data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In the Cartesian coordinate system \\( xOy \\), the area of the region corresponding to the set of points \\( K = \\{(x, y) \\mid (|x| + |3y| - 6)(|3x| + |y| - 6) \\leq 0 \\} \\) is ________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "35900",
"split": "train"
} |
math | [
{
"content": "The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\sqrt{3}",
"style": "rule"
} | {
"index": "35901",
"split": "train"
} |
math | [
{
"content": "James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins? Please output the final answer within \\box... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "35902",
"split": "train"
} |
math | [
{
"content": "A four-digit number satisfies the following conditions:\n(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;\n(2) When divided by 9, the remainder is 8.\nFind the smallest odd four-digit number that satisfies ... | {
"ground_truth": "1979",
"style": "rule"
} | {
"index": "35903",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the coefficients of $P(x)$ if $\\left(20 x^{27}+2 x^{2}+1\\right) P(x)=2001 x^{2001}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "35904",
"split": "train"
} |
math | [
{
"content": "The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "35905",
"split": "train"
} |
math | [
{
"content": "Dolly, Molly and Polly each can walk at $6 \\mathrm{~km} / \\mathrm{h}$. Their one motorcycle, which travels at $90 \\mathrm{~km} / \\mathrm{h}$, can accommodate at most two of them at once (and cannot drive by itself!). Let $t$ hours be the time taken for all three of them to reach a point 135 km... | {
"ground_truth": "t<3.9",
"style": "rule"
} | {
"index": "35906",
"split": "train"
} |
math | [
{
"content": "1. Given $\\sin\\alpha + \\cos\\alpha = \\frac{7}{13}$, with $\\alpha \\in (0, \\pi)$, find the value of $\\tan\\alpha$.\n2. Find the minimum value for $y=\\sin 2x + 2\\sqrt{2}\\cos\\left(\\frac{\\pi}{4}+x\\right)+3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "35907",
"split": "train"
} |
math | [
{
"content": "The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either G or K... | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "35908",
"split": "train"
} |
math | [
{
"content": "Real numbers \\(x\\) and \\(y\\) satisfy the following equations: \\(x=\\log_{10}(10^{y-1}+1)-1\\) and \\(y=\\log_{10}(10^{x}+1)-1\\). Compute \\(10^{x-y}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{101}{110}",
"style": "rule"
} | {
"index": "35909",
"split": "train"
} |
math | [
{
"content": "A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4:40 a.m.",
"style": "rule"
} | {
"index": "35910",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "35911",
"split": "train"
} |
math | [
{
"content": "Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of the quadrilateral $A C E G$ be $m$. Calculate the value of $\\frac{m}{n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "35912",
"split": "train"
} |
math | [
{
"content": "Every day at noon, a scheduled ship departs from Moscow to Astrakhan and from Astrakhan to Moscow. The ship traveling from Moscow takes exactly four days to reach Astrakhan, then stays there for two days, and at noon two days after its arrival in Astrakhan, it departs back to Moscow. The ship trav... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "35913",
"split": "train"
} |
math | [
{
"content": "Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3059",
"style": "rule"
} | {
"index": "35914",
"split": "train"
} |
math | [
{
"content": "Define $F(x, y, z) = x \\times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8 \\cdot \\sqrt[3]{2}",
"style": "rule"
} | {
"index": "35915",
"split": "train"
} |
math | [
{
"content": "Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35916",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$ . Given that $\\frac{a}{b}+\\frac{b}{a}=98$ , compute $\\sqrt c$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "202",
"style": "rule"
} | {
"index": "35917",
"split": "train"
} |
math | [
{
"content": "What is the area and perimeter of the smallest square that can contain a circle with a radius of 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "35918",
"split": "train"
} |
math | [
{
"content": "The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "35919",
"split": "train"
} |
math | [
{
"content": "For how many three-digit positive integers is the sum of the digits equal to $5?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35920",
"split": "train"
} |
math | [
{
"content": "Given that the center of circle $C$ lies on the $x$-axis and circle $C$ is tangent to the line $x + \\sqrt{3}y + n = 0$ at the point $(\\frac{3}{2}, \\frac{\\sqrt{3}}{2})$, find:\n\n1. The value of $n$ and the equation of circle $C$.\n2. If circle $M: x^2 + (y - \\sqrt{15})^2 = r^2 (r > 0)$ is tan... | {
