data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the line $y=kx+1$ intersects the parabola $C: x^2=4y$ at points $A$ and $B$, and a line $l$ is parallel to $AB$ and is tangent to the parabola $C$ at point $P$, find the minimum value of the area of triangle $PAB$ minus the length of $AB$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{64}{27}",
"style": "rule"
} | {
"index": "38400",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfies $a_n + (-1)^{n+1}a_{n+1} = 2n - 1$, find the sum of the first 40 terms, $S_{40}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "780",
"style": "rule"
} | {
"index": "38401",
"split": "train"
} |
math | [
{
"content": "Find the smallest composite number that has no prime factors less than 10. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "38402",
"split": "train"
} |
math | [
{
"content": "Simplify the expression \\(\\frac{\\operatorname{tg}\\left(\\frac{5}{4} \\pi - 4 \\alpha\\right) \\sin^{2}\\left(\\frac{5}{4} \\pi + 4 \\alpha\\right)}{1 - 2 \\cos^{2} 4 \\alpha}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "38403",
"split": "train"
} |
math | [
{
"content": "The probability that a randomly chosen divisor of $25!$ is odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{23}",
"style": "rule"
} | {
"index": "38404",
"split": "train"
} |
math | [
{
"content": "Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$, $0^{}_{}<r<1$, that have a repeating decimal expansion in the form $0.abcabcabc\\ldots=0.\\overline{abc}$, where the digits $a^{}_{}$, $b^{}_{}$, and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{}$ as fractions ... | {
"ground_truth": "660",
"style": "rule"
} | {
"index": "38405",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{cases} |\\ln x|, & (0 < x\\leqslant e^{3}) \\\\ e^{3}+3-x, & (x > e^{3})\\end{cases}$, there exist $x\\_1 < x\\_2 < x\\_3$ such that $f(x\\_1)=f(x\\_2)=f(x\\_3)$. Find the maximum value of $\\frac{f(x\\_3)}{x\\_2}$. Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{1}{e}",
"style": "rule"
} | {
"index": "38406",
"split": "train"
} |
math | [
{
"content": "The polynomial $x^3 -ax^2 + bx -2010$ has three positive integer roots. What is the smallest possible value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "38407",
"split": "train"
} |
math | [
{
"content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "38408",
"split": "train"
} |
math | [
{
"content": "How many multiples of 15 are there between 40 and 240? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "38409",
"split": "train"
} |
math | [
{
"content": "Compute \\[\\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": "38410",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\cos^2x+2\\sqrt{3}\\sin x\\cos x+a$, and when $x\\in\\left[0, \\frac{\\pi}{2}\\right]$, the minimum value of $f(x)$ is $2$,\n\n$(1)$ Find the value of $a$, and determine the intervals where $f(x)$ is monotonically increasing;\n\n$(2)$ First, transform the graph of the fu... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "38411",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(1,4)$, if $p(\\xi > 4)=0.1$, then $p(-2 \\leqslant \\xi \\leqslant 4)=$ _____ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "38412",
"split": "train"
} |
math | [
{
"content": "Calculate: \n(1) $2\\log_{2}{10}+\\log_{2}{0.04}$ \n(2) $(\\log_{4}{3}+\\log_{8}{3})\\cdot(\\log_{3}{5}+\\log_{9}{5})\\cdot(\\log_{5}{2}+\\log_{25}{2})$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{15}{8}",
"style": "rule"
} | {
"index": "38413",
"split": "train"
} |
math | [
{
"content": "Find the phase shift of the graph of $y = \\sin (3x - \\pi).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "38414",
"split": "train"
} |
math | [
{
"content": "On Tuesday, I worked $t+1$ hours and earned $3t-3$ dollars per hour. My friend Andrew worked $3t-5$ hours but only earned $t+2$ dollars an hour. At the end of the day, I had earned two dollars more than he had. What is the value of $t$? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38415",
"split": "train"
} |
math | [
{
"content": "A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Please output th... | {
"ground_truth": "817",
"style": "rule"
} | {
"index": "38416",
"split": "train"
} |
math | [
{
"content": "In $\\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. It is known that $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. Calculate the length of segment $BC$. Please output the final answer within ... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "38417",
