data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the distance between the vertices of the hyperbola given by the equation $4x^2 + 16x - 9y^2 + 18y - 23 = 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{30}",
"style": "rule"
} | {
"index": "36600",
"split": "train"
} |
math | [
{
"content": "A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally schedul... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "36601",
"split": "train"
} |
math | [
{
"content": "There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and cut with a straight line, not passing through its vertices, into two sheets. After 60 moves, it turned out that all the sheets are triangles or hexagons. How many hexagons are there? Please output the final ans... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "36602",
"split": "train"
} |
math | [
{
"content": "In a triangle, one of the sides is equal to 6, the radius of the inscribed circle is 2, and the radius of the circumscribed circle is 5. Find the perimeter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "36603",
"split": "train"
} |
math | [
{
"content": "Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $1+2i$. Gracie chooses $-1+i$. How far apart are Gracie and Joe's points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "36604",
"split": "train"
} |
math | [
{
"content": "If $3 \\in \\{a, a^2 - 2a\\}$, then the value of the real number $a$ is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "36605",
"split": "train"
} |
math | [
{
"content": "Given the complex number $z$ that satisfies $$z= \\frac {1-i}{i}$$ (where $i$ is the imaginary unit), find $z^2$ and $|z|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt {2}",
"style": "rule"
} | {
"index": "36606",
"split": "train"
} |
math | [
{
"content": "Thirteen blue and six green hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of green tiles with the same size and shape as the others, what will be the difference between the total number of green tiles and the total number of blue tiles in th... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "36607",
"split": "train"
} |
math | [
{
"content": "What is $\\left(\\dfrac{9819156364}{-24635745744574}\\right)^0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "36608",
"split": "train"
} |
math | [
{
"content": "Given several rectangular prisms with edge lengths of $2, 3,$ and $5$, aligned in the same direction to form a cube with an edge length of $90$, how many small rectangular prisms does one diagonal of the cube intersect? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "36609",
"split": "train"
} |
math | [
{
"content": "Given that $x$, $y$, $z$ are positive real numbers, find the maximum value of $\\dfrac{xy+yz}{x^{2}+y^{2}+z^{2}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "36610",
"split": "train"
} |
math | [
{
"content": "Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "36611",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\(ABC\\) with an area of 1. Points \\(P\\), \\(Q\\), and \\(R\\) are taken on the medians \\(AK\\), \\(BL\\), and \\(CN\\) respectively such that \\(AP = PK\\), \\(BQ : QL = 1 : 2\\), and \\(CR : RN = 5 : 4\\). Find the area of triangle \\(PQR\\). Please output the final answer w... | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "36612",
"split": "train"
} |
math | [
{
"content": "The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$? Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "553",
"style": "rule"
} | {
"index": "36613",
"split": "train"
} |
math | [
{
"content": "The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 17\\qquad\\textbf{(E)}\\ 18$ Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36614",
"split": "train"
} |
math | [
{
"content": "Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collecte... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "36615",
"split": "train"
} |
math | [
{
"content": "A pedestrian departed from point \\( A \\) to point \\( B \\). After walking 8 km, a second pedestrian left point \\( A \\) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \\( B \\), and both pedestrians arrived at point \\( B ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "36616",
"split": "train"
} |
math | [
{
"content": "For each positive integer $k$ , let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$ . Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$ . Determine the number of digits in $n$ . Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "224",
"style": "rule"
} | {
"index": "36617",
"split": "train"
} |
math | [
{
"content": "The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "36618",
"split": "train"
} |
math | [
{
"content": "How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "36619",
"split": "train"
} |
math | [
{
"content": "An airplane has three sections: First Class (24 seats), Business Class ($25\\%$ of the total number of seats) and Economy ($\\frac{2}{3}$ of the total number of seats). How many seats does the plane have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "36620",
