data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given a sequence of natural numbers $\\left\\{x_{n}\\right\\}$ defined by:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\quad n=1,2,3,\\cdots\n$$\nIf an element of the sequence is 1000, what is the minimum possible value of $a+b$? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8000",
"split": "train"
} |
math | [
{
"content": "A store purchased a batch of New Year cards at a price of 21 cents each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how many cents did the store earn in total? Please output the final answer within ... | {
"ground_truth": "470",
"style": "rule"
} | {
"index": "8001",
"split": "train"
} |
math | [
{
"content": "Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "8002",
"split": "train"
} |
math | [
{
"content": "A number in the set $\\{50, 51, 52, 53, ... , 999\\}$ is randomly selected. What is the probability that it is a two-digit number? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{19}",
"style": "rule"
} | {
"index": "8003",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 6.\\]Find the minimum value of $x^3 y^2 z.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{108}",
"style": "rule"
} | {
"index": "8004",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at $O$ . If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$ , compute the minimum possible value of the area of $ABCD$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "8005",
"split": "train"
} |
math | [
{
"content": "Let $X$ be the number of sequences of integers $a_{1}, a_{2}, \\ldots, a_{2047}$ that satisfy all of the following properties: - Each $a_{i}$ is either 0 or a power of 2 . - $a_{i}=a_{2 i}+a_{2 i+1}$ for $1 \\leq i \\leq 1023$ - $a_{1}=1024$. Find the remainder when $X$ is divided by 100 . Please ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "8006",
"split": "train"
} |
math | [
{
"content": "Given $f\\left(x\\right)=e^{x}-ax+\\frac{1}{2}{x}^{2}$, where $a \\gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\\left(x\\right)$ at the point $\\left(0,f\\left(0\\right)\\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the function $f\\left(x\\rig... | {
"ground_truth": "1 + \\frac{1}{e}",
"style": "rule"
} | {
"index": "8007",
"split": "train"
} |
math | [
{
"content": "Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\\omega_{1}, \\omega_{2}, \\ldots$ of circles such that $\\omega_{1}$ is the incircle of $T$ and $\\omega_{i+1}$ is tangent to $\\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the ratio of the ra... | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "8008",
"split": "train"
} |
math | [
{
"content": "Given there are $50$ students in a class, a systematic sampling method is used to select $10$ students from these $50$. The $50$ students are randomly numbered from $1$ to $50$ and then evenly divided into $10$ groups in numerical order, with each group containing $5$ students. If the number drawn... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "8009",
"split": "train"
} |
math | [
{
"content": "An integer greater than 9 and less than 100 is randomly chosen. What is the probability that its digits are different? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{10}",
"style": "rule"
} | {
"index": "8010",
"split": "train"
} |
math | [
{
"content": "On the game show $\\text{\\emph{Wheel of Fortune II}}$, you observe a spinner with the labels [\"Bankrupt\", \"$\\$700$\", \"$\\$900$\", \"$\\$200$\", \"$\\$3000$\", \"$\\$800$\"]. Given that each region has equal area, determine the probability of earning exactly $\\$2400$ in your first three spi... | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "8011",
"split": "train"
} |
math | [
{
"content": "While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $\\$14$ each, $4$-pound rocks worth $\\$11$ each, and $1$-pound rocks worth $\\$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he ca... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "8012",
"split": "train"
} |
math | [
{
"content": "Natural numbers \\( m \\) and \\( n \\) are such that \\( m > n \\), \\( m \\) is not divisible by \\( n \\), and \\( m \\) has the same remainder when divided by \\( n \\) as \\( m + n \\) has when divided by \\( m - n \\).\n\nFind the ratio \\( m : n \\). Please output the final answer within \\... | {
"ground_truth": "5/2",
"style": "rule"
} | {
"index": "8013",
"split": "train"
