data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given positive integers $a$ and $b$ are members of a set where $a \\in \\{2, 3, 5, 7\\}$ and $b \\in \\{2, 4, 6, 8\\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \\cdot a - a \\cdot b$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "38200",
"split": "train"
} |
math | [
{
"content": "Sally the snail sits on the $3 \\times 24$ lattice of points $(i, j)$ for all $1 \\leq i \\leq 3$ and $1 \\leq j \\leq 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2,1)$, com... | {
"ground_truth": "4096",
"style": "rule"
} | {
"index": "38201",
"split": "train"
} |
math | [
{
"content": "If the fractional equation $\\frac{3}{{x-2}}+1=\\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "38202",
"split": "train"
} |
math | [
{
"content": "In the diagram, a rectangular ceiling \\( P Q R S \\) measures \\( 6 \\mathrm{~m} \\) by \\( 4 \\mathrm{~m} \\) and is to be completely covered using 12 rectangular tiles, each measuring \\( 1 \\mathrm{~m} \\) by \\( 2 \\mathrm{~m} \\). If there is a beam, \\( T U \\), that is positioned so that \... | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "38203",
"split": "train"
} |
math | [
{
"content": "\\(AB\\) is the diameter of a circle; \\(BC\\) is a tangent; \\(D\\) is the point where line \\(AC\\) intersects the circle. It is known that \\(AD = 32\\) and \\(DC = 18\\). Find the radius of the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38204",
"split": "train"
} |
math | [
{
"content": "A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where the bug is located, uses no edge multiple times, and uses at most two of the edges adjacent to any particular face. Find the number of healthy paths. Please output the final ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38205",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that\n\n$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2, 1, 4, 1) \\text{ and } (2, 1, 1, 4)",
"style": "rule"
} | {
"index": "38206",
"split": "train"
} |
math | [
{
"content": "Given $l_{1}$: $ρ \\sin (θ- \\frac{π}{3})= \\sqrt {3}$, $l_{2}$: $ \\begin{cases} x=-t \\\\ y= \\sqrt {3}t \\end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse $\\frac{x^{2}}{4}+y... | {
"ground_truth": "\\frac{15}{8}",
"style": "rule"
} | {
"index": "38207",
"split": "train"
} |
math | [
{
"content": "Given $a-b=4$ and $b+c=2$, determine the value of $a^2+b^2+c^2-ab+bc+ca$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38208",
"split": "train"
} |
math | [
{
"content": "An integer is called \"octal\" if it is divisible by $8$ or if at least one of its digits is $8$ . \nHow many integers between $1$ and $100$ are octal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38209",
"split": "train"
} |
math | [
{
"content": "Let $a = \\pi/2008$. Find the smallest positive integer $n$ such that\\[2[\\cos(a)\\sin(a) + \\cos(4a)\\sin(2a) + \\cos(9a)\\sin(3a) + \\cdots + \\cos(n^2a)\\sin(na)]\\]is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "251",
"style": "rule"
} | {
"index": "38210",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with a common ratio $q \\in (1,2)$, and that $a_{n}$ is a positive integer for $1 \\leq n \\leq 6$, find the minimum value of $a_{6}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "38211",
"split": "train"
} |
math | [
{
"content": "Given $\\tan \\alpha = -\\frac{1}{2}$, find the value of $\\frac{1+2\\sin \\alpha \\cos \\alpha}{\\sin^2 \\alpha - \\cos^2 \\alpha}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "38212",
"split": "train"
} |
math | [
{
"content": "Let $x$, $y$, and $z$ be positive integers such that $x + y + z = 12$. What is the smallest possible value of $(x+y+z)\\left(\\frac{1}{x+y}+\\frac{1}{x+z}+\\frac{1}{y+z}\\right)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "38213",
"split": "train"
} |
math | [
{
"content": "Solve \\[\\frac{x^2+x+1}{x+1}=x+2\\]for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac12",
"style": "rule"
} | {
"index": "38214",
"split": "train"
} |
math | [
{
"content": "A small village has $n$ people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that ev... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "38215",
"split": "train"
} |
math | [
{
"content": "On a table, there are 30 coins: 23 ten-ruble coins and 7 five-ruble coins. Out of these, 20 coins are heads up, and the remaining 10 are tails up. What is the smallest number \\( k \\) such that among any randomly chosen \\( k \\) coins, there will always be at least one ten-ruble coin that is hea... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38216",
"split": "train"
} |
math | [
{
