data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given a circle $C: (x-1)^2+(y-2)^2=25$, and a line $l: (2m+1)x+(m+1)y-7m-4=0$. If the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "39500",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha}{\\beta^2}$ is a real number and $|\\alpha - \\beta| = 2 \\sqrt{3}.$ Find $|\\alpha|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39501",
"split": "train"
} |
math | [
{
"content": "By what common fraction does $0.\\overline{81}$ exceed $0.81$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{1100}",
"style": "rule"
} | {
"index": "39502",
"split": "train"
} |
math | [
{
"content": "Define $x \\star y=\\frac{\\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute $$((\\cdots((2007 \\star 2006) \\star 2005) \\star \\cdots) \\star 1)$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{15}}{9}",
"style": "rule"
} | {
"index": "39503",
"split": "train"
} |
math | [
{
"content": "In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ . Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "39504",
"split": "train"
} |
math | [
{
"content": "$(1)$ Solve the inequality: $3A_{x}^{3}≤2A_{x+1}^{2}+6A_{x}^{2}$;<br/>$(2)$ Find the value of $C_{n}^{5-n}+C_{n+1}^{9-n}$;<br/>$(3)$ Given $\\frac{1}{C_{5}^{m}}-\\frac{1}{C_{6}^{m}}=\\frac{7}{10C_{7}^{m}}$, find $C_{8}^{m}$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "39505",
"split": "train"
} |
math | [
{
"content": "The members of a band are arranged in a rectangular formation. When they are arranged in 8 rows, there are 2 positions unoccupied in the formation. When they are arranged in 9 rows, there are 3 positions unoccupied. How many members are in the band if the membership is between 100 and 200? Please ... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "39506",
"split": "train"
} |
math | [
{
"content": "Given real numbers $b_0, b_1, \\dots, b_{2019}$ with $b_{2019} \\neq 0$, let $z_1,z_2,\\dots,z_{2019}$ be the roots in the complex plane of the polynomial \\[ P(z) = \\sum_{k=0}^{2019} b_k z^k. \\] Let $\\mu = (|z_1| + \\cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\\dot... | {
"ground_truth": "2019^{-1/2019}",
"style": "rule"
} | {
"index": "39507",
"split": "train"
} |
math | [
{
"content": "A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39508",
"split": "train"
} |
math | [
{
"content": "In the diagram, point $D$ is one-third of the way from $A$ to $B$, and point $E$ is the midpoint of $\\overline{BC}$. Given $A(0,8)$, $B(0,0)$, and $C(10,0)$, find the sum of the slope and $y$-intercept of the line passing through points $C$ and $D$. Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{12}{5}",
"style": "rule"
} | {
"index": "39509",
"split": "train"
} |
math | [
{
"content": "Let $x$, $y$, $z$ be the three sides of a triangle, and let $\\xi$, $\\eta$, $\\zeta$ be the angles opposite them, respectively. If $x^2 + y^2 = 2023z^2$, find the value of\n\\[\\frac{\\cot \\zeta}{\\cot \\xi + \\cot \\eta}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "1011",
"style": "rule"
} | {
"index": "39510",
"split": "train"
} |
math | [
{
"content": "A rare genetic disorder is present in one out of every 1000 people in a certain community, and it typically shows no outward symptoms. There is a specialized blood test to detect this disorder. For individuals who have this disorder, the test consistently yields a positive result. For those withou... | {
"ground_truth": "0.019",
"style": "rule"
} | {
"index": "39511",
"split": "train"
} |
math | [
{
"content": "What is $10 \\cdot \\left(\\frac{1}{2} + \\frac{1}{5} + \\frac{1}{10}\\right)^{-1}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{2}",
"style": "rule"
} | {
"index": "39512",
"split": "train"
} |
math | [
{
"content": "Given that $a>0$, $b>1$, and $a+b=2$, find the minimum value of $$\\frac{1}{2a}+\\frac{2}{b-1}$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "39513",
"split": "train"
} |
math | [
{
"content": "The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "39514",
"split": "train"
} |
math | [
{
"content": "Line segment $\\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ 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);\... | {
"ground_truth": "\\left( -\\frac{3}{7}, \\frac{10}{7} \\right)",
"style": "rule"
} | {
"index": "39515",
"split": "train"
} |
math | [
{
"content": "Two strips of width 2 overlap at an angle of 60 degrees inside a rectangle of dimensions 4 units by 3 units. Find the area of the overlap, considering that the angle is measured from the horizontal line of the rectangle.\nA) $\\frac{2\\sqrt{3}}{3}$\nB) $\\frac{8\\sqrt{3}}{9}$\nC) $\\frac{4\\sqrt{3... | {
"ground_truth": "\\frac{4\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "39516",
"split": "train"
} |
math | [
{
"content": "Given a parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$, directrix $x=-1$, and a point $M$ on the directrix below the $x$-axis. A line $l$ passing through $M$ and $F$ intersects with $C$ at point $N$, and $F$ is the midpoint of segment $MN$.\n\n1. Find the equations of parabola $C$ and line $l$.\... | {