} |
math | [
{
"content": "Given that Bob was instructed to subtract 5 from a certain number and then divide the result by 7, but instead subtracted 7 and then divided by 5, yielding an answer of 47, determine what his answer would have been had he worked the problem correctly. Please output the final answer within \\boxed{... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "8014",
"split": "train"
} |
math | [
{
"content": "In a circle centered at $O$, points $A$ and $C$ lie on the circle, each of $\\overline{BA}$ and $\\overline{BC}$ are tangent to the circle. Triangle $ABC$ is isosceles with $AB = BC$ and $\\angle ABC = 100^\\circ$. The circle intersects $\\overline{BO}$ at $D$. Determine $\\frac{BD}{BO}$.\nA) $\\f... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8015",
"split": "train"
} |
math | [
{
"content": "A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated two independent times... | {
"ground_truth": "$\\frac{7}{16}$",
"style": "rule"
} | {
"index": "8016",
"split": "train"
} |
math | [
{
"content": "The standard enthalpy of formation (ΔH_f°) of a substance is equal to the heat effect of the formation reaction of 1 mole of the substance from simple substances in their standard states (at 1 atm pressure and a given temperature). Therefore, it is necessary to find the heat effect of the reaction... | {
"ground_truth": "-82.9",
"style": "rule"
} | {
"index": "8017",
"split": "train"
} |
math | [
{
"content": "Given the value \\(\\left(\\frac{11}{12}\\right)^{2}\\), determine the interval in which this value lies. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8018",
"split": "train"
} |
math | [
{
"content": "Indicate the integer closest to the number: \\(\\sqrt{2012-\\sqrt{2013 \\cdot 2011}}+\\sqrt{2010-\\sqrt{2011 \\cdot 2009}}+\\ldots+\\sqrt{2-\\sqrt{3 \\cdot 1}}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "8019",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{m} = (a\\cos x, \\cos x)$ and $\\overrightarrow{n} = (2\\cos x, b\\sin x)$, with a function $f(x) = \\overrightarrow{m} \\cdot \\overrightarrow{n}$ that satisfies $f(0) = 2$ and $f\\left(\\frac{\\pi}{3}\\right) = \\frac{1}{2} + \\frac{\\sqrt{3}}{2}$, \n(1) If $x \\... | {
"ground_truth": "-\\frac{4 + \\sqrt{7}}{3}",
"style": "rule"
} | {
"index": "8020",
"split": "train"
} |
math | [
{
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "8021",
"split": "train"
} |
math | [
{
"content": "(a) A natural number $n$ is less than 120. What is the largest remainder that the number 209 can give when divided by $n$?\n\n(b) A natural number $n$ is less than 90. What is the largest remainder that the number 209 can give when divided by $n$? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "8022",
"split": "train"
} |
math | [
{
"content": "If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{x}{x-1}",
"style": "rule"
} | {
"index": "8023",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is $0+5+1+3=9$). Find the number of dates in the year 2021 with digit sum equal to the positive integer $n$. Please output the final answe... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "8024",
"split": "train"
} |
math | [
{
"content": "What is the largest five-digit number whose digits add up to 20? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99200",
"style": "rule"
} | {
"index": "8025",
"split": "train"
} |
math | [
{
"content": "Undecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of th... | {
"ground_truth": "\\frac{27 \\sqrt{6}}{8} \\pi",
"style": "rule"
} | {
"index": "8026",
"split": "train"
} |
math | [
{
"content": "Calculate $\\left(\\frac{1}{3}\\right)^{6} \\div \\left(\\frac{2}{5}\\right)^{-4} + \\frac{1}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{455657}{911250}",
"style": "rule"
} | {
"index": "8027",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD$, $\\overline{CE}$ bisects angle $C$ (no trisection this time), where $E$ is on $\\overline{AB}$, $F$ is still on $\\overline{AD}$, but now $BE=10$, and $AF=5$. Find the area of $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "8028",
"split": "train"
} |
math | [
{
"content": "Given $\\log_9 \\Big(\\log_4 (\\log_3 x) \\Big) = 1$, calculate the value of $x^{-2/3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3^{-174762.6667}",
"style": "rule"
} | {
"index": "8029",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$ . Find $b - a$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "223",
