data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "$A,B,C$ are points in the plane such that $\\angle ABC=90^\\circ$ . Circles with diameters $BA$ and $BC$ meet at $D$ . If $BA=20$ and $BC=21$ , then the length of segment $BD$ can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Wh... | {
"ground_truth": "449",
"style": "rule"
} | {
"index": "39600",
"split": "train"
} |
math | [
{
"content": "Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "39601",
"split": "train"
} |
math | [
{
"content": "Equilateral triangle $\\triangle ABC$ is inscribed in circle $\\omega$ with radius $18.$ Circle $\\omega_A$ is tangent to sides $\\overline{AB}$ and $\\overline{AC}$ and is internally tangent to $\\omega.$ Circles $\\omega_B$ and $\\omega_C$ are defined analogously. Circles $\\omega_A,$ $\\omega_B... | {
"ground_truth": "378",
"style": "rule"
} | {
"index": "39602",
"split": "train"
} |
math | [
{
"content": "Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "39603",
"split": "train"
} |
math | [
{
"content": "Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \\ge 4. $$ What is the maximum possible size for the set $S?$ $$ \\mathrm a. ~ 12\\qquad \\mathrm b.~14\\qquad \\mathrm c. ~16 \\qquad \\math... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "39604",
"split": "train"
} |
math | [
{
"content": "Calculate the surface integrals of the first kind:\na) \\(\\iint_{\\sigma}|x| dS\\), where \\(\\sigma\\) is defined by \\(x^2 + y^2 + z^2 = 1\\), \\(z \\geqslant 0\\).\n\nb) \\(\\iint_{\\sigma} (x^2 + y^2) dS\\), where \\(\\sigma\\) is defined by \\(x^2 + y^2 = 2z\\), \\(z = 1\\).\n\nc) \\(\\iint_... | {
"ground_truth": "3\\sqrt{2} \\pi",
"style": "rule"
} | {
"index": "39605",
"split": "train"
} |
math | [
{
"content": "What is the area of the smallest square that can contain a circle of radius 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "39606",
"split": "train"
} |
math | [
{
"content": "A club has 20 members and needs to choose 2 members to be co-presidents and 1 member to be a treasurer. In how many ways can the club select its co-presidents and treasurer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3420",
"style": "rule"
} | {
"index": "39607",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\dfrac {x^{2}}{a^{2}}- \\dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at points $M$ and $N$.... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "39608",
"split": "train"
} |
math | [
{
"content": "What is the sum of the largest and smallest prime factors of 990? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39609",
"split": "train"
} |
math | [
{
"content": "All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-88",
"style": "rule"
} | {
"index": "39610",
"split": "train"
} |
math | [
{
"content": "Find all values of $r$ such that $\\lfloor r \\rfloor + r = 12.2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "r=6.2",
"style": "rule"
} | {
"index": "39611",
"split": "train"
} |
math | [
{
"content": "How many subsets of two elements can be removed from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$ so that the mean (average) of the remaining numbers is 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "39612",
"split": "train"
} |
math | [
{
"content": "Let $p(x)=x^4-4x^3+2x^2+ax+b$ . Suppose that for every root $\\lambda$ of $p$ , $\\frac{1}{\\lambda}$ is also a root of $p$ . Then $a+b=$ [list=1]\n[*] -3\n[*] -6\n[*] -4\n[*] -8\n[/list] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "39613",
"split": "train"
} |
math | [
{
"content": "A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\\frac{AP}{AB} =\\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circl... | {
"ground_truth": "205/1681",
"style": "rule"
} | {
"index": "39614",
"split": "train"
} |
math | [
{
"content": "If $\\frac{1}{9}+\\frac{1}{18}=\\frac{1}{\\square}$, what is the number that replaces the $\\square$ to make the equation true? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "39615",
"split": "train"
} |
math | [
{
"content": "There are two distinguishable flagpoles, and there are $17$ flags, of which $11$ are identical red flags, and $6$ are identical white flags. Determine the number of distinguishable arrangements using all of the flags such that each flagpole has at least one flag and no two white flags on either po... | {
"ground_truth": "164",
"style": "rule"
} | {
"index": "39616",
"split": "train"
} |
math | [
{
"content": "Without using a calculator, find the largest prime factor of $15^4+2\\times15^2+1-14^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "211",
"style": "rule"
} | {