"split": "train"
} |
math | [
{
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{15})$ be a permutation of $(1,2,3,\\ldots,15)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \\mathrm{\\ and \\ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$\nAn example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find the number of such per... | {
"ground_truth": "3003",
"style": "rule"
} | {
"index": "36621",
"split": "train"
} |
math | [
{
"content": "Find the product of all positive integral values of $n$ such that $n^2-35n+306= p$ for some prime number $p$. Note that there is at least one such $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "304",
"style": "rule"
} | {
"index": "36622",
"split": "train"
} |
math | [
{
"content": "There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$ . What is the maximum possible number of cups in the kitchen? Please output the final answer within \\box... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "36623",
"split": "train"
} |
math | [
{
"content": "A circle is tangent to sides \\( AB \\) and \\( AD \\) of rectangle \\( ABCD \\) and intersects side \\( DC \\) at a single point \\( F \\) and side \\( BC \\) at a single point \\( E \\).\n\nFind the area of trapezoid \\( AFCB \\) if \\( AB = 32 \\), \\( AD = 40 \\), and \\( BE = 1 \\). Please ou... | {
"ground_truth": "1180",
"style": "rule"
} | {
"index": "36624",
"split": "train"
} |
math | [
{
"content": "Yan is somewhere between his office and a concert hall. To get to the concert hall, he can either walk directly there, or walk to his office and then take a scooter to the concert hall. He rides 5 times as fast as he walks, and both choices take the same amount of time. What is the ratio of Yan's ... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "36625",
"split": "train"
} |
math | [
{
"content": "Determine the sum of all prime numbers $p$ for which there exists no integer solution $x$ to the congruence $5(8x+2)\\equiv 3\\pmod{p}$, and there exists no integer solution $y$ to the congruence $3(10y+3)\\equiv 2\\pmod{p}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36626",
"split": "train"
} |
math | [
{
"content": "The angle bisector of angle \\(ABC\\) forms an angle with its sides that is three times smaller than the adjacent angle to \\(ABC\\). Find the measure of angle \\(ABC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "36627",
"split": "train"
} |
math | [
{
"content": "Let $A_{10}$ denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by $x$ and $y$, respectively. Given that the product of the radii of these two circles is $15 / 2$, and that the distance between their cen... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "36628",
"split": "train"
} |
math | [
{
"content": "Given $\\sqrt[3]{2.37} \\approx 1.333$ and $\\sqrt[3]{23.7} \\approx 2.872$, determine the approximate value of $\\sqrt[3]{2370}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13.33",
"style": "rule"
} | {
"index": "36629",
"split": "train"
} |
math | [
{
"content": "The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\\sqrt{3},~5,$ and $\\sqrt{37},$ as shown, is $\\frac{m\\sqrt{p}}{n},$ where $m,~n,$ and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisibl... | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "36630",
"split": "train"
} |
math | [
{
"content": "Rounded to 2 decimal places, what is $\\frac{7}{9}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.78",
"style": "rule"
} | {
"index": "36631",
"split": "train"
} |
math | [
{
"content": "Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? Please output the final answer withi... | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "36632",
"split": "train"
} |
math | [
{
"content": "Find the root $x$ of the equation $\\log x = 4 - x$ where $x \\in (k, k+1)$, and $k \\in \\mathbb{Z}$. What is the value of $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "k = 3",
"style": "rule"
} | {
"index": "36633",
"split": "train"
} |
math | [
{
"content": "Suppose $11^5\\equiv n\\pmod 9$, where $0\\le n<9$.\n\nWhat is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36634",
"split": "train"
} |
math | [
{
"content": "For all positive integers $n$, let $g(n)=\\log_{3003} n^3$. Find $g(7)+g(11)+g(13)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "36635",
"split": "train"
} |
math | [
{
"content": "How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "36636",
"split": "train"
} |
math | [
{
"content": "Adam filled a $3 \\times 3$ table with the numbers from 1 to 9 as follows:\n\n| 7 | 6 | 4 |\n| :--- | :--- | :--- |\n| 1 | 2 | 8 |\n| 9 | 3 | 5 |\n\nFor this arrangement, the sum of the numbers along every side of the table remains the same. Adam found that the numbers can be arranged differently,... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36637",
"split": "train"
} |
math | [
{