"style": "rule"
} | {
"index": "8030",
"split": "train"
} |
math | [
{
"content": "Given \\((n+1)^{\\alpha+1}-n^{\\alpha+1} < n^{\\alpha}(\\alpha+1) < n^{\\alpha+1}-(n-1)^{\\alpha+1}, -1 < \\alpha < 0\\). Let \\(x = \\sum_{k=4}^{10^{6}} \\frac{1}{\\sqrt[3]{k}}\\), find the integer part of \\(x\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "146",
"style": "rule"
} | {
"index": "8031",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, point $P\\left( \\frac{1}{2},\\cos^2\\theta\\right)$ is on the terminal side of angle $\\alpha$, and point $Q(\\sin^2\\theta,-1)$ is on the terminal side of angle $\\beta$, and $\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}=-\\frac{1}{2}$. \n$(1)$ Find ... | {
"ground_truth": "-\\frac{\\sqrt{10}}{10}",
"style": "rule"
} | {
"index": "8032",
"split": "train"
} |
math | [
{
"content": "On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\\angle BEC = 30^\\circ$. \n\nFind the ratio of the sides of rectangle $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}/2",
"style": "rule"
} | {
"index": "8033",
"split": "train"
} |
math | [
{
"content": "A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\\frac1{1985}$. Ple... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "8034",
"split": "train"
} |
math | [
{
"content": "Milly has twelve socks, three of each color: red, blue, green, and yellow. She randomly draws five socks. What is the probability that she has exactly two pairs of socks of different colors and one sock of a third color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{22}",
"style": "rule"
} | {
"index": "8035",
"split": "train"
} |
math | [
{
"content": "Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "8036",
"split": "train"
} |
math | [
{
"content": "Given an integer \\( n \\geq 2 \\), let \\( f(n) \\) be the second largest positive divisor of \\( n \\). For example, \\( f(12)=6 \\) and \\( f(13)=1 \\). Determine the largest positive integer \\( n \\) such that \\( f(n)=35 \\). Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "175",
"style": "rule"
} | {
"index": "8037",
"split": "train"
} |
math | [
{
"content": "The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "8038",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the value of $x$. Please ou... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "8039",
"split": "train"
} |
math | [
{
"content": "The coefficient of $x^3$ in the expansion of $(2x-1)\\left( \\frac {1}{x}+x\\right)^{6}$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8040",
"split": "train"
} |
math | [
{
"content": "Calculate $\\frac{1}{2} \\cdot \\frac{3}{5} \\cdot \\frac{7}{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{21}{110}",
"style": "rule"
} | {
"index": "8041",
"split": "train"
} |
math | [
{
"content": "The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence? Express your answer in simplest radical form. [asy]\nsize(150); defaultpen(linewidth(2));\nreal loc = 0;\nfor(int i = 0; i < 11; ++i) {\n\nif(i == 4)\n\nlabel(... | {
"ground_truth": "\\sqrt{55}",
"style": "rule"
} | {
"index": "8042",
"split": "train"
} |
math | [
{
"content": "Suppose point P is on the curve $y=x^2+1$ (where $x \\geq 0$), and point Q is on the curve $y=\\sqrt{x-1}$ (where $x \\geq 1$). Then the minimum value of the distance $|PQ|$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "8043",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^2 + c^2 - b^2 = ac$, $c=2$, and point $G$ satisfies $|\\overrightarrow{BG}| = \\frac{\\sqrt{19}}{3}$ and $\\overrightarrow{BG} = \\frac{1}{3}(\\overrightarrow{BA} + \\overrightarrow{... | {
"ground_truth": "\\frac{3 \\sqrt{21}}{14}",
"style": "rule"
} | {
"index": "8044",
"split": "train"
} |
math | [
{
"content": "Today, Ivan the Confessor prefers continuous functions \\(f:[0,1] \\rightarrow \\mathbb{R}\\) satisfying \\(f(x)+f(y) \\geq|x-y|\\) for all pairs \\(x, y \\in[0,1]\\). Find the minimum of \\(\\int_{0}^{1} f\\) over all preferred functions. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "8045",
"split": "train"
} |
math | [
{
"content": "Let \\( A \\) be a subset of \\(\\{1, 2, 3, \\ldots, 2019\\}\\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \\( A \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "505",
"style": "rule"
} | {
"index": "8046",
"split": "train"
} |
math | [
{