"index": "39617",
"split": "train"
} |
math | [
{
"content": "Let \\(a_{1}, a_{2}, \\cdots, a_{n}\\) be an increasing sequence of positive integers. For a positive integer \\(m\\), define\n\\[b_{m}=\\min \\left\\{n \\mid a_{n} \\geq m\\right\\} (m=1,2, \\cdots),\\]\nthat is, \\(b_{m}\\) is the smallest index \\(n\\) such that \\(a_{n} \\geq m\\). Given \\(a_... | {
"ground_truth": "42399",
"style": "rule"
} | {
"index": "39618",
"split": "train"
} |
math | [
{
"content": "Let function $G(n)$ denote the number of solutions to the equation $\\cos x = \\sin nx$ on the interval $[0, 2\\pi]$. For each integer $n$ greater than 2, what is the sum $\\sum_{n=3}^{100} G(n)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10094",
"style": "rule"
} | {
"index": "39619",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39620",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "39621",
"split": "train"
} |
math | [
{
"content": "The slope angle of the tangent line to the curve y=\\frac{1}{3}x^{3} at x=1 is what value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "39622",
"split": "train"
} |
math | [
{
"content": "A basketball player made the following number of successful free throws in 10 successive games: 8, 17, 15, 22, 14, 12, 24, 10, 20, and 16. He attempted 10, 20, 18, 25, 16, 15, 27, 12, 22, and 19 free throws in those respective games. Calculate both the median number of successful free throws and t... | {
"ground_truth": "90.91\\%",
"style": "rule"
} | {
"index": "39623",
"split": "train"
} |
math | [
{
"content": "Compute the sum \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "39624",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 5.$ Find the maximum value of\n\\[\\sqrt{2x + 1} + \\sqrt{2y + 1} + \\sqrt{2z + 1}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{39}",
"style": "rule"
} | {
"index": "39625",
"split": "train"
} |
math | [
{
"content": "Given a plane $\\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:\n(1) If $m \\parallel \\alpha$ and $n \\subseteq \\alpha$, then $m \\parallel n$.\n(2) If $m \\parallel \\alpha$ and $n \\parallel \\alpha$, then $m \\parallel n$.\n(3) If $m \\para... | {
"ground_truth": "(4)",
"style": "rule"
} | {
"index": "39626",
"split": "train"
} |
math | [
{
"content": "Given a triangular pyramid where two of the three lateral faces are isosceles right triangles and the third face is an equilateral triangle with a side length of 1, calculate the volume of this triangular pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{12}",
"style": "rule"
} | {
"index": "39627",
"split": "train"
} |
math | [
{
"content": "Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "280",
"style": "rule"
} | {
"index": "39628",
"split": "train"
} |
math | [
{
"content": "The bases \\( AB \\) and \\( CD \\) of the trapezoid \\( ABCD \\) are equal to 65 and 31 respectively, and its lateral sides are mutually perpendicular. Find the dot product of the vectors \\( \\overrightarrow{AC} \\) and \\( \\overrightarrow{BD} \\). Please output the final answer within \\boxed{... | {
"ground_truth": "-2015",
"style": "rule"
} | {
"index": "39629",
"split": "train"
} |
math | [
{
"content": "A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p}}{n}$, where $m$, $... | {
"ground_truth": "113",
"style": "rule"
} | {
"index": "39630",
"split": "train"
} |
math | [
{
"content": "Marsha now has two numbers, $a$ and $b$. When she divides $a$ by $60$ she gets a remainder of $58$. When she divides $b$ by $90$ she gets a remainder of $84$. What remainder does she get when she divides $a+b$ by $30$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "39631",
"split": "train"
} |
math | [
{
"content": "What is the sum of all values of $z$ such that $z^2=12z-7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39632",
"split": "train"
} |
math | [
{
"content": "Let $$Q(x) = \\left(\\frac{x^{24} - 1}{x - 1}\\right)^2 - x^{23}$$ and this polynomial has complex zeros in the form $z_k = r_k[\\cos(2\\pi\\alpha_k) + i\\sin(2\\pi\\alpha_k)]$, for $k = 1, 2, ..., 46$, where $0 < \\alpha_1 \\le \\alpha_2 \\le \\ldots \\le \\alpha_{46} < 1$ and $r_k > 0$. Find $\\... | {
"ground_truth": "\\frac{121}{575}",
"style": "rule"
} | {
"index": "39633",
"split": "train"
} |
math | [
{
"content": "Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. H... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "39634",
"split": "train"
} |
math | [
{
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follo... | {
"ground_truth": "99\\%",
"style": "rule"
} | {
"index": "39635",
"split": "train"
} |
math | [
{