"ground_truth": "\\frac {16}{3}",
"style": "rule"
} | {
"index": "39517",
"split": "train"
} |
math | [
{
"content": "The Aces are playing the Kings in a playoff series, where the first team to win 5 games wins the series. Each game's outcome leads to the Aces winning with a probability of $\\dfrac{7}{10}$, and there are no draws. Calculate the probability that the Aces win the series. Please output the final ans... | {
"ground_truth": "90\\%",
"style": "rule"
} | {
"index": "39518",
"split": "train"
} |
math | [
{
"content": "Let $i_{1}, i_{2}, \\cdots, i_{n}$ be a permutation of the set $\\{1, 2, \\cdots, n\\}$. If there exist $k < l$ and $i_{k} > i_{l}$, then the pair $\\left(i_{k}, i_{l}\\right)$ is called an inversion, and the total number of inversions in the permutation is called the inversion count of this permu... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "39519",
"split": "train"
} |
math | [
{
"content": "The integer sequence \\(a_1, a_2, a_3, \\dots\\) is defined as follows: \\(a_1 = 1\\). For \\(n \\geq 1\\), \\(a_{n+1}\\) is the smallest integer greater than \\(a_n\\) such that for all \\(i, j, k \\in \\{1, 2, \\dots, n+1\\}\\), the condition \\(a_i + a_j \\neq 3a_k\\) is satisfied. Find the val... | {
"ground_truth": "66016",
"style": "rule"
} | {
"index": "39520",
"split": "train"
} |
math | [
{
"content": "Given the circle with radius $6\\sqrt{2}$, diameter $\\overline{AB}$, and chord $\\overline{CD}$ intersecting $\\overline{AB}$ at point $E$, where $BE = 3\\sqrt{2}$ and $\\angle AEC = 60^{\\circ}$, calculate $CE^2+DE^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "39521",
"split": "train"
} |
math | [
{
"content": "Let $N,O$ be functions such that $N(x) = 2\\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(3))))))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39522",
"split": "train"
} |
math | [
{
"content": "In a right triangle, one leg is 24 inches, and the other leg is 10 inches more than twice the shorter leg. Calculate the area of the triangle and find the length of the hypotenuse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3940}",
"style": "rule"
} | {
"index": "39523",
"split": "train"
} |
math | [
{
"content": "If $a$ is a constant such that $4x^2 + 14x + a$ is the square of a binomial, then what is $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{49}{4}",
"style": "rule"
} | {
"index": "39524",
"split": "train"
} |
math | [
{
"content": "From a \\(6 \\times 6\\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square centimeters. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "39525",
"split": "train"
} |
math | [
{
"content": "Given that $\\binom{23}{3}=1771$, $\\binom{23}{4}=8855$, and $\\binom{23}{5}=33649$, find $\\binom{25}{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53130",
"style": "rule"
} | {
"index": "39526",
"split": "train"
} |
math | [
{
"content": "An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39527",
"split": "train"
} |
math | [
{
"content": "It is known that solution A has a salinity of $8\\%$, and solution B has a salinity of $5\\%$. After mixing both solutions, the resulting salinity is $6.2\\%$. What will be the concentration when a quarter of solution A is mixed with a sixth of solution B? Please output the final answer within \\b... | {
"ground_truth": "$6.5\\%$",
"style": "rule"
} | {
"index": "39528",
"split": "train"
} |
math | [
{
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{119.84}{\\pi^2}",
"style": "rule"
} | {
"index": "39529",
"split": "train"
} |
math | [
{
"content": "Let the complex number $z = \\cos\\tfrac{1}{1000} + i \\sin\\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\\tfrac{1}{2}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "524",
"style": "rule"
} | {
"index": "39530",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), side \\(BC\\) is equal to 5. A circle passes through vertices \\(B\\) and \\(C\\) and intersects side \\(AC\\) at point \\(K\\), where \\(CK = 3\\) and \\(KA = 1\\). It is known that the cosine of angle \\(ACB\\) is \\(\\frac{4}{5}\\). Find the ratio of the radius of this ci... | {
"ground_truth": "\\frac{10\\sqrt{10} + 25}{9}",
"style": "rule"
} | {
"index": "39531",
"split": "train"
} |
math | [
{
"content": "How many sequences of ten binary digits are there in which neither two zeroes nor three ones ever appear in a row? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "39532",
"split": "train"
} |
math | [
{
"content": "Write 2.24 as a simplified fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{56}{25}",
"style": "rule"
} | {
"index": "39533",
"split": "train"
} |
math | [
{
"content": "By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39534",
"split": "train"
} |
math | [
{
"content": "The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{12}{25}",
"style": "rule"
} | {
"index": "39535",
"split": "train"
} |
math | [
{
"content": "How many divisors of $9!$ are multiples of $10$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "39536",
"split": "train"
} |
math | [