"content": "If the inequality \\( \\ln(ax) + ax \\leq x + e^x \\) holds for all \\( x \\) and given real numbers \\( a \\), what is the maximum value of \\( a \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "e",
"style": "rule"
} | {
"index": "8047",
"split": "train"
} |
math | [
{
"content": "The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "8048",
"split": "train"
} |
math | [
{
"content": "Find the focus of the parabola $x = -\\frac{1}{12} y^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-3,0)",
"style": "rule"
} | {
"index": "8049",
"split": "train"
} |
math | [
{
"content": "Kiana has two older twin siblings, and their ages, together with her own, multiply together to equal 162. Calculate the sum of their ages. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "8050",
"split": "train"
} |
math | [
{
"content": "The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{15 \\sqrt{7}}{4}",
"style": "rule"
} | {
"index": "8051",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of\n\\[f(x) = 3x - x^3\\]for $0 \\le x \\le \\sqrt{3}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "8052",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a\\_n\\}$ satisfies $a\\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{b_{n}\\}$, then the minimum value of $\\{\\frac{S_{n}+4}{n}\\}$ is ___. Please output the final answer within \\b... | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "8053",
"split": "train"
} |
math | [
{
"content": "Given $\\cos(α-β)= \\frac{3}{5}$, $\\sin β= -\\frac{5}{13}$, and $α∈(0, \\frac{π}{2})$, $β∈(-\\frac{π}{2},0)$, find $\\sin α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{33}{65}",
"style": "rule"
} | {
"index": "8054",
"split": "train"
} |
math | [
{
"content": "Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30} \\cdot 5^6 \\cdot 3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8055",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$? Please output the final answer within... | {
"ground_truth": "\\frac{500}{3}",
"style": "rule"
} | {
"index": "8056",
"split": "train"
} |
math | [
{
"content": "Find the product of two approximate numbers: $0.3862 \\times 0.85$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.33",
"style": "rule"
} | {
"index": "8057",
"split": "train"
} |
math | [
{
"content": "What is the following value when expressed as a common fraction: $$\\frac{1}{3^{1}}+\\frac{1}{3^{2}}+\\frac{1}{3^{3}}+\\frac{1}{3^{4}}+\\frac{1}{3^{5}}+\\frac{1}{3^{6}}?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{364}{729}",
"style": "rule"
} | {
"index": "8058",
"split": "train"
} |
math | [
{
"content": "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "8059",
"split": "train"
} |
math | [
{
"content": "$r$ and $s$ are integers such that $3 r \\geq 2 s-3 \\text { and } 4 s \\geq r+12$. What is the smallest possible value of $r / s$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "8060",
"split": "train"
} |
math | [
{
"content": "Geoff and Trevor each roll a fair six-sided die. What is the probability that the product of the numbers they roll is even? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac34",
"style": "rule"
} | {
"index": "8061",
"split": "train"
} |
math | [
{
"content": "Find the least odd prime factor of $2019^8+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "97",
"style": "rule"
} | {
"index": "8062",
"split": "train"
} |
math | [
{
"content": "Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "211",
"style": "rule"
} | {
"index": "8063",
"split": "train"
} |
math | [
{
"content": "Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\\sin x$ and $\\cos x$, where $x$ is a real number such that $0<x<\\frac{\\pi}{2}$. The length of the hypotenuse is Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "8064",
"split": "train"
} |
math | [
{
"content": "Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\\triangle BDF$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi \\sqrt{3}}{12}",
"style": "rule"
} | {
"index": "8065",
"split": "train"
} |
math | [
{
"content": "Let \\( X = \\{0, a, b, c\\} \\) and \\( M(X) = \\{ f \\mid f: X \\rightarrow X \\} \\) be the set of all functions from \\( X \\) to itself. Define the addition operation \\( \\oplus \\) on \\( X \\) as given in the following table:\n\\[\n\\begin{array}{|c|c|c|c|c|}\n\\hline\n\\oplus & 0 & a & b ... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "8066",
"split": "train"
} |
math | [
{