"content": "Xiao Hua plays a certain game where each round can be played several times freely. Each score in a round is one of the numbers $8$, $a$ (a natural number), or $0$. The total score for a round is the sum of all individual scores in that round. Xiao Hua has achieved the following total scores in som... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39636",
"split": "train"
} |
math | [
{
"content": "Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is ... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "39637",
"split": "train"
} |
math | [
{
"content": "If the integer solutions to the system of inequalities\n\\[\n\\begin{cases}\n9x - a \\geq 0, \\\\\n8x - b < 0\n\\end{cases}\n\\]\nare only 1, 2, and 3, how many ordered pairs \\((a, b)\\) of integers satisfy this system? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "39638",
"split": "train"
} |
math | [
{
"content": "If \\( a, b, c \\) are real numbers such that \\( |a-b|=1 \\), \\( |b-c|=1 \\), \\( |c-a|=2 \\) and \\( abc=60 \\), calculate the value of \\( \\frac{a}{bc} + \\frac{b}{ca} + \\frac{c}{ab} - \\frac{1}{a} - \\frac{1}{b} - \\frac{1}{c} \\). Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "39639",
"split": "train"
} |
math | [
{
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "39640",
"split": "train"
} |
math | [
{
"content": "A wire has a length of 6 meters and has 5 nodes that divide the wire into 6 equal parts. If a node is randomly selected to cut the wire, what is the probability that both resulting pieces will have lengths not less than 2 meters? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "39641",
"split": "train"
} |
math | [
{
"content": "Given that Jessica uses 150 grams of lemon juice and 100 grams of sugar, and there are 30 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water contains no calories, compute the total number of calories in 300 grams of her lemonade. Please output the final answer w... | {
"ground_truth": "152.1",
"style": "rule"
} | {
"index": "39642",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sqrt{3}\\sin x\\cos x-\\cos (\\pi +2x)$.\n\n(1) Find the interval(s) where $f(x)$ is monotonically increasing.\n\n(2) In $\\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(C)=1,c=\\sqrt{3},a+b=2\\sqrt{3}$, find the ar... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "39643",
"split": "train"
} |
math | [
{
"content": "In the complex plane, let the vertices \\( A \\) and \\( B \\) of triangle \\( \\triangle AOB \\) correspond to the complex numbers \\( \\alpha \\) and \\( \\beta \\), respectively, and satisfy the conditions: \\( \\beta = (1 + i)\\alpha \\) and \\( |\\alpha - 2| = 1 \\). \\( O \\) is the origin. ... | {
"ground_truth": "9/2",
"style": "rule"
} | {
"index": "39644",
"split": "train"
} |
math | [
{
"content": "Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $\\angle BAC=90^{\\circ}$, the area of the side face $BCC_{1}B_{1}$ is $16$. Find the minimum value of the radius of the circumscribed sphere of the right triangular prism $ABC-A_{1}B_{1}C_{1}$. Please output the final answer within \\box... | {
"ground_truth": "2 \\sqrt {2}",
"style": "rule"
} | {
"index": "39645",
"split": "train"
} |
math | [
{
"content": "In an isosceles triangle \\(ABC\\), the base \\(AC\\) is equal to 1, and the angle \\(\\angle ABC\\) is \\(2 \\arctan \\frac{1}{2}\\). Point \\(D\\) lies on the side \\(BC\\) such that the area of triangle \\(ABC\\) is four times the area of triangle \\(ADC\\). Find the distance from point \\(D\\)... | {
"ground_truth": "\\frac{\\sqrt{265}}{32}",
"style": "rule"
} | {
"index": "39646",
"split": "train"
} |
math | [
{
"content": "Let $x$ be the least real number greater than $1$ such that $\\sin(x) = \\sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39647",
"split": "train"
} |
math | [
{
"content": "The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39648",
"split": "train"
} |
math | [
{
"content": "In circle $\\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_{1}$ and $M_{2}$ respectively, such that $P M_{1}=15$ and $P M_{2}=20$. Line $M_{1} M_{2}$ intersects $\\omega$ at points $A$ and $B$, with $M_{1}$ between $A$ and $M_{2}$. Compute the largest ... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39649",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=e^{x}-mx^{3}$ ($m$ is a nonzero constant). \n$(1)$ If the function $f(x)$ is increasing on $(0,+\\infty)$, find the range of real numbers for $m$. \n$(2)$ If $f_{n+1}(x)$ ($n\\in \\mathbb{N}$) represents the derivative of $f_{n}(x)$, where $f_{0}(x)=f(x)$, and when $m=1$... | {
"ground_truth": "n = 8",
