible value of the sum $A + B + C$? Please output the final answer within \\boxed{}.... | {
"ground_truth": "297",
"style": "rule"
} | {
"index": "34856",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = a\\sin(\\pi x + \\alpha) + b\\cos(\\pi x + \\beta)$, and it is known that $f(2001) = 3$, find the value of $f(2012)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "34857",
"split": "train"
} |
math | [
{
"content": "A solid cube of side length \\(4 \\mathrm{~cm}\\) is cut into two pieces by a plane that passed through the midpoints of six edges. To the nearest square centimetre, the surface area of each half cube created is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "34858",
"split": "train"
} |
math | [
{
"content": "Calculate: $(\\sqrt{3}+\\sqrt{2})(\\sqrt{3}-\\sqrt{2})^{2}=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}-\\sqrt{2}",
"style": "rule"
} | {
"index": "34859",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units, what is the distance from $D$ to the midpoint of segment $EF$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "34860",
"split": "train"
} |
math | [
{
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter $10\\text{ cm}.$ A side view of the first four rows of each of the two different methods of packing is shown below.\n\n[asy]\ndraw(circle((1,1),1),black+linewidth(1));\ndraw(circle((... | {
"ground_truth": "10 + 5\\sqrt{3}",
"style": "rule"
} | {
"index": "34861",
"split": "train"
} |
math | [
{
"content": "Add $46.913$ to $58.27$ and round your answer to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105.18",
"style": "rule"
} | {
"index": "34862",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $(-3 \\frac{3}{8})^{- \\frac{2}{3}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "34863",
"split": "train"
} |
math | [
{
"content": "How can you weigh 1 kg of grain on a balance scale using two weights, one weighing 300 g and the other 650 g? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "34864",
"split": "train"
} |
math | [
{
"content": "The third and twentieth terms of an arithmetic sequence are 10 and 65, respectively. What is the thirty-second term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "103.8235294118",
"style": "rule"
} | {
"index": "34865",
"split": "train"
} |
math | [
{
"content": "It is known that the only solution to the equation\n\n$$\n\\pi / 4 = \\operatorname{arcctg} 2 + \\operatorname{arcctg} 5 + \\operatorname{arcctg} 13 + \\operatorname{arcctg} 34 + \\operatorname{arcctg} 89 + \\operatorname{arcctg}(x / 14)\n$$\n\nis a natural number. Find it. Please output the final... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "34866",
"split": "train"
} |
math | [
{
"content": "Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{30}",
"style": "rule"
} | {
"index": "34867",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, we have $\\angle A = 90^\\circ$ and $\\sin B = \\frac{4}{7}$. Find $\\cos C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac47",
"style": "rule"
} | {
"index": "34868",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\\mathcal{T}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{90}{11}",
"style": "rule"
} | {
"index": "34869",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite sum $\\sum_{n=1}^{\\infty}\\frac{n}{n^4+4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac 3 8",
"style": "rule"
} | {
"index": "34870",
"split": "train"
} |
math | [
{
"content": "The price of each item at the Gauss Gadget Store has been reduced by $20 \\%$ from its original price. An MP3 player has a sale price of $\\$ 112$. What would the same MP3 player sell for if it was on sale for $30 \\%$ off of its original price? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "34871",
"split": "train"
} |
math | [
{
"content": "A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much sausage is left if bo... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "34872",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34873",
"split": "train"
} |
math | [
{
"content": "Let \\(P(x)\\) be a quadratic polynomial with real coefficients. Suppose that \\(P(1)=20, P(-1)=22\\), and \\(P(P(0))=400\\). Compute the largest possible value of \\(P(10)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2486",
"style": "rule"
} | {
"index": "34874",
"split": "train"
} |
math | [
{
"content": "Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{5}",
"style": "rule"
} | {
"index": "34875",
"split": "train"
} |
math | [
{
"content": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum... | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "34876",
"split": "train"
} |
math | [
{
"content": "Calculate: $\\frac{{\\cos190°(1+\\sqrt{3}\\tan10°)}}{{\\sin290°\\sqrt{1-\\cos40°}}}=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "34877",
"split": "train"
} |
math | [
{
"content": "A regular hexagon \\(ABCDEF\\) is inscribed in a circle with a radius of \\(3+\\sqrt{3}\\). Find the radius of the inscribed circle of triangle \\(ACD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "34878",