"split": "train"
} |
math | [
{
"content": "[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP(\"B\",(0,0),SW);MP(\"A\",(0,2),NW);MP(\"D\",(2,2),NE);MP(\"C\",(2,0),SE); MP(\"E\",(0,1),W);MP(\"F\",(1,0),S);MP(\"H\",(2/3,2/3),E);MP(\"I\",(2/5,6/5),N); dot((1,0));dot((0,1));dot((... | {
"ground_truth": "\\frac{7}{15}",
"style": "rule"
} | {
"index": "38418",
"split": "train"
} |
math | [
{
"content": "Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \\ldots + a_{100}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2525",
"style": "rule"
} | {
"index": "38419",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a square of side length 5, and let $E$ be the midpoint of side $A B$. Let $P$ and $Q$ be the feet of perpendiculars from $B$ and $D$ to $C E$, respectively, and let $R$ be the foot of the perpendicular from $A$ to $D Q$. The segments $C E, B P, D Q$, and $A R$ partition $A B C D$ ... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38420",
"split": "train"
} |
math | [
{
"content": "Given four points \\(O, A, B, C\\) on a plane, such that \\(OA = 4\\), \\(OB = 3\\), \\(OC = 2\\), and \\(\\overrightarrow{OB} \\cdot \\overrightarrow{OC} = 3\\), find the maximum value of the area \\(S_{\\triangle ABC}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "38421",
"split": "train"
} |
math | [
{
"content": "Let $u$ and $v$ be real numbers such that \\[(u + u^2 + u^3 + \\cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \\cdots + v^{10}) + 10v^{11} = 8.\\] Determine, with proof, which of the two numbers, $u$ or $v$ , is larger. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ v \\]",
"style": "rule"
} | {
"index": "38422",
"split": "train"
} |
math | [
{
"content": "Given $$f(x)= \\begin{cases} f(x+1), x < 4 \\\\ ( \\frac {1}{2})^{x}, x\\geq4\\end{cases}$$, find $f(\\log_{2}3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{24}",
"style": "rule"
} | {
"index": "38423",
"split": "train"
} |
math | [
{
"content": "The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38424",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n=\\frac{n^2+n}{2}+1$, find the sum of the first 99 terms of the sequence ${\\frac{1}{a_n a_{n+1}}}$, denoted as $T_{99}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{37}{50}",
"style": "rule"
} | {
"index": "38425",
"split": "train"
} |
math | [
{
"content": "Please write an irrational number that is greater than -3 and less than -2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{5}",
"style": "rule"
} | {
"index": "38426",
"split": "train"
} |
math | [
{
"content": "How many different 7-digit positive integers exist? (Note that we don't allow \"7-digit\" integers that start with 0, such as 0123456; this is actually a 6-digit integer.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9,\\!000,\\!000",
"style": "rule"
} | {
"index": "38427",
"split": "train"
} |
math | [
{
"content": "If $\\cos 2^{\\circ} - \\sin 4^{\\circ} -\\cos 6^{\\circ} + \\sin 8^{\\circ} \\ldots + \\sin 88^{\\circ}=\\sec \\theta - \\tan \\theta$ , compute $\\theta$ in degrees.\n\n*2015 CCA Math Bonanza Team Round #10* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "38428",
"split": "train"
} |
math | [
{
"content": "In a bag containing 7 apples and 1 orange, the probability of randomly picking an apple is ______, and the probability of picking an orange is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "38429",
"split": "train"
} |
math | [
{
"content": "If $f(x)=\\sqrt{x-3}$, what is the smallest real number $x$ in the domain of $f(f(x))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38430",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on the sides $YZ$, $ZX$, and $XY$, respectively. Given that lines $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\\frac{XP}{PX'}+\\frac{YP}{PY'}+\\frac{ZP}{PZ'}=100$, find the product $\\frac{XP}{PX'}\\cdot \\frac{YP}{PY'}\\cdot \\... | {
"ground_truth": "102",
"style": "rule"
} | {
"index": "38431",
"split": "train"
} |
math | [
{
"content": "John went to the bookstore and purchased 20 notebooks totaling $62. Some notebooks were priced at $2 each, some at $5 each, and some at $6 each. John bought at least one notebook of each type. Let x be the number of $2 notebooks, y be the number of $5 notebooks, and z be the number of $6 notebooks... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "38432",