"content": "How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "8067",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the answer to this problem. Box $B$ initially contains $n$ balls, and Box $A$ contains half as many balls as Box $B$. After 80 balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\\frac{p}{q}$, where $p, q$ are positive integers with $\\operatorname... | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "8068",
"split": "train"
} |
math | [
{
"content": "Given real numbers \\\\(x\\\\) and \\\\(y\\\\) satisfy the equation \\\\((x-3)^{2}+y^{2}=9\\\\), find the minimum value of \\\\(-2y-3x\\\\) \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3\\sqrt{13}-9",
"style": "rule"
} | {
"index": "8069",
"split": "train"
} |
math | [
{
"content": "Given \\(\\sin^{2}(18^{\\circ}) + \\cos^{2}(63^{\\circ}) + \\sqrt{2} \\sin(18^{\\circ}) \\cdot \\cos(63^{\\circ}) =\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8070",
"split": "train"
} |
math | [
{
"content": "The equation $y = -16t^2 + 80t$ describes the height (in feet) of a projectile launched from the ground at 80 feet per second. At what $t$ will the projectile reach 36 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth. Please output the final answer w... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "8071",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, side $AC = 900$ and side $BC = 600$. Points $K$ and $L$ are located on segment $AC$ and segment $AB$ respectively, such that $AK = CK$, and $CL$ is the angle bisector of $\\angle C$. Let point $P$ be the intersection of line segments $BK$ and $CL$. Point $M$ is located on line $... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "8072",
"split": "train"
} |
math | [
{
"content": "Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some ... | {
"ground_truth": "280",
"style": "rule"
} | {
"index": "8073",
"split": "train"
} |
math | [
{
"content": "Two circles, circle $A$ with radius 2 and circle $B$ with radius 1.5, are to be constructed with the following process: The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ to $(3,0)$. The center of circle $B$ is chosen uniformly and at random, and indep... | {
"ground_truth": "0.96",
"style": "rule"
} | {
"index": "8074",
"split": "train"
} |
math | [
{
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (Assume that the graph has the standard unit scale.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "8075",
"split": "train"
} |
math | [
{
"content": "Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Halfway between the poles, there is an additional smaller pole of 10 feet high. The lines go from the top of each main pole to the foot of the opposite main pole, intersecting somewhere above the smaller pole. Determine th... | {
"ground_truth": "22.5",
"style": "rule"
} | {
"index": "8076",
"split": "train"
} |
math | [
{
"content": "What is $10 \\cdot \\left(\\frac{1}{2} + \\frac{1}{5} + \\frac{1}{10}\\right)^{-1}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{2}",
"style": "rule"
} | {
"index": "8077",
"split": "train"
} |
math | [
{
"content": "Some squares of a $n \\times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "3n^{2}-5n+2",
"style": "rule"
} | {
"index": "8078",
"split": "train"
} |
math | [
{
"content": "A man born in the first half of the nineteenth century was $x$ years old in the year $x^2$. He was born in: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1806",
"style": "rule"
} | {
"index": "8079",
"split": "train"
} |
math | [
{
"content": "In a trapezoid, the lengths of the bases are 5 and 15, and the lengths of the diagonals are 12 and 16. Find the area of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "8080",
"split": "train"
} |
math | [
{
"content": "The two figures shown are made of unit squares. What is the positive difference of the perimeters, in units?\n\n[asy]\ndraw((0,0)--(0,1)--(5,1)--(5,0)--cycle,linewidth(1));\ndraw((1,0)--(1,2)--(4,2)--(4,0),linewidth(1));\ndraw((2,-1)--(2,3)--(3,3)--(3,-1)--cycle,linewidth(1));\ndraw((7,0)--(7,2)--... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "8081",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with common difference $d \\neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\\{a_n^2\\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\\{b_n\\}$ with a common ratio $q$ that is a positive rational n... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8082",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\sin \\theta+\\cos \\theta,1)$ and $\\overrightarrow{b}=(5,1)$, which are orthogonal, and $\\theta \\in (0,\\pi)$, find the value of $\\tan \\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "8083",