"content": "In the figure, $\\overline{AB}$ and $\\overline{CD}$ are diameters of the circle with center $O$, $\\overline{AB} \\perp \\overline{CD}$, and chord $\\overline{DF}$ intersects $\\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is Please output the final answer within \\... | {
"ground_truth": "24 \\pi",
"style": "rule"
} | {
"index": "38217",
"split": "train"
} |
math | [
{
"content": "The numbers 1 to 2031 are written on a blackboard.\n1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.\n2. Then select another number on the blackboard, find the absolute value of its difference from the previous absolute value ob... | {
"ground_truth": "2030",
"style": "rule"
} | {
"index": "38218",
"split": "train"
} |
math | [
{
"content": "Starting with a pair of adult rabbits, the number of pairs of adult rabbits after one year can be calculated using the recurrence relation of the Fibonacci sequence, and the result is given by $F(13). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "38219",
"split": "train"
} |
math | [
{
"content": "A circle of radius $r$ has chords $\\overline{AB}$ of length $12$ and $\\overline{CD}$ of length $9$. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\\angle{APD}=90^\\circ$ and $BP=10$, det... | {
"ground_truth": "221",
"style": "rule"
} | {
"index": "38220",
"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": "38221",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+n.\\,$ Please... | {
"ground_truth": "450",
"style": "rule"
} | {
"index": "38222",
"split": "train"
} |
math | [
{
"content": "If $\\frac{2013 \\times 2013}{2014 \\times 2014 + 2012} = \\frac{n}{m}$ (where $m$ and $n$ are coprime natural numbers), then what is the value of $m + n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1343",
"style": "rule"
} | {
"index": "38223",
"split": "train"
} |
math | [
{
"content": "Use the five digits $0$, $1$, $2$, $3$, $4$ to form integers that satisfy the following conditions:\n(I) All four-digit integers;\n(II) Five-digit integers without repetition that are greater than $21000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "38224",
"split": "train"
} |
math | [
{
"content": "What is the degree measure of the supplement of the complement of a 42-degree angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "38225",
"split": "train"
} |
math | [
{
"content": "There are \\( n \\) pieces of paper, each containing 3 different positive integers no greater than \\( n \\). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "38226",
"split": "train"
} |
math | [
{
"content": "A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "38227",
"split": "train"
} |
math | [
{
"content": "For any real number $a$, let $\\left[a\\right]$ denote the largest integer not exceeding $a$. For example, $\\left[4\\right]=4$, $[\\sqrt{3}]=1$. Now, for the number $72$, the following operations are performed: $72\\stackrel{1st}{→}[\\sqrt{72}]=8\\stackrel{2nd}{→}[\\sqrt{8}]=2\\stackrel{3rd}{→}[\... | {
"ground_truth": "6560",
"style": "rule"
} | {
"index": "38228",
"split": "train"
} |
math | [
{
"content": "A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (both horizontal and vertical) and containing at least 7 black squares, can be drawn on the checkerboard? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "38229",
"split": "train"
} |
math | [
{
"content": "Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "676",
"style": "rule"
} | {
"index": "38230",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\overrightarrow {BA} \\cdot \\overrightarrow {AC} = 6$, $b-c=2$, and $\\tan A = -\\sqrt {15}$, find the length of the altitude drawn from $A$ to side $BC$. Please out... | {
"ground_truth": "\\frac{3\\sqrt{15}}{4}",
"style": "rule"
} | {
"index": "38231",
"split": "train"
} |
math | [
{
"content": "Given that \\( n \\) is a positive integer and \\( n \\leq 144 \\), ten questions of the type \"Is \\( n \\) smaller than \\( a \\)?\" are allowed. The answers are provided with a delay such that the answer to the \\( i \\)-th question is given only after the \\((i+1)\\)-st question is asked for \... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "38232",
"split": "train"
} |
math | [
{
"content": "Mrs. Crabapple now teaches two different classes of British Literature. Her first class has 12 students and meets three times a week, while her second class has 9 students and meets twice a week. How many different sequences of crabapple recipients are possible in a week for both classes combined?... | {
"ground_truth": "139,968",
"style": "rule"
} | {
"index": "38233",
"split": "train"
} |
math | [
{
"content": "Find the largest prime factor of $11236$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "38234",