"split": "train"
} |
math | [
{
"content": "Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38433",
"split": "train"
} |
math | [
{
"content": "The ratio of the areas of two squares is $\\frac{32}{63}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$? Please output the final answer... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "38434",
"split": "train"
} |
math | [
{
"content": "The expression \\[(x+y+z)^{2006}+(x-y-z)^{2006}\\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1{,}008{,}016",
"style": "rule"
} | {
"index": "38435",
"split": "train"
} |
math | [
{
"content": "Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38436",
"split": "train"
} |
math | [
{
"content": "In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.\n\nOf the three equations\n\\begin{equation*} \\label{eq:1} d-s=1, \\qquad ds=1,... | {
"ground_truth": "I, II and III",
"style": "rule"
} | {
"index": "38437",
"split": "train"
} |
math | [
{
"content": "In the figure below, all corner angles are right angles and each number represents the unit-length of the segment which is nearest to it. How many square units of area does the figure have?\n\n[asy]\n\ndraw((0,0)--(12,0)--(12,5)--(8,5)--(8,4)--(5,4)\n--(5,6)--(0,6)--(0,0));\n\nlabel(\"6\",(0,3),W)... | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "38438",
"split": "train"
} |
math | [
{
"content": "According to the data shown, how much cheaper, in cents, is the cheaper store's price for Camera $X$ compared to the more expensive? \\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Store}& \\textbf{Sale Price for Camera $X$} \\\\ \\hline\nSuper Savers & $\\$9$~off the list price~$\\$39.96$ \\\\ \\hline... | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "38439",
"split": "train"
} |
math | [
{
"content": "How many perfect squares are between 100 and 500? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38440",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \\dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\dfrac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38441",
"split": "train"
} |
math | [
{
"content": "Given that $\\overrightarrow{OA}=(-2,4)$, $\\overrightarrow{OB}=(-a,2)$, $\\overrightarrow{OC}=(b,0)$, $a > 0$, $b > 0$, and $O$ is the coordinate origin. If points $A$, $B$, and $C$ are collinear, find the minimum value of $\\frac{1}{a}+\\frac{1}{b}$. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{3 + 2\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38442",
"split": "train"
} |
math | [
{
"content": "An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$? Plea... | {
"ground_truth": "\\frac{8080}{199}",
"style": "rule"
} | {
"index": "38443",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be relatively prime positive integers such that $\\dfrac ab=\\dfrac1{2^1}+\\dfrac2{3^2}+\\dfrac3{2^3}+\\dfrac4{3^4}+\\dfrac5{2^5}+\\dfrac6{3^6}+\\cdots$, where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also i... | {
"ground_truth": "689",
"style": "rule"
} | {
"index": "38444",
"split": "train"
} |
math | [
{
"content": "A collection of five positive integers has mean 4.4, unique mode 3 and median 4. If an 8 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "38445",
"split": "train"
} |
math | [
{
"content": "We write the equation on the board:\n\n$$\n(x-1)(x-2) \\ldots(x-2016) = (x-1)(x-2) \\ldots(x-2016)\n$$\n\nWe want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this? Please ... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "38446",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ that satisfies: $a_{n+2}=\\begin{cases} 2a_{n}+1 & (n=2k-1, k\\in \\mathbb{N}^*) \\\\ (-1)^{\\frac{n}{2}} \\cdot n & (n=2k, k\\in \\mathbb{N}^*) \\end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \\leqslant 2046$. Please outp... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38447",
"split": "train"
} |
math | [
{
"content": "On the sides \\( AB, BC \\), and \\( AC \\) of triangle \\( ABC \\), points \\( M, N, \\) and \\( K \\) are taken respectively so that \\( AM:MB = 2:3 \\), \\( AK:KC = 2:1 \\), and \\( BN:NC = 1:2 \\). In what ratio does the line \\( MK \\) divide the segment \\( AN \\)? Please output the final an... | {
"ground_truth": "6:7",
"style": "rule"
} | {
"index": "38448",
"split": "train"
} |
math | [
{