"ground_truth": "2\\sqrt{19}",
"style": "rule"
} | {
"index": "35921",
"split": "train"
} |
math | [
{
"content": "At a nursery, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1003}{2}",
"style": "rule"
} | {
"index": "35922",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer that is divisible by three distinct primes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "35923",
"split": "train"
} |
math | [
{
"content": "In the rhombus \\(ABCD\\), the angle \\(BCD\\) is \\(135^{\\circ}\\), and the sides are 8. A circle touches the line \\(CD\\) and intersects side \\(AB\\) at two points located 1 unit away from \\(A\\) and \\(B\\). Find the radius of this circle. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{41 \\sqrt{2}}{16}",
"style": "rule"
} | {
"index": "35924",
"split": "train"
} |
math | [
{
"content": "Let the operation $\\#$ be defined as $\\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$ and $c$. What is the value of $\\#(1, 2, 3)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "35925",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\cfrac{\\left\\lceil\\cfrac{17}{7}-\\left\\lceil\\cfrac{27}{17}\\right\\rceil\\right\\rceil}{\\left\\lceil\\cfrac{27}{7}+\\left\\lceil\\cfrac{7\\cdot17}{27}\\right\\rceil\\right\\rceil}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "35926",
"split": "train"
} |
math | [
{
"content": "Given that the three sides of triangle $\\triangle ABC$ are $a$, $a+3$, and $a+6$, and the largest angle is twice the smallest angle, calculate the cosine value of the smallest angle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "35927",
"split": "train"
} |
math | [
{
"content": "For a real number \\( x \\), let \\( [x] \\) denote the greatest integer less than or equal to \\( x \\). Find the positive integer \\( n \\) such that \\(\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2} n\\right] = 1994\\). Plea... | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "35928",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\\cos B= \\frac{1}{4}$.\n(1) If $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}=4$, find the value of $b$;\n(2) If $\\sin A= \\frac{\\sqrt{6}}{4}$, find the value of $\\sin C$. Please o... | {
"ground_truth": "\\frac{3\\sqrt{6}}{8}",
"style": "rule"
} | {
"index": "35929",
"split": "train"
} |
math | [
{
"content": "A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the gr... | {
"ground_truth": "8 \\frac{3}{4}",
"style": "rule"
} | {
"index": "35930",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2} - y^2 = 1$ $(a > 0)$ and the line $l$: $x + y = 1$, which intersect at two distinct points $A$ and $B$.\n\n1. Find the range of values for $a$.\n2. Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\\overrightarrow{PA} = \\frac{5}{12}\\... | {
"ground_truth": "\\frac{17}{13}",
"style": "rule"
} | {
"index": "35931",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = |\\overrightarrow{b}| = 1$ and $|3\\overrightarrow{a} - 2\\overrightarrow{b}| = \\sqrt{7}$,\n\n(I) Find the magnitude of the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$;\n\n(II) Fi... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "35932",
"split": "train"
} |
math | [
{
"content": "The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "35933",
"split": "train"
} |
math | [
{
"content": "In the equation $|x-7| -3 = -2$, what is the product of all possible values of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "35934",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a rectangle with $AB = CD = 2$ . A circle centered at $O$ is tangent to $BC$ , $CD$ , and $AD$ (and hence has radius $1$ ). Another circle, centered at $P$ , is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$ . If line $AT$ is tangent to both ci... | {
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "35935",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x\\ln x-x$, find the monotonic intervals and the extreme values of the function $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35936",
"split": "train"
} |
math | [
{
"content": "Find the sum of the squares of the solutions to\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": "35937",
"split": "train"
} |
math | [
{
"content": "$\\triangle ABC$ is similar to $\\triangle DEF$ . What is the number of centimeters in the length of $\\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,0)--(8,-2)--(5,4)--cycle);\nlabel(\"8cm\",(2.5,2),NW);\nlabel(\"5cm\",(6.1,1),NE);\ndraw((12,0)--(18,-1.5... | {
"ground_truth": "4.8",
"style": "rule"
} | {
"index": "35938",
"split": "train"
} |
math | [
{