"split": "train"
} |
math | [
{
"content": "Johny's father tells him: \"I am twice as old as you will be seven years from the time I was thrice as old as you were\". What is Johny's age? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "38235",
"split": "train"
} |
math | [
{
"content": "Given $f(x)= \\frac{a\\cdot 2^{x}+a-2}{2^{x}+1}$ ($x\\in\\mathbb{R}$), if $f(x)$ satisfies $f(-x)+f(x)=0$,\n(1) find the value of the real number $a$ and $f(3)$;\n(2) determine the monotonicity of the function and provide a proof. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "38236",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to the angles $A$, $B$, and $C$ are labeled as $a$, $b$, and $c$ respectively. Given $a=5, B= \\frac {\\pi}{3},$ and $\\cos A= \\frac {11}{14}$, find the area $S$ of the triangle $ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "10 \\sqrt {3}",
"style": "rule"
} | {
"index": "38237",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number \\( n \\) for which \\( (n+1)(n+2)(n+3)(n+4) \\) is divisible by 1000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "38238",
"split": "train"
} |
math | [
{
"content": "Given \\(\\alpha, \\beta \\in [0, \\pi]\\), find the maximum value of \\((\\sin \\alpha + \\sin (\\alpha + \\beta)) \\cdot \\sin \\beta\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8\\sqrt{3}}{9}",
"style": "rule"
} | {
"index": "38239",
"split": "train"
} |
math | [
{
"content": "For non-zero real numbers \\( x, y, z, w \\), if\n$$\n\\frac{6 x y + 5 y z + 6 z w}{x^{2} + y^{2} + z^{2} + w^{2}} \\leq f,\n$$\n\nfind the minimum value of \\( f \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9/2",
"style": "rule"
} | {
"index": "38240",
"split": "train"
} |
math | [
{
"content": "In a math interest class, the teacher gave a problem for everyone to discuss: \"Given real numbers $a$, $b$, $c$ not all equal to zero satisfying $a+b+c=0$, find the maximum value of $\\frac{|a+2b+3c|}{\\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}.\"$ Jia quickly offered his opinion: Isn't this just the Cauchy... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "38241",
"split": "train"
} |
math | [
{
"content": "A die has faces numbered $1$, $2$, $3$, $3$, $4$, and $4$. Another die has faces numbered $2$, $3$, $5$, $6$, $7$, and $8$. Determine the probability that the sum of the top two numbers when both dice are rolled will be either $6$, $8$, or $10$.\nA) $\\frac{1}{36}$\nB) $\\frac{1}{18}$\nC) $\\frac{... | {
"ground_truth": "\\frac{11}{36}",
"style": "rule"
} | {
"index": "38242",
"split": "train"
} |
math | [
{
"content": "How many different $4\\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "38243",
"split": "train"
} |
math | [
{
"content": "A farmer has a right-angled triangular farm with legs of lengths 3 and 4. At the right-angle corner, the farmer leaves an unplanted square area $S$. The shortest distance from area $S$ to the hypotenuse of the triangle is 2. What is the ratio of the area planted with crops to the total area of the... | {
"ground_truth": "$\\frac{145}{147}$",
"style": "rule"
} | {
"index": "38244",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(A B C D\\) is a rectangle, \\(P\\) is on \\(B C\\), \\(Q\\) is on \\(C D\\), and \\(R\\) is inside \\(A B C D\\). Also, \\(\\angle P R Q = 30^\\circ\\), \\(\\angle R Q D = w^\\circ\\), \\(\\angle P Q C = x^\\circ\\), \\(\\angle C P Q = y^\\circ\\), and \\(\\angle B P R = z^\\cir... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "38245",
"split": "train"
} |
math | [
{
"content": "Find $\\tan G$ in the right triangle where side GH is 20 units, side FG is 25 units, and ∠H is the right angle.\n\n[asy]\npair H,F,G;\nH = (0,0);\nG = (20,0);\nF = (0,25);\ndraw(F--G--H--F);\ndraw(rightanglemark(F,H,G,20));\nlabel(\"$H$\",H,SW);\nlabel(\"$G$\",G,SE);\nlabel(\"$F$\",F,N);\nlabel(\"... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "38246",
"split": "train"
} |
math | [
{
"content": "If you add 2 to the last digit of the quotient, you get the penultimate digit. If you add 2 to the third digit from the right of the quotient, you get the fourth digit from the right. For example, the quotient could end in 9742 or 3186.\n\nWe managed to find only one solution. Please output the fi... | {
"ground_truth": "9742",
"style": "rule"
} | {
"index": "38247",
"split": "train"
} |
math | [
{
"content": "When the numbers \\(\\sqrt{5}, 2.1, \\frac{7}{3}, 2.0 \\overline{5}, 2 \\frac{1}{5}\\) are arranged in order from smallest to largest, the middle number is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\frac{1}{5}",
"style": "rule"
} | {