"content": "If $\\alpha \\in (0, \\frac{\\pi}{2})$, and $\\tan 2\\alpha = \\frac{\\cos \\alpha}{2-\\sin \\alpha}$, then find the value of $\\tan \\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{15}}{15}",
"style": "rule"
} | {
"index": "36638",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB = 13$, $BC = 15$, and $CA = 14$. Point $D$ is on $\\overline{BC}$ with $CD = 6$. Point $E$ is on $\\overline{BC}$ such that $\\angle BAE = \\angle CAD$. Find $BE.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2535}{463}",
"style": "rule"
} | {
"index": "36639",
"split": "train"
} |
math | [
{
"content": "Consider a modified sequence rule:\n1) If a number is 30 or less, triple the number.\n2) If a number is more than 30, subtract 15 from it.\n\nLet $G$ be the first number in a sequence generated by the new rule. $G$ is a \"magic number\" if 18 is not a term in the sequence that starts with $G$. Det... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36640",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "4022",
"style": "rule"
} | {
"index": "36641",
"split": "train"
} |
math | [
{
"content": "On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "36642",
"split": "train"
} |
math | [
{
"content": "Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \\in \\mathbb{R}$, the range of $f(x)$ is $[0, +\\infty)$. \n(1) Find the expression for $f(x)$. \n(2) Let $g(x) = f(x) - 2kx$, where $k \\in \\mathbb{R}$. \n (i) If $g(x)$ is monotonic on $x \\in [-2, 2... | {
"ground_truth": "k = 6",
"style": "rule"
} | {
"index": "36643",
"split": "train"
} |
math | [
{
"content": "Given that \\( b \\) is an even number between 1 and 11 (inclusive), and \\( c \\) is any natural number, determine the number of quadratic equations \\( x^{2} + b x + c = 0 \\) that have two distinct real roots. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "36644",
"split": "train"
} |
math | [
{
"content": "Given a square $A B C D$ on a plane, find the minimum of the ratio $\\frac{O A + O C}{O B + O D}$, where $O$ is an arbitrary point on the plane. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "36645",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x) = x^3 + 3x^2 + 6x + 14 \\), and \\( f(a) = 1 \\), \\( f(b) = 19 \\), find the value of \\( a + b \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "36646",
"split": "train"
} |
math | [
{
"content": "If $\\lfloor{\\sqrt{x}}\\rfloor=6$, how many possible integer values of $x$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "36647",
"split": "train"
} |
math | [
{
"content": "Find the product of $218_9 \\cdot 5_9$. Express your answer in base 9. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1204_9",
"style": "rule"
} | {
"index": "36648",
"split": "train"
} |
math | [
{
"content": "Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "294",
"style": "rule"
} | {
"index": "36649",
"split": "train"
} |
math | [
{
"content": "They paid 100 rubles for a book and still need to pay as much as they would need to pay if they had paid as much as they still need to pay. How much does the book cost? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "36650",
"split": "train"
} |
math | [
{
"content": "Given that $13^{-1} \\equiv 29 \\pmod{47}$, find $34^{-1} \\pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "36651",
"split": "train"
} |
math | [
{
"content": "In a bus station in the city, there are 10 waiting seats arranged in a row. Now, if 4 passengers randomly choose some seats to wait, the number of ways to arrange them so that there are exactly 5 consecutive empty seats is $\\boxed{480}$. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "480",
"style": "rule"
} | {
"index": "36652",
"split": "train"
} |
math | [
{
"content": "Among the natural numbers from 1 to 1000, there are a total of number 7s. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "36653",
"split": "train"
} |
math | [
{
"content": "A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converg... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "36654",
"split": "train"
} |
math | [
{
"content": "Compute the unique positive integer $n$ such that $\\frac{n^{3}-1989}{n}$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "36655",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\), \\(b\\), \\(c\\) be real numbers such that \\(9a^2 + 4b^2 + 25c^2 = 1\\). Find the maximum value of\n\\[\n10a + 3b + 5c.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{134}",
"style": "rule"
} | {
"index": "36656",
"split": "train"
} |
math | [
{
"content": "In the diagram, what is the value of $y$?\n\n[asy]\ndraw((0,0)--(18,0),black+linewidth(1));\ndraw((18,0)--(18,-6),black+linewidth(1));\ndraw((0,0)--(4,6)--(18,-6),black+linewidth(1));\ndraw((18,0)--(18,-0.5)--(17.5,-0.5)--(17.5,0)--cycle,black+linewidth(1));\nlabel(\"$80^{\\circ}$\",(4.5,5),S);\nl... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "36657",
"split": "train"
} |
math | [
{
"content": "We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\\le i\\le k}$ with $$1=d_1<d_2<\\dots <d_k=n\\quad (k\\ge 6).$$\nFind all positive integers $n$ such that $$n=d_5^2+d_6^2.$$ Please output the final answer ... | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "36658",