"style": "rule"
} | {
"index": "39650",
"split": "train"
} |
math | [
{
"content": "Let $h(x) = \\sqrt[3]{\\frac{2x+5}{5}}$. For what value of $x$ will $h(3x) = 3(h(x))$? Express your answer in simplest form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{65}{24}",
"style": "rule"
} | {
"index": "39651",
"split": "train"
} |
math | [
{
"content": "Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \\(10 \\, \\text{m/s}\\) and Betty runs at \\(15 \\, \\text{m/s}\\). Betty gives Alice a 40 meter head start before they both start running. After 15 minutes,... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39652",
"split": "train"
} |
math | [
{
"content": "Five friends did gardening for their local community and earned $15, $22, $28, $35, and $50 respectively. They decide to share their total earnings equally. How much money must the friend who earned $50 contribute to the pool?\nA) $10 \nB) $15 \nC) $20 \nD) $25 \nE) $35 Please output the f... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "39653",
"split": "train"
} |
math | [
{
"content": "Find the arithmetic mean of the reciprocals of the first three prime numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{31}{90}",
"style": "rule"
} | {
"index": "39654",
"split": "train"
} |
math | [
{
"content": "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}\\rfloor = \\cdots =... | {
"ground_truth": "553",
"style": "rule"
} | {
"index": "39655",
"split": "train"
} |
math | [
{
"content": "Given Aniyah has a $5 \\times 7$ index card and if she shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card in square inches if she shortens the length of the other side by $2$ inches. Please output the final ... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "39656",
"split": "train"
} |
math | [
{
"content": "A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments measuring 6 cm and 7 cm. Calculate the area of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "39657",
"split": "train"
} |
math | [
{
"content": "Given that $8!=40320$, what is the value of $8!\\div3!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6720",
"style": "rule"
} | {
"index": "39658",
"split": "train"
} |
math | [
{
"content": "In the trapezoid \\(ABCD\\), the base \\(AB\\) is three times longer than the base \\(CD\\). On the base \\(CD\\), point \\(M\\) is taken such that \\(MC = 2MD\\). \\(N\\) is the intersection point of lines \\(BM\\) and \\(AC\\). Find the ratio of the area of triangle \\(MNC\\) to the area of the ... | {
"ground_truth": "\\frac{1}{33}",
"style": "rule"
} | {
"index": "39659",
"split": "train"
} |
math | [
{
"content": "Find the sum: $1+2+3+4+\\dots +48+49$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1225",
"style": "rule"
} | {
"index": "39660",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "39661",
"split": "train"
} |
math | [
{
"content": "Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\\mathbf{v}=(v_1,\\ldots,v_{2022})$ and $\\mathbf{w}=(w_1,\\ldots,w_{2022})$ that she has already written, and apply one of the following operations to... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "39662",
"split": "train"
} |
math | [
{
"content": "Find the focal length of the hyperbola that shares the same asymptotes with the hyperbola $\\frac{x^{2}}{9} - \\frac{y^{2}}{16} = 1$ and passes through the point $A(-3, 3\\sqrt{2})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "39663",
"split": "train"
} |
math | [
{
"content": "A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "39664",
"split": "train"
} |
math | [
{
"content": "A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "39665",
"split": "train"
} |
math | [
{
"content": "A factory must filter its emissions before discharging them. The relationship between the concentration of pollutants $p$ (in milligrams per liter) and the filtration time $t$ (in hours) during the filtration process is given by the equation $p(t) = p_0e^{-kt}$. Here, $e$ is the base of the natura... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39666",
"split": "train"
} |
math | [
{
"content": "Given sets A = {-2, 1, 2} and B = {-1, 1, 3}, calculate the probability that a line represented by the equation ax - y + b = 0 will pass through the fourth quadrant. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "39667",
"split": "train"
} |
math | [
{
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "39668",
"split": "train"
} |
math | [
{
"content": "How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39669",
"split": "train"
} |
math | [
{