"split": "train"
} |
math | [
{
"content": "A coin is tossed 10 times. Find the probability that two heads never appear consecutively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9/64",
"style": "rule"
} | {
"index": "34879",
"split": "train"
} |
math | [
{
"content": "The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the o... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34880",
"split": "train"
} |
math | [
{
"content": "For an arithmetic sequence \\(a_1, a_2, a_3, \\dots\\), let\n\\[ S_n = a_1 + a_2 + a_3 + \\dots + a_n, \\]\nand let\n\\[ T_n = S_1 + S_2 + S_3 + \\dots + S_n. \\]\nIf you know the value of \\( S_{2023}, \\) then you can uniquely determine the value of \\( T_n \\) for some integer \\( n \\). What i... | {
"ground_truth": "3034",
"style": "rule"
} | {
"index": "34881",
"split": "train"
} |
math | [
{
"content": "Suppose that $f$ is a function from $\\mathbb{R}$ to $\\mathbb{R}$ such that \\[ f(x) + f\\left( 1 - \\frac{1}{x} \\right) = \\arctan x \\] for all real $x \\neq 0$. (As usual, $y = \\arctan x$ means $-\\pi/2 < y < \\pi/2$ and $\\tan y = x$.) Find \\[ \\int_0^1 f(x)\\,dx. \\] Please output the fin... | {
"ground_truth": "\\frac{3\\pi}{8}",
"style": "rule"
} | {
"index": "34882",
"split": "train"
} |
math | [
{
"content": "What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34883",
"split": "train"
} |
math | [
{
"content": "The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": "34884",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 870^\\circ$ and $\\cos 870^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "34885",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "34886",
"split": "train"
} |
math | [
{
"content": "A standard die is rolled six times. What is the probability that the product of all six rolls is odd? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{64}",
"style": "rule"
} | {
"index": "34887",
"split": "train"
} |
math | [
{
"content": "If $x \\ne 0$ or $4$ and $y \\ne 0$ or $6$, then $\\frac{2}{x} + \\frac{3}{y} = \\frac{1}{2}$ is equivalent to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4y}{y-6}=x",
"style": "rule"
} | {
"index": "34888",
"split": "train"
} |
math | [
{
"content": "In an isosceles right triangle \\( \\triangle ABC \\), \\( CA = CB = 1 \\). Let point \\( P \\) be any point on the boundary of \\( \\triangle ABC \\). Find the maximum value of \\( PA \\cdot PB \\cdot PC \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "34889",
"split": "train"
} |
math | [
{
"content": "A workshop produces a type of instrument with a fixed cost of $7500. Each additional unit of the instrument requires an additional investment of $100. The total revenue function is given by: $H(x) = \\begin{cases} 400x - x^2, (0 \\leq x \\leq 200) \\\\ 40000, (x > 200) \\end{cases}$, where $x$ is ... | {
"ground_truth": "15000",
"style": "rule"
} | {
"index": "34890",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive integer \\( m \\) for which \\( m^3 - 90 \\) is divisible by \\( m + 9 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34891",
"split": "train"
} |
math | [
{
"content": "The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$90^\\circ$",
"style": "rule"
} | {
"index": "34892",
"split": "train"
} |
math | [
{
"content": "One student has 6 mathematics books, and another has 8. In how many ways can they exchange three books? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1120",
"style": "rule"
} | {
"index": "34893",
"split": "train"
} |
math | [
{
"content": "Consider the sequence \\(\\left\\{a_{n}\\right\\}\\) defined by \\(a_{0}=\\frac{1}{2}\\) and \\(a_{n+1}=a_{n}+\\frac{a_{n}^{2}}{2023}\\) for \\(n = 0, 1, \\ldots\\). Find the integer \\(k\\) such that \\(a_{k} < 1 < a_{k+1}\\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2023",
"style": "rule"
} | {
"index": "34894",
"split": "train"
} |
math | [
{
"content": "What is the area of rectangle \\( PQRS \\) if the perimeter of rectangle \\( TVWY \\) is 60? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "34895",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x \\in [0,1]$, $f(x)=x^{3}$. Also, for all $x \\in \\mathbb{R}$, $f(x)=f(2-x)$. Determine the value of $f(2017.5)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{8}",
"style": "rule"
} | {
"index": "34896",
"split": "train"
} |
math | [
{
"content": "Mountain bikes are a popular means of transportation for high school students, with a huge market and fierce competition. The total sales of type $A$ bikes operated by a certain brand dealer last year was $50,000. This year, the selling price per bike is $400 less than last year. If the quantity s... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34897",
"split": "train"
} |
math | [
{
"content": "Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "x - 3",
"style": "rule"
} | {
"index": "34898",
"split": "train"
} |
math | [
{
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "115",
"style": "rule"
} | {
"index": "34899",
"split": "train"
} |
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