"split": "train"
} |
math | [
{
"content": "Let set $I=\\{1,2,3,4,5,6\\}$, and sets $A, B \\subseteq I$. If set $A$ contains 3 elements, set $B$ contains at least 2 elements, and all elements in $B$ are not less than the largest element in $A$, calculate the number of pairs of sets $A$ and $B$ that satisfy these conditions. Please output th... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "36659",
"split": "train"
} |
math | [
{
"content": "If $\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{c}$, and $\\mathbf{d}$ are unit vectors, find the largest possible value of\n\\[\n\\|\\mathbf{a} - \\mathbf{b}\\|^2 + \\|\\mathbf{a} - \\mathbf{c}\\|^2 + \\|\\mathbf{a} - \\mathbf{d}\\|^2 + \\|\\mathbf{b} - \\mathbf{c}\\|^2 + \\|\\mathbf{b} - \\mathbf{d}\... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36660",
"split": "train"
} |
math | [
{
"content": "Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$. Pleas... | {
"ground_truth": "n^2 + 2",
"style": "rule"
} | {
"index": "36661",
"split": "train"
} |
math | [
{
"content": "What is the least prime factor of $7^4 - 7^3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "36662",
"split": "train"
} |
math | [
{
"content": "When $\\sqrt[4]{2^7\\cdot3^3}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "218",
"style": "rule"
} | {
"index": "36663",
"split": "train"
} |
math | [
{
"content": "Given that the angles A, B, C of triangle ABC correspond to the sides a, b, c respectively, and vectors $\\overrightarrow {m}$ = (a, $- \\sqrt {3}b$) and $\\overrightarrow {n}$ = (cosA, sinB), and $\\overrightarrow {m}$ is parallel to $\\overrightarrow {n}$.\n(1) Find angle A.\n(2) If $a = \\sqrt{... | {
"ground_truth": "\\frac{5\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "36664",
"split": "train"
} |
math | [
{
"content": "Let positive integers \\( a, b, c, d \\) satisfy \\( a > b > c > d \\) and \\( a+b+c+d=2004 \\), \\( a^2 - b^2 + c^2 - d^2 = 2004 \\). Find the minimum value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "503",
"style": "rule"
} | {
"index": "36665",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \\sqrt{5}c$ and $\\cos C = \\frac{3}{5}$. \n$(Ⅰ)$ Find the value of $\\sin A$. \n$(Ⅱ)$ If $b = 11$, find the area of $\\triangle ABC$. Please output the final a... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "36666",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{P}_{1}, \\mathcal{P}_{2}, \\mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36667",
"split": "train"
} |
math | [
{
"content": "A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the mornin... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36668",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "36669",
"split": "train"
} |
math | [
{
"content": "A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers\nand $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers\n$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$? Please output the final ... | {
"ground_truth": "1/3 < c \\leq 1",
"style": "rule"
} | {
"index": "36670",
"split": "train"
} |
math | [
{
"content": "Five points, no three of which are collinear, are given. Calculate the least possible value of the number of convex polygons whose some corners are formed by these five points. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36671",
"split": "train"
} |
math | [
{
"content": "The value of $1.000 + 0.101 + 0.011 + 0.001$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.113",
"style": "rule"
} | {
"index": "36672",
"split": "train"
} |
math | [
{
"content": "Solve the following cryptarithm ensuring that identical letters correspond to identical digits:\n\n$$\n\\begin{array}{r}\n\\text { К O Ш К A } \\\\\n+ \\text { К O Ш К A } \\\\\n\\text { К O Ш К A } \\\\\n\\hline \\text { С О Б А К А }\n\\end{array}\n$$ Please output the final answer within \\boxe... | {
"ground_truth": "50350",
"style": "rule"
} | {
"index": "36673",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}, AB=10, \\angle{A}=30^\\circ$ , and $\\angle{C=45^\\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\\perp{BC}$. T... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "36674",
"split": "train"
} |
math | [
{
"content": "An ellipse has a major axis of length 12 and a minor axis of 10. Using one focus as a center, an external circle is tangent to the ellipse. Find the radius of the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{11}",
"style": "rule"
} | {
"index": "36675",
"split": "train"
} |
math | [
{
"content": "Square $EFGH$ has sides of length 4. A point $P$ on $EH$ is such that line segments $FP$ and $GP$ divide the square’s area into four equal parts. Find the length of segment $FP$.\nA) $2\\sqrt{3}$\nB) $3$\nC) $2\\sqrt{5}$\nD) $4$\nE) $2\\sqrt{7}$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "36676",
"split": "train"
} |
math | [
{