"content": "If there are exactly $3$ integer solutions for the inequality system about $x$: $\\left\\{\\begin{array}{c}6x-5≥m\\\\ \\frac{x}{2}-\\frac{x-1}{3}＜1\\end{array}\\right.$, and the solution to the equation about $y$: $\\frac{y-2}{3}=\\frac{m-2}{3}+1$ is a non-negative number, find the sum of all inte... | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "39670",
"split": "train"
} |
math | [
{
"content": "Point A is the intersection of the unit circle and the positive half of the x-axis, and point B is in the second quadrant. Let θ be the angle formed by the rays OA and OB. Given sin(θ) = 4/5, calculate the value of sin(π + θ) + 2sin(π/2 - θ) divided by 2tan(π - θ). Please output the final answer w... | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "39671",
"split": "train"
} |
math | [
{
"content": "How many ways are there to label the faces of a regular octahedron with the integers 18, using each exactly once, so that any two faces that share an edge have numbers that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflectio... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39672",
"split": "train"
} |
math | [
{
"content": "For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three ... | {
"ground_truth": "577",
"style": "rule"
} | {
"index": "39673",
"split": "train"
} |
math | [
{
"content": "Find the sum of the roots of $\\tan^2 x - 8\\tan x + \\sqrt{2} = 0$ that are between $x=0$ and $x=2\\pi$ radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "39674",
"split": "train"
} |
math | [
{
"content": "Evaluate\n\\[\\begin{vmatrix} \\cos \\alpha \\cos \\beta & \\cos \\alpha \\sin \\beta & -\\sin \\alpha \\\\ -\\sin \\beta & \\cos \\beta & 0 \\\\ \\sin \\alpha \\cos \\beta & \\sin \\alpha \\sin \\beta & \\cos \\alpha \\end{vmatrix}.\\] Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39675",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), find the radius of the circumscribed circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{85}{8}",
"style": "rule"
} | {
"index": "39676",
"split": "train"
} |
math | [
{
"content": "The denominator of a geometric progression \\( b_{n} \\) is \\( q \\), and for some natural \\( n \\geq 2 \\),\n\n$$\n\\log_{4} b_{2}+\\log_{4} b_{3}+\\ldots+\\log_{4} b_{n}=4 \\cdot \\log_{4} b_{1}\n$$\n\nFind the smallest possible value of \\( \\log_{q} b_{1}^{2} \\), given that it is an integer... | {
"ground_truth": "-30",
"style": "rule"
} | {
"index": "39677",
"split": "train"
} |
math | [
{
"content": "Compute the value of $1^{25}+2^{24}+3^{23}+\\ldots+24^{2}+25^{1}$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\\left\\lfloor 25 \\mathrm{~min}\\left(\\left(\\frac{A}{C}\\right)^{2},\\left(\\frac{C}{A}\\right)^{2}\\right)\\right\\rfloor$. Please ou... | {
"ground_truth": "66071772829247409",
"style": "rule"
} | {
"index": "39678",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)$ is defined on $\\mathbb{R}$ and is not identically zero, and for any real numbers $x$, $y$, it satisfies: $f(2)=2$, $f(xy)=xf(y)+yf(x)$, $a_{n}= \\dfrac {f(2^{n})}{2^{n}}(n\\in\\mathbb{N}^{*})$, $b_{n}= \\dfrac {f(2^{n})}{n}(n\\in\\mathbb{N}^{*})$, consider the follo... | {
"ground_truth": "(2)(3)(4)",
"style": "rule"
} | {
"index": "39679",
"split": "train"
} |
math | [
{
"content": "Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the value of $a$ when $a=$____, such that the minimum value of the algebraic expression $\\frac{{2{a^2}+1}}{{ab}}-2$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "39680",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a}, \\mathbf{b},$ and $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 2,$ $\\|\\mathbf{b}\\| = 3,$ and $\\|\\mathbf{c}\\| = 6,$ and\n\\[\\mathbf{a} + 2\\mathbf{b} + \\mathbf{c} = \\mathbf{0}.\\]\nCompute $\\mathbf{a} \\cdot \\mathbf{b} + \\mathbf{a} \\cdot \\mathbf{c} + \\mat... | {
"ground_truth": "-19",
"style": "rule"
} | {
"index": "39681",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to $\\angle A$, $\\angle B$, $\\angle C$ respectively. It is known that $a$, $b$, $c$ are in geometric progression, and $a^{2}-c^{2}=ac-bc$,\n(1) Find the measure of $\\angle A$;\n(2) Find the value of $\\frac{b\\sin B}{c}$. ... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "39682",
"split": "train"
} |
math | [
{
"content": "When Lisa squares her favorite $2$ -digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$ -digit number. What is Lisa's favorite $2$ -digit number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "39683",
"split": "train"
} |
math | [
{
"content": "The quadratic $8x^2 - 48x - 320$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-387",
"style": "rule"
} | {