"content": "One year ago, the number of years in Jane's age was a perfect square, and one year from now, her age will be a perfect cube. How many years old is Jane? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "35939",
"split": "train"
} |
math | [
{
"content": "It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored with two colors), th... | {
"ground_truth": "2017^2",
"style": "rule"
} | {
"index": "35940",
"split": "train"
} |
math | [
{
"content": "Given that the polynomial \\(x^2 - kx + 24\\) has only positive integer roots, find the average of all distinct possibilities for \\(k\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35941",
"split": "train"
} |
math | [
{
"content": "Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \\pi h^3$ cubic units. What is the value of $N$?\n\n... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "35942",
"split": "train"
} |
math | [
{
"content": "An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "35943",
"split": "train"
} |
math | [
{
"content": "A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the s... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "35944",
"split": "train"
} |
math | [
{
"content": "Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "35945",
"split": "train"
} |
math | [
{
"content": "Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his ... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "35946",
"split": "train"
} |
math | [
{
"content": "If the inequality $x^{2}+ax+1 \\geqslant 0$ holds for all $x \\in (0, \\frac{1}{2}]$, find the minimum value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{2}",
"style": "rule"
} | {
"index": "35947",
"split": "train"
} |
math | [
{
"content": "The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{8}",
"style": "rule"
} | {
"index": "35948",
"split": "train"
} |
math | [
{
"content": "If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "35949",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) \\) be the polynomial \\( (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) \\) where \\( a_1, a_2, a_3, a_4, \\) and \\( a_5 \\) are distinct integers. Given that \\( f(104) = 2012 \\), evaluate \\( a_1 + a_2 + a_3 + a_4 + a_5 \\). Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35950",
"split": "train"
} |
math | [
{
"content": "Mark has $\\frac{3}{4}$ of a dollar and Carolyn has $\\frac{3}{10}$ of a dollar. How many dollars do they have altogether? (Give your answer as a decimal.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\$1.05",
"style": "rule"
} | {
"index": "35951",
"split": "train"
} |
math | [
{
"content": "If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "35952",
"split": "train"
} |
math | [
{
"content": "What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.8));\n\nxaxis(0,7,Ticks(1.0,NoZero));\nyaxis(0,7,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);... | {
"ground_truth": "67.35\\%",
"style": "rule"
} | {
"index": "35953",
"split": "train"
} |
math | [
{
"content": "How many total days were there in the years 2001 through 2004? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1461",
"style": "rule"
} | {
"index": "35954",
"split": "train"
} |
math | [
{
"content": "A region $S$ in the complex plane is defined by \\begin{align*}\nS = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.\n\\end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac34 + \\frac34i\\right)z$ is also in $S$? Please output t... | {
"ground_truth": "\\frac 79",
"style": "rule"
} | {
"index": "35955",
"split": "train"
} |
math | [
{
"content": "On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers rang... | {
"ground_truth": "15180",
"style": "rule"
} | {
"index": "35956",
"split": "train"
} |
math | [
{
"content": "23. Two friends, Marco and Ian, are talking about their ages. Ian says, \"My age is a zero of a polynomial with integer coefficients.\"\nHaving seen the polynomial \\( p(x) \\) Ian was talking about, Marco exclaims, \"You mean, you are seven years old? Oops, sorry I miscalculated! \\( p(7) = 77 \\... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "35957",
"split": "train"
} |
math | [
{
"content": "From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers. The probability that these $2$ numbers are coprime is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "35958",
"split": "train"
} |
math | [
{
"content": "In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "35959",
"split": "train"
} |
math | [
{
"content": "Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is $9$ times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$? Please output t... | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "35960",
"split": "train"
} |
math | [
{