{
"content": "Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize h... | {
"ground_truth": "\\frac{7}{36}",
"style": "rule"
} | {
"index": "39537",
"split": "train"
} |
math | [
{
"content": "What is the value of $x$ if $P Q S$ is a straight line and $\\angle P Q R=110^{\\circ}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39538",
"split": "train"
} |
math | [
{
"content": "\"The Nine Chapters on the Mathematical Art\" is the first mathematical monograph in China, which includes the following problem: \"There is a gold rod, 5 feet in length. Cutting 1 foot from the base, it weighs 4 jin. Cutting 1 foot from the tip, it weighs 2 jin. How much does the gold rod weigh?\... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39539",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=3, B C=4$, and $C A=5$. Let $A_{1}, A_{2}$ be points on side $B C$, $B_{1}, B_{2}$ be points on side $C A$, and $C_{1}, C_{2}$ be points on side $A B$. Suppose that there exists a point $P$ such that $P A_{1} A_{2}, P B_{1} B_{2}$, and $P C_{1} C_{2}$ are congru... | {
"ground_truth": "\\frac{12+22 \\sqrt{3}}{15}",
"style": "rule"
} | {
"index": "39540",
"split": "train"
} |
math | [
{
"content": "There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>... | {
"ground_truth": "2014",
"style": "rule"
} | {
"index": "39541",
"split": "train"
} |
math | [
{
"content": "Given a function $f(x) = \\frac {1}{2}\\sin x - \\frac {\\sqrt {3}}{2}\\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\\frac {1}{2}, 1]$. Find the maximum value of $b-a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4\\pi}{3}",
"style": "rule"
} | {
"index": "39542",
"split": "train"
} |
math | [
{
"content": "A container holds $47\\frac{2}{3}$ cups of sugar. If one recipe requires $1\\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31\\frac{7}{9}",
"style": "rule"
} | {
"index": "39543",
"split": "train"
} |
math | [
{
"content": "A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{120}{17}",
"style": "rule"
} | {
"index": "39544",
"split": "train"
} |
math | [
{
"content": "The midpoints of the sides of a regular octagon \\( ABCDEFGH \\) are joined to form a smaller octagon. What fraction of the area of \\( ABCDEFGH \\) is enclosed by the smaller octagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39545",
"split": "train"
} |
math | [
{
"content": "Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is th... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "39546",
"split": "train"
} |
math | [
{
"content": "In Flower Town, there are $99^{2}$ residents, some of whom are knights (who always tell the truth) and others are liars (who always lie). The houses in the town are arranged in the cells of a $99 \\times 99$ square grid (totaling $99^{2}$ houses, arranged on 99 vertical and 99 horizontal streets).... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "39547",
"split": "train"
} |
math | [
{
"content": "Three positive integers differ from each other by at most 6. The product of these three integers is 2808. What is the smallest integer among them? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39548",
"split": "train"
} |
math | [
{
"content": "The angle that has the same terminal side as $- \\frac{\\pi}{3}$ is $\\frac{\\pi}{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{3}",
"style": "rule"
} | {
"index": "39549",
"split": "train"
} |
math | [
{
"content": "On the side AB of triangle ABC with a $100^{\\circ}$ angle at vertex C, points P and Q are taken such that $AP = BC$ and $BQ = AC$. Let M, N, and K be the midpoints of segments AB, CP, and CQ respectively. Find the angle $NMK$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "39550",
"split": "train"
} |
math | [
{
"content": "The line segment connecting $(-2,7)$ and $(3,11)$ can be parameterized by the equations\n\\begin{align*}\nx &= at + b, \\\\\ny &= ct + d,\n\\end{align*}where $0 \\le t \\le 1,$ and $t = 0$ corresponds to the point $(-2,7).$ Find $a^2 + b^2 + c^2 + d^2.$ Please output the final answer within \\box... | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "39551",
"split": "train"
} |
math | [
{
"content": "How many ways are there to partition $7$ students into the groups of $2$ or $3$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "39552",
"split": "train"
} |
math | [
{
"content": "$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies? Please output the... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39553",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\\text{th}}$ roots of unity? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39554",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{p}$ and $\\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $30^\\circ.$ Find the area of the parallelogram whose diagonals correspond to $\\mathbf{p} + 2 \\mathbf{q}$ and $2 \\mathbf{p} + \\mathbf{q}.$ Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "39555",
"split": "train"
} |
math | [
{
"content": "A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?\n[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1))... | {