"content": "In the convex pentagon $ABCDE$, $\\angle A = \\angle B = 120^{\\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7\\sqrt{3}",
"style": "rule"
} | {
"index": "36677",
"split": "train"
} |
math | [
{
"content": "Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:\n\n\n- The sum of the fractions is equal to $2$ .\n- The sum of the numerators of the fractions is equal to $1000$ .\n\nIn how many ways can Pedro do this? Please output the final answer within... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "36678",
"split": "train"
} |
math | [
{
"content": "The skeletal structure of circumcircumcircumcoronene, a hydrocarbon with the chemical formula $\\mathrm{C}_{150} \\mathrm{H}_{30}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hyd... | {
"ground_truth": "267227532",
"style": "rule"
} | {
"index": "36679",
"split": "train"
} |
math | [
{
"content": "Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43 \\frac{7}{11}",
"style": "rule"
} | {
"index": "36680",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 500 are congruent to 7 (mod 13)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "36681",
"split": "train"
} |
math | [
{
"content": "A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyramid. Please output the... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "36682",
"split": "train"
} |
math | [
{
"content": "In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a d... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "36683",
"split": "train"
} |
math | [
{
"content": "Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "36684",
"split": "train"
} |
math | [
{
"content": "Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "36685",
"split": "train"
} |
math | [
{
"content": "Given that $|\\vec{a}|=1$, $|\\vec{b}|=2$, and $(\\vec{a}+\\vec{b})\\cdot \\vec{b}=3$, find the angle between $\\vec{b}$ and $\\vec{a}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "36686",
"split": "train"
} |
math | [
{
"content": "How many distinct digits can appear as the units digit of an integral perfect-square number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "36687",
"split": "train"
} |
math | [
{
"content": "Adults made up $\\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. Please output the final answer ... | {
"ground_truth": "154",
"style": "rule"
} | {
"index": "36688",
"split": "train"
} |
math | [
{
"content": "The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36689",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=-3x^2+6x$, let ${S_n}$ be the sum of the first $n$ terms of the sequence ${{a_n}}$. The points $(n, {S_n})$ (where $n \\in \\mathbb{N}^*$) lie on the curve $y=f(x)$.\n\n(I) Find the general formula for the terms of the sequence ${{a_n}}$.\n\n(II) If ${b_n}={(\\frac{1}{2})^... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36690",
"split": "train"
} |
math | [
{
"content": "Express $361_9 + 4C5_{13}$ as a base 10 integer, where $C$ denotes the digit whose value is 12 in base 13. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1135",
"style": "rule"
} | {
"index": "36691",
"split": "train"
} |
math | [
{
"content": "Find $w$, such that $5^65^w=25$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "36692",
"split": "train"
} |
math | [
{
"content": "Find $x$ such that $\\log_x 49 = \\log_2 32$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7^{2/5}",
"style": "rule"
} | {
"index": "36693",
"split": "train"
} |
math | [
{
"content": "A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square? Please output the final answer with... | {
"ground_truth": "4 - 2\\sqrt{3}",
"style": "rule"
} | {
"index": "36694",
"split": "train"
} |
math | [
{
"content": "Given the function $f\\left(x\\right)=x^{3}+ax^{2}+bx-4$ and the tangent line equation $y=x-4$ at point $P\\left(2,f\\left(2\\right)\\right)$.<br/>$(1)$ Find the values of $a$ and $b$;<br/>$(2)$ Find the extreme values of $f\\left(x\\right)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{58}{27}",
"style": "rule"
} | {
"index": "36695",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ with pairwise angles of $60^\\circ$, and $|\\overrightarrow{a}|=|\\overrightarrow{b}|=|\\overrightarrow{c}|=1$, find $|\\overrightarrow{a}+\\overrightarrow{b}-\\overrightarrow{c}|$. Please output the final answer wit... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "36696",
"split": "train"
} |
math | [
{
"content": "A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "36697",
"split": "train"
} |
math | [
{
"content": "Given $\\sin\\theta + \\cos\\theta = \\frac{3}{4}$, where $\\theta$ is an angle of a triangle, find the value of $\\sin\\theta - \\cos\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{23}}{4}",
"style": "rule"
} | {
"index": "36698",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=a^{2}\\sin 2x+(a-2)\\cos 2x$, if its graph is symmetric about the line $x=-\\frac{\\pi}{8}$, determine the maximum value of $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "36699",
"split": "train"
} |
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