"content": "Compute the integral: \\(\\int_{0}^{\\pi / 2}\\left(\\sin ^{2}(\\sin x) + \\cos ^{2}(\\cos x)\\right) \\,dx\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "38449",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\\{b_{n}\\}$ is an arithmetic sequence with a common difference of $-1$. \n$(1)$ Investigation: Determine whether the sequence $\\{a_{n}-n\\}$ is an arithmetic sequence or a geometric sequence, and provi... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38450",
"split": "train"
} |
math | [
{
"content": "In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\\circ}$. Find the angle between this side and the other diagonal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "38451",
"split": "train"
} |
math | [
{
"content": "A student recorded the exact percentage frequency distribution for a set of measurements, as shown below.\nHowever, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$?\n\\begin{tabular}{c c}\\text{measured value}&\\text{percent frequ... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38452",
"split": "train"
} |
math | [
{
"content": "The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "38453",
"split": "train"
} |
math | [
{
"content": "How many consecutive \"0\"s are there at the end of the product \\(5 \\times 10 \\times 15 \\times 20 \\times \\cdots \\times 2010 \\times 2015\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "398",
"style": "rule"
} | {
"index": "38454",
"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": "14",
"style": "rule"
} | {
"index": "38455",
"split": "train"
} |
math | [
{
"content": "Find the greatest natural number $n$ such that $n\\leq 2008$ and $(1^2+2^2+3^2+\\cdots + n^2)\\left[(n+1)^2+(n+2)^2+(n+3)^2+\\cdots + (2n)^2\\right]$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1921",
"style": "rule"
} | {
"index": "38456",
"split": "train"
} |
math | [
{
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "38457",
"split": "train"
} |
math | [
{
"content": "How many edges does a square-based pyramid have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38458",
"split": "train"
} |
math | [
{
"content": "Vasya and Petya live in the mountains and like to visit each other. They ascend the mountain at a speed of 3 km/h and descend at a speed of 6 km/h (there are no flat sections of the road). Vasya calculated that it takes him 2 hours and 30 minutes to go to Petya, and 3 hours and 30 minutes to retur... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "38459",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $a^{2}+c^{2}=ac+b^{2}$, $b= \\sqrt{3}$, and $a\\geqslant c$. The minimum value of $2a-c$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38460",
"split": "train"
} |
math | [
{
"content": "Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "38461",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=0$, for any $k\\in N^*$, $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ form an arithmetic sequence with a common difference of $k$. If $b_n= \\dfrac {(2n+1)^{2}}{a_{2n+1}}$, calculate the sum of the first $10$ terms of the sequence $\\{b_n\\}$. Please output the fin... | {
"ground_truth": "\\dfrac {450}{11}",
"style": "rule"
} | {
"index": "38462",
"split": "train"
} |
math | [
{
"content": "Many of the students in M. Gamache's class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was $7:4$. There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total? Please output the final... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "38463",
"split": "train"
} |
math | [
{
"content": "A function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfies: $f(0)=0$ and $$\\left|f\\left((n+1) 2^{k}\\right)-f\\left(n 2^{k}\\right)\\right| \\leq 1$$ for all integers $k \\geq 0$ and $n$. What is the maximum possible value of $f(2019)$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38464",
"split": "train"
} |
math | [
{
"content": "Evaluate the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{987} + 1\\right)^{8/3}$ using the Binomial Expansion. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "666",
"style": "rule"
} | {
"index": "38465",
"split": "train"
} |
math | [
{
"content": "When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "205",
"style": "rule"
} | {
"index": "38466",
"split": "train"
} |
math | [
{
"content": "Let \\( M = \\{1, 2, \\cdots, 17\\} \\). If there exist four distinct numbers \\( a, b, c, d \\in M \\) such that \\( a + b \\equiv c + d \\pmod{17} \\), then \\( \\{a, b\\} \\) and \\( \\{c, d\\} \\) are called a balanced pair of the set \\( M \\). Find the number of balanced pairs in the set \\(... | {