"content": "Given the polynomial $f(x) = 4x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$, find the value of $V_1$ when calculating $f(5)$ using the Horner's Method. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "35961",
"split": "train"
} |
math | [
{
"content": "Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determine the number of swe... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "35962",
"split": "train"
} |
math | [
{
"content": "Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. Please output the final ans... | {
"ground_truth": "\\sqrt{17}-1",
"style": "rule"
} | {
"index": "35963",
"split": "train"
} |
math | [
{
"content": "If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\\leqslant x\\leqslant n\\left(m \\lt n\\right)$, and satisfies $M-N=2$, then the function is called the \"range function\" on $m\\leqslant x\\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "35964",
"split": "train"
} |
math | [
{
"content": "Let \\((a,b,c,d)\\) be an ordered quadruple of integers, each in the set \\(\\{-2, -1, 0, 1, 2\\}\\). Determine the count of such quadruples for which \\(a\\cdot d - b\\cdot c\\) is divisible by 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "35965",
"split": "train"
} |
math | [
{
"content": "Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "35966",
"split": "train"
} |
math | [
{
"content": "Determine if there exists a positive integer \\( m \\) such that the equation\n\\[\n\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}+\\frac{1}{abc}=\\frac{m}{a+b+c}\n\\]\nhas infinitely many solutions in positive integers \\( (a, b, c) \\). Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35967",
"split": "train"
} |
math | [
{
"content": "Let $(a_1,a_2,\\ldots, a_{13})$ be a permutation of $(1, 2, \\ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ . Define the *weight*... | {
"ground_truth": "13703",
"style": "rule"
} | {
"index": "35968",
"split": "train"
} |
math | [
{
"content": "What is the length of the segment of the number line whose endpoints satisfy $|x-\\sqrt[5]{16}|=3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "35969",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(x^{4}+y^{2}+\\frac{1}{2xy})^{7}$, the constant term is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{105}{16}",
"style": "rule"
} | {
"index": "35970",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the angle bisector \\(BL\\) is drawn. Find the area of the triangle, given that \\(AL = 2\\), \\(BL = \\sqrt{30}\\), and \\(CL = 5\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7\\sqrt{39}}{4}",
"style": "rule"
} | {
"index": "35971",
"split": "train"
} |
math | [
{
"content": "A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35972",
"split": "train"
} |
math | [
{
"content": "Find the ones digit of $22^{22(11^{11})}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "35973",
"split": "train"
} |
math | [
{
"content": "One piece of string is 1.5 inches long and another piece of string is 4.5 inches long. What is the average length, in inches, of these two pieces of string? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "35974",
"split": "train"
} |
math | [
{
"content": "Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{4:10 P.M.}",
"style": "rule"
} | {
"index": "35975",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\\overline{BC}, \\overline{CA},$ and $\\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\\overline{DE}, \\overline{EF},$ and $\\overline{FD},$ respectively, with the property that $P$ is on $\\overline{CQ}, Q$ i... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "35976",
"split": "train"
} |
math | [
{
"content": "Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle. Please output the final answer ... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "35977",
"split": "train"
} |
math | [
{
"content": "Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis a... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "35978",
"split": "train"
} |
math | [
{
"content": "A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?\nA) 12\nB) 15\nC) 18\nD) 20\nE) 24 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35979",
"split": "train"
} |
math | [
{
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);\n[b](ii)[/b] each row and each column contains exactly 25 kings.\n\nFind the number of such arrangemen... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "35980",
"split": "train"
} |
math | [
{
"content": "Calculate:\n$$\n202.2 \\times 89.8 - 20.22 \\times 186 + 2.022 \\times 3570 - 0.2022 \\times 16900\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18198",
"style": "rule"
} | {
"index": "35981",
"split": "train"
} |
math | [
{