"ground_truth": "\\frac{4-\\pi}{\\pi}",
"style": "rule"
} | {
"index": "39556",
"split": "train"
} |
math | [
{
"content": "Susie Q has 1000 dollars to invest. She invests some of the money at the Pretty Penny Bank, which compounds annually at 3 percent. She invests the rest of the money at the Five and Dime Bank, which compounds annually at 5 percent. After two years, Susie has a total of $\\$1090.02$. How much di... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "39557",
"split": "train"
} |
math | [
{
"content": "In a particular game, each of $4$ players rolls a standard $8$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again continuing until one player wins. Hugo is one of the players. What is the probability t... | {
"ground_truth": "\\frac{27}{128}",
"style": "rule"
} | {
"index": "39558",
"split": "train"
} |
math | [
{
"content": "Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group? Please output the final answer ... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "39559",
"split": "train"
} |
math | [
{
"content": "Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "37.5",
"style": "rule"
} | {
"index": "39560",
"split": "train"
} |
math | [
{
"content": "Four cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 2, the third card is a 3, and the fourth card is a 4? Assume that the dealing is done without replacement. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{16}{405525}",
"style": "rule"
} | {
"index": "39561",
"split": "train"
} |
math | [
{
"content": "Given that $f(x) = x^{2}-2x+5$ and $g(x) =x+3$, what is the value of $f(g(5)) -g(f(5))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39562",
"split": "train"
} |
math | [
{
"content": "The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \\( F \\) is greater than \\( \\frac{1}{12} \\), the prob... | {
"ground_truth": "\\frac{7}{48}",
"style": "rule"
} | {
"index": "39563",
"split": "train"
} |
math | [
{
"content": "How many times does 24 divide into 100! (factorial)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39564",
"split": "train"
} |
math | [
{
"content": "Let \\( S = \\{1, 2, \\ldots, 1963\\} \\). What is the maximum number of elements that can be chosen from \\( S \\) such that the sum of any two chosen numbers is not divisible by their difference? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "655",
"style": "rule"
} | {
"index": "39565",
"split": "train"
} |
math | [
{
"content": "The decimal representation of \\(\\frac{1}{20^{20}}\\) consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "39566",
"split": "train"
} |
math | [
{
"content": "A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southwa... | {
"ground_truth": "\\frac{15}{77}",
"style": "rule"
} | {
"index": "39567",
"split": "train"
} |
math | [
{
"content": "A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division? Please... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "39568",
"split": "train"
} |
math | [
{
"content": "Compute the definite integral:\n\n$$\n\\int_{1}^{2} \\frac{x+\\sqrt{3 x-2}-10}{\\sqrt{3 x-2}+7} d x\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{22}{27}",
"style": "rule"
} | {
"index": "39569",
"split": "train"
} |
math | [
{
"content": "Let $m \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.\nNote: a partition of $S$ is a pair... | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "39570",
"split": "train"
} |
math | [
{
"content": "A six-digit number can be tripled by reducing the first digit by three and appending a three at the end. What is this number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "428571",
"style": "rule"
} | {
"index": "39571",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "39572",
"split": "train"
} |
math | [
{
"content": "Form a three-digit number using the digits 4, 5, and 6. The probability that the number formed is a common multiple of 3 and 5 is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "39573",
"split": "train"
} |
math | [
{
"content": "Let $(a_n)_{n \\ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \\ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \\lim_{n \\to \\infty} \\left( \\frac{a_{n+1}}{a_n} \\right)^n. $$ Please output the final answer within \\boxed{}."... | {
"ground_truth": "e",
"style": "rule"
} | {
"index": "39574",
"split": "train"
} |
math | [
{
"content": "There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.\n\n(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.\n\n(2) Given that among the 3 selected questions, there are ... | {
"ground_truth": "\\dfrac{93}{125}",
"style": "rule"
} | {
"index": "39575",
"split": "train"
} |
math | [
{
"content": "The union of sets \\( A \\) and \\( B \\) is \\( A \\cup B = \\{a_1, a_2, a_3\\} \\). When \\( A \\neq B \\), \\((A, B)\\) and \\((B, A)\\) are considered different pairs. How many such pairs \\((A, B)\\) exist? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "39576",
"split": "train"
} |
math | [
{
"content": "Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed. Please output the final answer within \\boxe... | {