"index": "39684",
"split": "train"
} |
math | [
{
"content": "Lulu has a quadratic of the form $x^2+bx+44$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Lulu is able to rewrite this quadratic in the form $(x+m)^2+8$. What is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39685",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\\overline{QR}$. What is the sum of the radii of the circles inscribed in $\\triangle PQS$ and $\\triangle PRS$?\nA) $\\frac{14\\sqrt{5}}{13}$\nB) $\\frac{14\\sqrt{5}}{6.5 + \\sqrt{29}}$\nC) $\\frac{12\\sqrt{4}}{8.5}$\... | {
"ground_truth": "\\frac{14\\sqrt{5}}{6.5 + \\sqrt{29}}",
"style": "rule"
} | {
"index": "39686",
"split": "train"
} |
math | [
{
"content": "Point \\( O \\) is located on side \\( AC \\) of triangle \\( ABC \\) such that \\( CO : CA = 2 : 3 \\). When this triangle is rotated by a certain angle around point \\( O \\), vertex \\( B \\) moves to vertex \\( C \\), and vertex \\( A \\) moves to point \\( D \\), which lies on side \\( AB \\)... | {
"ground_truth": "1/6",
"style": "rule"
} | {
"index": "39687",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a given positive integer. Solve the system \n\\[x_1 + x_2^2 + x_3^3 + \\cdots + x_n^n = n,\\]\n\\[x_1 + 2x_2 + 3x_3 + \\cdots + nx_n = \\frac{n(n+1)}{2}\\]\nin the set of nonnegative real numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(x_1, x_2, \\ldots, x_n) = (1, 1, \\ldots, 1)",
"style": "rule"
} | {
"index": "39688",
"split": "train"
} |
math | [
{
"content": "Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written n... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "39689",
"split": "train"
} |
math | [
{
"content": "Let the natural number $N$ be a perfect square, which has at least three digits, its last two digits are not $00$, and after removing these two digits, the remaining number is still a perfect square. Then, the maximum value of $N$ is ____. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "1681",
"style": "rule"
} | {
"index": "39690",
"split": "train"
} |
math | [
{
"content": "Given that $\\theta$ is an angle in the fourth quadrant, and $\\sin\\theta + \\cos\\theta = \\frac{1}{5}$, find: \n(1) $\\sin\\theta - \\cos\\theta$; \n(2) $\\tan\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "39691",
"split": "train"
} |
math | [
{
"content": "In rectangle \\(ABCD\\), \\(A B: AD = 1: 2\\). Point \\(M\\) is the midpoint of \\(AB\\), and point \\(K\\) lies on \\(AD\\), dividing it in the ratio \\(3:1\\) starting from point \\(A\\). Find the sum of \\(\\angle CAD\\) and \\(\\angle AKM\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "39692",
"split": "train"
} |
math | [
{
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\\le x\\le 5$ and $1\\le y\\le 5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2158",
"style": "rule"
} | {
"index": "39693",
"split": "train"
} |
math | [
{
"content": "Given \\( x, y, z \\in \\mathbf{R} \\) such that \\( x^2 + y^2 + xy = 1 \\), \\( y^2 + z^2 + yz = 2 \\), \\( x^2 + z^2 + xz = 3 \\), find \\( x + y + z \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3 + \\sqrt{6}}",
"style": "rule"
} | {
"index": "39694",
"split": "train"
} |
math | [
{
"content": "Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "39695",
"split": "train"
} |
math | [
{
"content": "Increasing the radius of a cylinder by $4$ units results in an increase in volume by $z$ cubic units. Increasing the height of the cylinder by $10$ units also results in an increase in volume by the same $z$ cubic units. If the original radius is $3$ units, what is the original height of the cylin... | {
"ground_truth": "2.25",
"style": "rule"
} | {
"index": "39696",
"split": "train"
} |
math | [
{
"content": "Find the largest possible subset of {1, 2, ... , 15} such that the product of any three distinct elements of the subset is not a square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39697",
"split": "train"
} |
math | [
{
"content": "Children get in for half the price of adults. The price for $8$ adult tickets and $7$ child tickets is $42$. If a group buys more than $10$ tickets, they get an additional $10\\%$ discount on the total price. Calculate the cost of $10$ adult tickets and $8$ child tickets. Please output the final a... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "39698",
"split": "train"
} |
math | [
{
"content": "Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \\ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness o... | {
"ground_truth": "1121",
"style": "rule"
} | {
"index": "39699",
"split": "train"
} |
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