"index": "38248",
"split": "train"
} |
math | [
{
"content": "If I roll a fair, regular six-sided die four times, what is the probability that I will roll the number $1$ exactly three times? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{324}",
"style": "rule"
} | {
"index": "38249",
"split": "train"
} |
math | [
{
"content": "Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "38250",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers. Find the maximum value of\n\\[\\frac{(x + y)^2}{x^2 + y^2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "38251",
"split": "train"
} |
math | [
{
"content": "Round 3967149.1587234 to the nearest integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3967149",
"style": "rule"
} | {
"index": "38252",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\\overline{AB}$ is tangent to segments $\\overline{AD}$ and $\\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=\n$\\text{(A) } 1369\\quad \\text{(B) } 1679\\q... | {
"ground_truth": "1679",
"style": "rule"
} | {
"index": "38253",
"split": "train"
} |
math | [
{
"content": "Given that $α,β∈(0,π), \\text{and } cosα= \\frac {3}{5}, cosβ=- \\frac {12}{13}$.\n(1) Find the value of $cos2α$,\n(2) Find the value of $sin(2α-β)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{253}{325}",
"style": "rule"
} | {
"index": "38254",
"split": "train"
} |
math | [
{
"content": "A three-digit number has distinct digits. By arbitrarily swapping the positions of its digits, five other three-digit numbers can be obtained. If the sum of these six three-digit numbers equals 2220, then among all the numbers that meet this condition, the smallest three-digit number is ____. Plea... | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "38255",
"split": "train"
} |
math | [
{
"content": "In the animal kingdom, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 of each of these three types of animals, divided into 100 groups. Each group has exactly 3 animals, with exactly 2 animals of one type and 1 animal of anothe... | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "38256",
"split": "train"
} |
math | [
{
"content": "For any positive integer $n$, let $\\tau(n)$ denote the number of positive divisors of $n$. If $n$ is a positive integer such that $\\frac{\\tau\\left(n^{2}\\right)}{\\tau(n)}=3$, compute $\\frac{\\tau\\left(n^{7}\\right)}{\\tau(n)}$. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "38257",
"split": "train"
} |
math | [
{
"content": "The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38258",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $\\angle BAC = 90^\\circ$. A circle is tangent to the sides $AB$ and $AC$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $BC$. Given that $AB = 6$, find the area of the portion of the circle th... | {
"ground_truth": "\\pi - 2",
"style": "rule"
} | {
"index": "38259",
"split": "train"
} |
math | [
{
"content": "Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25$, and $36$. What is the sum of the areas of the six rectangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "182",
"style": "rule"
} | {
"index": "38260",
"split": "train"
} |
math | [
{
"content": "Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part?\n- $z = -3$\n- $z = -2 + \\frac{1}{2}i$\n- $z = -\\frac{3}{2} + \\frac{3}{2}i$\n- $z = -1 + 2i$\n- $z = 3i$\nA) $-243$\nB) $-12.3125$\nC) $-168.75$\nD) $39$\nE) $0$ Please output the final answe... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "38261",
"split": "train"
} |
math | [
{
"content": "Segment $s_1$ has endpoints at $(4,1)$ and $(-8,5)$. Segment $s_2$ is obtained by translating $s_1$ by $2$ units to the right and $3$ units up. Find the midpoint of segment $s_2$. Express your answer as $(a,b)$ with $a$ and $b$ integers. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "(0,6)",
"style": "rule"
} | {
"index": "38262",
"split": "train"
} |
math | [
{
"content": "A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the amount of wages underp... | {
"ground_truth": "2.60",
"style": "rule"
} | {
"index": "38263",
"split": "train"
} |
math | [
{
"content": "Over the summer, a one-room apartment increased in price by 21%, a two-room apartment by 11%, and the total cost of both apartments by 15%. How many times cheaper is the one-room apartment compared to the two-room apartment? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "38264",
"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, 6, and 7. What is the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{202.2192}{\\pi^2}",
"style": "rule"
} | {
"index": "38265",
"split": "train"
} |
math | [
{