"ground_truth": "4019",
"style": "rule"
} | {
"index": "39577",
"split": "train"
} |
math | [
{
"content": "On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.\n\nDetermine the maximum possible number of positive numbers among these 2018 numbers.\n\n(Walther Janous) ... | {
"ground_truth": "1009",
"style": "rule"
} | {
"index": "39578",
"split": "train"
} |
math | [
{
"content": "What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49999",
"style": "rule"
} | {
"index": "39579",
"split": "train"
} |
math | [
{
"content": "The fourth power of $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3+2\\sqrt{2}",
"style": "rule"
} | {
"index": "39580",
"split": "train"
} |
math | [
{
"content": "In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of triangle $ABC$ . Pl... | {
"ground_truth": "12\\sqrt{34}",
"style": "rule"
} | {
"index": "39581",
"split": "train"
} |
math | [
{
"content": "In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\\overline{ES}$ and $\\overlin... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39582",
"split": "train"
} |
math | [
{
"content": "A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a \"convex number\". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that it is a \"convex nu... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "39583",
"split": "train"
} |
math | [
{
"content": "Given that non-negative real numbers \\( x \\) and \\( y \\) satisfy\n\\[ \nx^{2}+4y^{2}+4xy+4x^{2}y^{2}=32,\n\\]\nfind the maximum value of \\( \\sqrt{7}(x+2y)+2xy \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "39584",
"split": "train"
} |
math | [
{
"content": "Find the largest possible value of \\( n \\) such that there exist \\( n \\) consecutive positive integers whose sum is equal to 2010. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39585",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be real numbers such that $\\frac{\\sin x}{\\sin y} = 3$ and $\\frac{\\cos x}{\\cos y} = \\frac12$. The value of $\\frac{\\sin 2x}{\\sin 2y} + \\frac{\\cos 2x}{\\cos 2y}$ can be expressed in the form $\\frac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.... | {
"ground_truth": "107",
"style": "rule"
} | {
"index": "39586",
"split": "train"
} |
math | [
{
"content": "An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees capable of monitoring ... | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "39587",
"split": "train"
} |
math | [
{
"content": "Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \\[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\\geq (xyz)^{2}-xyz+1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "39588",
"split": "train"
} |
math | [
{
"content": "A line $x=k$ intersects the graph of $y=\\log_5 x$ and the graph of $y=\\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \\sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "39589",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the graph of the quadratic function $y=a\\left(x-h\\right)^{2}+k\\left(a \\lt 0\\right)$ passes through the points $A\\left(-3,m\\right)$ and $B\\left(-1,n\\right)$.<br/>$(1)$ When $m=n$, find the length of the line segment $AB$ and the value of $h... | {
"ground_truth": "-\\frac{1}{4}",
"style": "rule"
} | {
"index": "39590",
"split": "train"
} |
math | [
{
"content": "Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8. Please output the final answer within \\boxed{}... | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "39591",
"split": "train"
} |
math | [
{
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39592",
"split": "train"
} |
math | [
{
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not div... | {
"ground_truth": "554",
"style": "rule"
} | {
"index": "39593",
"split": "train"
} |
math | [
{
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 18? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39594",
"split": "train"
} |
math | [
{
"content": "Given $\\sin\\theta + \\cos\\theta = \\frac{1}{5}$, with $\\theta \\in (0,\\pi)$,\n1. Find the value of $\\tan\\theta$;\n2. Find the value of $\\frac{1+\\sin 2\\theta + \\cos 2\\theta}{1+\\sin 2\\theta - \\cos 2\\theta}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "39595",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\\alpha$, if $\\cos(\\alpha+\\beta)=\\frac{1}{3}$, where $\\beta \\in (0,\\pi)$, then $\\cos \\beta =\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{3 + 8\\sqrt{2}}{15}",
"style": "rule"
} | {
"index": "39596",
"split": "train"
} |
math | [
{
"content": "31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of the race. In the end, ... | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "39597",
"split": "train"
} |
math | [
{
"content": "Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is $\\frac{3}{5}$, and ... | {
"ground_truth": "\\frac{441}{1000}",
"style": "rule"
} | {
"index": "39598",
"split": "train"
} |
math | [
{
"content": "How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "39599",
"split": "train"
} |
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