"content": "During the \"Cool Summer Happy Shopping\" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ______. Please output t... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "35982",
"split": "train"
} |
math | [
{
"content": "Let $ a, b \\in \\mathbb{N}$ with $ 1 \\leq a \\leq b,$ and $ M \\equal{} \\left[\\frac {a \\plus{} b}{2} \\right].$ Define a function $ f: \\mathbb{Z} \\mapsto \\mathbb{Z}$ by\n\\[ f(n) \\equal{} \\begin{cases} n \\plus{} a, & \\text{if } n \\leq M, \\\\\nn \\minus{} b, & \\text{if } n >M. \\end... | {
"ground_truth": "\\frac {a + b}{\\gcd(a,b)}",
"style": "rule"
} | {
"index": "35983",
"split": "train"
} |
math | [
{
"content": "When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "35984",
"split": "train"
} |
math | [
{
"content": "Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\\sqrt{y}}=27$ and $(\\sqrt{x})^{y}=9$, compute $x y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16 \\sqrt[4]{3}",
"style": "rule"
} | {
"index": "35985",
"split": "train"
} |
math | [
{
"content": "When a number is tripled and then decreased by 5, the result is 16. What is the original number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "35986",
"split": "train"
} |
math | [
{
"content": "Out of two hundred ninth-grade students, $80\\%$ received excellent grades on the first exam, $70\\%$ on the second exam, and $59\\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams? Please output the final answer within \\boxed... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35987",
"split": "train"
} |
math | [
{
"content": "Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \\in \\{-1,0,1,2,3\\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35988",
"split": "train"
} |
math | [
{
"content": "Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8654232",
"style": "rule"
} | {
"index": "35989",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\\left\\{{\\begin{array}{l}{x=3-\\frac{{\\sqrt{3}}}{2}t,}\\\\{y=\\sqrt{3}-\\frac{1}{2}t}\\end{array}}\\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the p... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "35990",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \\begin{cases} x=-\\frac{1}{2}t \\\\ y=2+\\frac{\\sqrt{3}}{2}t \\end{cases} (t\\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\\rho=4\\cos\\theta$ is established with the o... | {
"ground_truth": "4+2\\sqrt{3}",
"style": "rule"
} | {
"index": "35991",
"split": "train"
} |
math | [
{
"content": "Find the number of sets $A$ that satisfy the three conditions: $\\star$ $A$ is a set of two positive integers $\\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\\star$ $A$ contains the number $30.$ Please output the final answer within \\boxed... | {
"ground_truth": "129",
"style": "rule"
} | {
"index": "35992",
"split": "train"
} |
math | [
{
"content": "A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to do this (answer with... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "35993",
"split": "train"
} |
math | [
{
"content": "Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "35994",
"split": "train"
} |
math | [
{
"content": "Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will b... | {
"ground_truth": "0.64",
"style": "rule"
} | {
"index": "35995",
"split": "train"
} |
math | [
{
"content": "A numerical sequence is defined by the conditions: \\( a_{1}=1 \\), \\( a_{n+1}=a_{n}+\\left \\lfloor \\sqrt{a_{n}} \\right \\rfloor \\).\n\nHow many perfect squares are among the first terms of this sequence that do not exceed 1,000,000? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35996",
"split": "train"
} |
math | [
{
"content": "Trapezoid $A B C D$, with bases $A B$ and $C D$, has side lengths $A B=28, B C=13, C D=14$, and $D A=15$. Let diagonals $A C$ and $B D$ intersect at $P$, and let $E$ and $F$ be the midpoints of $A P$ and $B P$, respectively. Find the area of quadrilateral $C D E F$. Please output the final answer ... | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "35997",
"split": "train"
} |
math | [
{
"content": "A positive integer \\( N \\) and \\( N^2 \\) end with the same sequence of digits \\(\\overline{abcd}\\), where \\( a \\) is a non-zero digit. Find \\(\\overline{abc}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "937",
"style": "rule"
} | {
"index": "35998",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{p}$ be the projection of $\\mathbf{v}$ onto $\\mathbf{w},$ and let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{v}.$ If $\\frac{\\|\\mathbf{p}\\|}{\\|\\mathbf{v}\\|} = \\frac{5}{7},$ then find $\\frac{\\|\\mathbf{q}\\|}{\\|\\mathbf{v}\\|}.$ Please output the final... | {
"ground_truth": "\\frac{25}{49}",
"style": "rule"
} | {
"index": "35999",
"split": "train"
} |
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