"split": "train"
} |
math | [
{
"content": "The area of a triangle is $4 \\sqrt{21}$, its perimeter is 24, and the segment of the angle bisector from one of the vertices to the center of the inscribed circle is $\\frac{\\sqrt{30}}{3}$. Find the longest side of the triangle. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "8084",
"split": "train"
} |
math | [
{
"content": "Suppose $x$ and $y$ are real numbers such that $-1<x<y<1$. Let $G$ be the sum of the geometric series whose first term is $x$ and whose ratio is $y$, and let $G^{\\prime}$ be the sum of the geometric series whose first term is $y$ and ratio is $x$. If $G=G^{\\prime}$, find $x+y$. Please output the... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "8085",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $\\dfrac{1}{2} + \\dfrac{1}{x} = \\dfrac{5}{6}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "8086",
"split": "train"
} |
math | [
{
"content": "We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{9}",
"style": "rule"
} | {
"index": "8087",
"split": "train"
} |
math | [
{
"content": "We make colored cubes according to the following specifications:\n\n1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.\n2. Each face's two triangles must be of different colors.\n3. Triangle... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8088",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "8089",
"split": "train"
} |
math | [
{
"content": "Given the sequence: 1, 3, 6, 10, 15, ..., find the 6th term. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "8090",
"split": "train"
} |
math | [
{
"content": "For real numbers $t \\neq 0,$ the point\n\\[(x,y) = \\left( \\frac{t + 1}{t}, \\frac{t - 1}{t} \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the fi... | {
"ground_truth": "\\text{(A)}",
"style": "rule"
} | {
"index": "8091",
"split": "train"
} |
math | [
{
"content": "Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and a... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "8092",
"split": "train"
} |
math | [
{
"content": "In a bag, there are $4$ red balls, $m$ yellow balls, and $n$ green balls. Now, two balls are randomly selected from the bag. Let $\\xi$ be the number of red balls selected. If the probability of selecting two red balls is $\\frac{1}{6}$ and the probability of selecting one red and one yellow ball ... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "8093",
"split": "train"
} |
math | [
{
"content": "Given a circle $O: x^{2}+y^{2}=4$.<br/>$(1)$ A tangent line is drawn from point $P(2,1)$ to circle $O$, find the equation of the tangent line $l$;<br/>$(2)$ Let $A$ and $B$ be the points where circle $O$ intersects the positive $x$-axis and positive $y$-axis, respectively. A moving point $Q$ satis... | {
"ground_truth": "\\frac{8\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "8094",
"split": "train"
} |
math | [
{
"content": "If $a=2 \\int_{-3}^{3} (x+|x|) \\, dx$, determine the total number of terms in the expansion of $(\\sqrt{x} - \\frac{1}{\\sqrt[3]{x}})^a$ where the power of $x$ is not an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "8095",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$, and $A$, $B$, $C$ form an arithmetic sequence.\n\n(I) If $b=7$ and $a+c=13$, find the area of $\\triangle ABC$.\n\n(II) Find the maximum value of $\\sqrt{3}\\sin A + \\sin(C - \\frac{\\pi}{6})$ and the size of ang... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "8096",
"split": "train"
} |
math | [
{
"content": "Let \\(b_n = 7^n + 9^n\\). Determine the remainder when \\(b_{86}\\) is divided by \\(50\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "8097",
"split": "train"
} |
math | [
{
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "8098",
"split": "train"
} |
math | [
{
"content": "John has two identical cups. Initially, he puts 6 ounces of tea into the first cup and 6 ounces of milk into the second cup. He then pours one-third of the tea from the first cup into the second cup and mixes thoroughly. After stirring, John then pours half of the mixture from the second cup back ... | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "8099",
"split": "train"
} |
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