"content": "A bag contains 6 red balls and 8 white balls. If 5 balls are randomly placed into Box $A$ and the remaining 9 balls are placed into Box $B$, what is the probability that the sum of the number of white balls in Box $A$ and the number of red balls in Box $B$ is not a prime number? (Answer with a num... | {
"ground_truth": "213/1001",
"style": "rule"
} | {
"index": "38266",
"split": "train"
} |
math | [
{
"content": "Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\\cdots 100$ .\n[list=1]\n[*] 20\n[*] 25\n[*] 15\n[*] 50\n[/list] Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38267",
"split": "train"
} |
math | [
{
"content": "Which number from the set $\\{1,2,3,4,5,6,7,8,9,10,11\\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38268",
"split": "train"
} |
math | [
{
"content": "A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01? Please output the final answer within \\boxed{}... | {
"ground_truth": "11475",
"style": "rule"
} | {
"index": "38269",
"split": "train"
} |
math | [
{
"content": "One out of every five Americans suffers from allergies. In a random sample of 250 Americans, how many would you expect to suffer from allergies? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "38270",
"split": "train"
} |
math | [
{
"content": "Given rectangle $R_1$ with one side $4$ inches and area $24$ square inches. Rectangle $R_2$ with diagonal $17$ inches is similar to $R_1$. Find the area of $R_2$ in square inches. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{433.5}{3.25}",
"style": "rule"
} | {
"index": "38271",
"split": "train"
} |
math | [
{
"content": "What is the value of $(2x + 5)^2$ when $x = 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "38272",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $\\angle C=90^{\\circ}, \\angle B=30^{\\circ}, AC=2$, $M$ is the midpoint of $AB$. Fold triangle $ACM$ along $CM$ such that the distance between points $A$ and $B$ becomes $2\\sqrt{2}$. Find the volume of the resulting triangular pyramid $A-BCM$. Please output the final answer w... | {
"ground_truth": "\\frac{2 \\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "38273",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of\n\\[\n\\frac{\\sin^4 x + \\cos^4 x + 2}{\\sin^6 x + \\cos^6 x + 2}\n\\]\nover all real values \\(x\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{9}",
"style": "rule"
} | {
"index": "38274",
"split": "train"
} |
math | [
{
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{23}{54}",
"style": "rule"
} | {
"index": "38275",
"split": "train"
} |
math | [
{
"content": "How many unique numbers can you get by multiplying two or more distinct members of the set $\\{1,2,3,5,7\\}$ together? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38276",
"split": "train"
} |
math | [
{
"content": "If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{175}{1296}",
"style": "rule"
} | {
"index": "38277",
"split": "train"
} |
math | [
{
"content": "Given distinct prime numbers $p$ and $q$ and a natural number $n \\geq 3$, find all $a \\in \\mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "-1 - pq \\text{ and } 1 + pq",
"style": "rule"
} | {
"index": "38278",
"split": "train"
} |
math | [
{
"content": "Given Joy has 50 thin rods, one each of every integer length from 1 cm through 50 cm, and rods with lengths 8 cm, 12 cm, and 25 cm are already placed on a table, determine the number of the remaining rods that can be chosen as the fourth rod to form a quadrilateral with positive area. Please outpu... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38279",
"split": "train"
} |
math | [
{
"content": "A function $f$ satisfies, for all nonnegative integers $x$ and $y$: - $f(0, x)=f(x, 0)=x$ - If $x \\geq y \\geq 0, f(x, y)=f(x-y, y)+1$ - If $y \\geq x \\geq 0, f(x, y)=f(x, y-x)+1$ Find the maximum value of $f$ over $0 \\leq x, y \\leq 100$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "38280",
"split": "train"
} |
math | [
{
"content": "A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars? Please output the final answer within \\boxed{}.... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38281",
"split": "train"
} |
math | [
{
"content": "When three consecutive integers are added, the total is 27. What is the result when the same three integers are multiplied? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "38282",
"split": "train"
} |
math | [
{
"content": "The non-negative difference between two numbers \\(a\\) and \\(b\\) is \\(a-b\\) or \\(b-a\\), whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence \\(88, 24, 64, 40, 24, \\ldots\\), each number after the second is obtained by f... | {
"ground_truth": "760",
"style": "rule"
} | {