"ground_truth": "476",
"style": "rule"
} | {
"index": "38467",
"split": "train"
} |
math | [
{
"content": "Given that $a, b, c$ are positive real numbers and $\\log _{a} b+\\log _{b} c+\\log _{c} a=0$, find the value of $\\left(\\log _{a} b\\right)^{3}+\\left(\\log _{b} c\\right)^{3}+\\left(\\log _{c} a\\right)^{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38468",
"split": "train"
} |
math | [
{
"content": "Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number) Please output the final a... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "38469",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = \\frac{x+1}{x-1}$. Then for $x^2 \\neq 1$, $f(-x)$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{f(x)}",
"style": "rule"
} | {
"index": "38470",
"split": "train"
} |
math | [
{
"content": "If the length of a rectangle is increased by $15\\%$ and the width is increased by $25\\%$, by what percent is the area increased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43.75\\%",
"style": "rule"
} | {
"index": "38471",
"split": "train"
} |
math | [
{
"content": "In rectangle $EFGH$, we have $E=(1,1)$, $F=(101,21)$, and $H=(3,y)$ for some integer $y$. What is the area of rectangle $EFGH$?\nA) 520\nB) 1040\nC) 2080\nD) 2600 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1040",
"style": "rule"
} | {
"index": "38472",
"split": "train"
} |
math | [
{
"content": "Given the triangular pyramid P-ABC, PA is perpendicular to the base ABC, AB=2, AC=AP, BC is perpendicular to CA. If the surface area of the circumscribed sphere of the triangular pyramid P-ABC is $5\\pi$, find the value of BC. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38473",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers \\(n \\le 500\\) that can be expressed in the form\n\\[\n\\lfloor x \\rfloor + \\lfloor 3x \\rfloor + \\lfloor 4x \\rfloor = n\n\\]\nfor some real number \\(x\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248",
"style": "rule"
} | {
"index": "38474",
"split": "train"
} |
math | [
{
"content": "Rhombus $ABCD$ has $\\angle BAD < 90^\\circ.$ There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to the lines $DA,AB,$ and $BC$ are $9,$ $5,$ and $16,$ respectively. Find the perimeter of $ABCD.$\nDiagram\n[asy] /* Made by MRENTHUSIASM; inspired by Math Jams. */ s... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "38475",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle $ABC$ in square units?\n\n[asy]\n\nfill((0,1)--(4,0)--(2,5)--cycle,lightgray);\n\nfor(int i=1; i < 5; ++i){\nfor(int k=1; k < 4; ++k){\ndraw((0,i)--(4,i),dashed);... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38476",
"split": "train"
} |
math | [
{
"content": "The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38477",
"split": "train"
} |
math | [
{
"content": "Call a fraction $\\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38478",
"split": "train"
} |
math | [
{
"content": "A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S}\\cup\\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S}\\cup\\{2001\\}$ is $27$ more than the mean of $\\mathcal{S}$. Find the mean of $\\mathcal{S}$. Pleas... | {
"ground_truth": "651",
"style": "rule"
} | {
"index": "38479",
"split": "train"
} |
math | [
{
"content": "Delete all perfect squares from the sequence of positive integers $1, 2, 3, \\cdots$. Find the 2003rd term of the new sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": "38480",
"split": "train"
} |
math | [
{
"content": "Give an example of a number $x$ for which the equation $\\sin 2017 x - \\operatorname{tg} 2016 x = \\cos 2015 x$ holds. Justify your answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "38481",
"split": "train"
} |
math | [
{
"content": "Let $F=\\log\\dfrac{1+x}{1-x}$. Find a new function $G$ by replacing each $x$ in $F$ by $\\dfrac{3x+x^3}{1+3x^2}$, and simplify.\nThe simplified expression $G$ is equal to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3F",
"style": "rule"
} | {
"index": "38482",
"split": "train"
} |
math | [
{
"content": "The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least 2 larger than an adjacent number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{3}",
"style": "rule"
} | {
"index": "38483",
"split": "train"
} |
math | [
{