"index": "38283",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38284",
"split": "train"
} |
math | [
{
"content": "Find \\(\\cos 2 \\alpha\\), given that \\(2 \\operatorname{ctg}^{2} \\alpha+7 \\operatorname{ctg} \\alpha+3=0\\) and the value of \\(\\alpha\\) satisfies the inequalities: \na) \\(\\frac{3 \\pi}{2}<\\alpha<\\frac{7 \\pi}{4}\\);\nb) \\(\\frac{7 \\pi}{4}<\\alpha<2 \\pi\\). Please output the final an... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "38285",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$ with $AB=AC,$ point $D$ lies strictly between $A$ and $C$ on side $\\overline{AC},$ and point $E$ lies strictly between $A$ and $B$ on side $\\overline{AB}$ such that $AE=ED=DB=BC.$ The degree measure of $\\angle ABC$ is $\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime p... | {
"ground_truth": "547",
"style": "rule"
} | {
"index": "38286",
"split": "train"
} |
math | [
{
"content": "A right circular cone is sliced into five equal-height sections by planes parallel to its base. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{37}{61}",
"style": "rule"
} | {
"index": "38287",
"split": "train"
} |
math | [
{
"content": "Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\\triangle ABC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "38288",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the product \\(x^{2} y^{2} z^{2} u\\) given the condition that \\(x, y, z, u \\geq 0\\) and:\n\n\\[ 2x + xy + z + yz u = 1 \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/512",
"style": "rule"
} | {
"index": "38289",
"split": "train"
} |
math | [
{
"content": "Distinct planes $p_1, p_2, \\dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \\bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38290",
"split": "train"
} |
math | [
{
"content": "Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\\frac{700}{99}= 7.\\overline{07}= 7.070707\\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special num... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38291",
"split": "train"
} |
math | [
{
"content": "Compute\n\n$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "88572",
"style": "rule"
} | {
"index": "38292",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 2:7.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA = (0,0);\nB = (5,1);\nP = ... | {
"ground_truth": "\\left( \\frac{7}{9}, \\frac{2}{9} \\right)",
"style": "rule"
} | {
"index": "38293",
"split": "train"
} |
math | [
{
"content": "Let $S$ and $S_{1}$ respectively be the midpoints of edges $AD$ and $B_{1}C_{1}$. A rotated cube is denoted by $A^{\\prime}B^{\\prime}C^{\\prime}D^{\\prime}A_{1}^{\\prime}B_{1}^{\\prime}C_{1}^{\\prime}D_{1}^{\\prime}$. The common part of the original cube and the rotated one is a polyhedron consis... | {
"ground_truth": "\\sqrt{2} - \\frac{2}{3}",
"style": "rule"
} | {
"index": "38294",
"split": "train"
} |
math | [
{
"content": "A point is randomly thrown on the segment \\([6, 11]\\) and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\(\\left(k^{2}-2k-24\\right)x^{2}+(3k-8)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\). Please output the final answer within \\boxed{}.... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "38295",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(2\\sin \\omega x,2\\cos \\omega x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos\\omega x,-\\cos\\omega x)$, where the function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$ has a minimum positive period of $6\\pi$. Find the value of the real number $\\omega$.... | {
"ground_truth": "-\\frac{16}{65}",
"style": "rule"
} | {
"index": "38296",
"split": "train"
} |
math | [
{
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; import solids; size(5c... | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "38297",
"split": "train"
} |
math | [
{
"content": "At Easter-Egg Academy, each student has two eyes, each of which can be eggshell, cream, or cornsilk. It is known that $30 \\%$ of the students have at least one eggshell eye, $40 \\%$ of the students have at least one cream eye, and $50 \\%$ of the students have at least one cornsilk eye. What per... | {
"ground_truth": "80 \\%",
"style": "rule"
} | {
"index": "38298",
"split": "train"
} |
math | [
{
"content": "A point is randomly thrown onto the segment [6, 11], and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\( \\left(k^{2}-2k-15\\right)x^{2}+(3k-7)x+2=0 \\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\). Please output the final answer within \\boxed{}."... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "38299",
"split": "train"
} |
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