"content": "Let \\( M \\) be a subset of the set \\(\\{1, 2, 3, \\cdots, 15\\}\\), and suppose that the product of any three different elements in \\( M \\) is not a perfect square. Let \\( |M| \\) denote the number of elements in the set \\( M \\). Find the maximum value of \\( |M| \\). Please output the fin... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38484",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha \\cos \\alpha = \\frac{1}{8}$, and $\\alpha$ is an angle in the third quadrant. Find $\\frac{1 - \\cos^2 \\alpha}{\\cos(\\frac{3\\pi}{2} - \\alpha) + \\cos \\alpha} + \\frac{\\sin(\\alpha - \\frac{7\\pi}{2}) + \\sin(2017\\pi - \\alpha)}{\\tan^2 \\alpha - 1}$. Please outpu... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38485",
"split": "train"
} |
math | [
{
"content": "Given a point $p$ and a line segment $l$, let $d(p, l)$ be the distance between them. Let $A, B$, and $C$ be points in the plane such that $A B=6, B C=8, A C=10$. What is the area of the region in the $(x, y)$-plane formed by the ordered pairs $(x, y)$ such that there exists a point $P$ inside tri... | {
"ground_truth": "\\frac{288}{5}",
"style": "rule"
} | {
"index": "38486",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\in \\mathbf{R}_{+} \\). If the function\n\\[ \nf(x)=\\frac{a}{x-1}+\\frac{1}{x-2}+\\frac{1}{x-6} \\quad (3 < x < 5) \n\\]\nachieves its maximum value at \\( x=4 \\), find the value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{9}{2}",
"style": "rule"
} | {
"index": "38487",
"split": "train"
} |
math | [
{
"content": "A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.\n$\\text{(A) } 12... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38488",
"split": "train"
} |
math | [
{
"content": "A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38489",
"split": "train"
} |
math | [
{
"content": "Given a fixed point P (-2, 0) and a line $l: (1+3\\lambda)x + (1+2\\lambda)y - (2+5\\lambda) = 0$, where $\\lambda \\in \\mathbb{R}$, find the maximum distance $d$ from point P to line $l$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "38490",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $a$ and $b$ such that\n\\[ {a^2+b\\over b^2-a}\\quad\\mbox{and}\\quad{b^2+a\\over a^2-b} \\]\nare both integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,2)(3,3)(1,2)(2,1)(2,3)(3,2)",
"style": "rule"
} | {
"index": "38491",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \\times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\boxed{1999}",
"style": "rule"
} | {
"index": "38492",
"split": "train"
} |
math | [
{
"content": "Let the set \\( T \\) consist of integers between 1 and \\( 2^{30} \\) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \\( T \\), what is the probability that it is divisible by 9? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "5/29",
"style": "rule"
} | {
"index": "38493",
"split": "train"
} |
math | [
{
"content": "The volumes of a regular tetrahedron and a regular octahedron have equal edge lengths. Find the ratio of their volumes without calculating the volume of each polyhedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "38494",
"split": "train"
} |
math | [
{
"content": "For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\\sin x = \\sin (n+1)x$ on the interval $[0, 2\\pi]$. Calculate $\\sum_{n=2}^{100} G(n)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10296",
"style": "rule"
} | {
"index": "38495",
"split": "train"
} |
math | [
{
"content": "Solve for the positive integer(s) \\( n \\) such that \\( \\phi\\left(n^{2}\\right) = 1000 \\phi(n) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "38496",
"split": "train"
} |
math | [
{
"content": "For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\\tfrac{N}{2}.$ Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2k^3 + 3k^2 + 3k",
"style": "rule"
} | {
"index": "38497",
"split": "train"
} |
math | [
{
"content": "The number \\(n\\) is a three-digit positive integer and is the product of the three factors \\(x\\), \\(y\\), and \\(5x+2y\\), where \\(x\\) and \\(y\\) are integers less than 10 and \\((5x+2y)\\) is a composite number. What is the largest possible value of \\(n\\) given these conditions? Please ... | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "38498",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\\cdots2^{(2n+1)/7}$ is greater than $1000$? \n(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.) Please output the final answer... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38499",
"split": "train"
} |
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