data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given that the angle between the unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is obtuse, the minimum value of $|\\overrightarrow{b} - t\\overrightarrow{a}| (t \\in \\mathbb{R})$ is $\\frac{\\sqrt{3}}{2}$, and $(\\overrightarrow{c} - \\overrightarrow{a}) \\cdot (\\overrightarrow{c} ... | {
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}",
"style": "rule"
} | {
"index": "5000",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "330",
"style": "rule"
} | {
"index": "5001",
"split": "train"
} |
math | [
{
"content": "Suppose\n\\[\\frac{1}{x^3 - 3x^2 - 13x + 15} = \\frac{A}{x-1} + \\frac{B}{x-3} + \\frac{C}{(x-3)^2}\\]\nwhere $A$, $B$, and $C$ are real constants. What is $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "5002",
"split": "train"
} |
math | [
{
"content": "Given the set $A={m+2,2m^2+m}$, find the value of $m$ if $3\\in A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "5003",
"split": "train"
} |
math | [
{
"content": "Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5004",
"split": "train"
} |
math | [
{
"content": "In the coordinate plane, a parallelogram $O A B C$ is drawn such that its center is at the point $\\left(\\frac{19}{2}, \\frac{15}{2}\\right)$, and the points $A, B,$ and $C$ have natural number coordinates. Find the number of such parallelograms. (Here, $O$ denotes the origin - the point $(0,0)$;... | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "5005",
"split": "train"
} |
math | [
{
"content": "Given \\( A_{1}, A_{2}, \\cdots, A_{n} \\) are \\( n \\) non-empty subsets of the set \\( A=\\{1,2,3, \\cdots, 10\\} \\), if for any \\( i, j \\in \\{1,2,3, \\cdots, n\\} \\), we have \\( A_{i} \\cup A_{j} \\neq A \\), then the maximum value of \\( n \\) is \\(\\qquad\\). Please output the final a... | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "5006",
"split": "train"
} |
math | [
{
"content": "The volume of a rectangular solid each of whose side, front, and bottom faces are $12\\text{ in}^{2}$, $8\\text{ in}^{2}$, and $6\\text{ in}^{2}$ respectively is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$24\\text{ in}^{3}$",
"style": "rule"
} | {
"index": "5007",
"split": "train"
} |
math | [
{
"content": "A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches deep. What is the hei... | {
"ground_truth": "\\frac{1}{2}+\\frac{\\sqrt{93}}{6}",
"style": "rule"
} | {
"index": "5008",
"split": "train"
} |
math | [
{
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\\triangle DEF$. Please output... | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "5009",
"split": "train"
} |
math | [
{
"content": "What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "5010",
"split": "train"
} |
math | [
{
"content": "In a certain sequence the first term is $a_1 = 2007$ and the second term is $a_2 = 2008.$ Furthermore, the values of the remaining terms are chosen so that\n\\[a_n + a_{n + 1} + a_{n + 2} = n\\]for all $n \\ge 1.$ Determine $a_{1000}.$ Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "2340",
"style": "rule"
} | {
"index": "5011",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\\cos B+b\\sin A$.\n(1) Find $A$;\n(2) If $a=2$ and $b=c$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}+1",
"style": "rule"
} | {
"index": "5012",
"split": "train"
} |
math | [
{
"content": "Find the number of six-digit palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9000",
"style": "rule"
} | {
"index": "5013",
"split": "train"
} |
math | [
{
"content": "The volume of a hemisphere is $\\frac{500}{3}\\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\pi \\times 250^{2/3}",
"style": "rule"
} | {
"index": "5014",
"split": "train"
} |
math | [
{
"content": "In a New Year's cultural evening of a senior high school class, there was a game involving a box containing 6 cards of the same size, each with a different idiom written on it. The idioms were: 意气风发 (full of vigor), 风平浪静 (calm and peaceful), 心猿意马 (restless), 信马由缰 (let things take their own course)... | {
"ground_truth": "\\dfrac{2}{5}",
"style": "rule"
} | {
"index": "5015",
"split": "train"
} |
math | [
{
"content": "The length of a circular race track is $1 \\mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ on the track, and the... | {
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "5016",
"split": "train"
} |
math | [
{
"content": "Using the digits 0, 1, 2, 3, 4, how many even three-digit numbers can be formed if each digit can be used more than once, and the number must be greater than 200? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "5017",
"split": "train"
} |
math | [
{
"content": "Shari walks at a constant rate of 3 miles per hour. After 1.5 hours, how many miles did she walk? 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": "5018",
"split": "train"
} |
math | [
{
"content": "Given three integers \\( x, y, z \\) satisfying \\( x + y + z = 100 \\) and \\( x < y < 2z \\), what is the minimum value of \\( z \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "5019",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}, AB=10, \\angle{A}=30^\\circ$ , and $\\angle{C=45^\\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\\perp{BC}$. T... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "5020",
"split": "train"
} |
math | [
{
"content": "Find the maximum of the expression\n$$\n|| \\ldots|| x_{1}-x_{2}\\left|-x_{3}\\right|-\\ldots\\left|-x_{2023}\\right|,\n$$\nwhere \\( x_{1}, x_{2}, \\ldots, x_{2023} \\) are distinct natural numbers between 1 and 2023. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "5021",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\\cos(B-C) - 2\\sin B\\sin C = -\\frac{1}{2}$.\n1. Find the measure of angle $A$.\n2. When $a = 5$ and $b = 4$, find the area of $\\triangle ABC$. Please output the final answer with... | {
"ground_truth": "2\\sqrt{3} + \\sqrt{39}",
"style": "rule"
} | {
"index": "5022",
"split": "train"
} |
math | [
{
"content": "Let $g(x) = 12x + 5$. Find the sum of all $x$ that satisfy the equation $g^{-1}(x) = g((3x)^{-1})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "5023",
"split": "train"
} |
math | [
{
"content": "Bees, in processing flower nectar into honey, remove a significant amount of water. Research has shown that nectar usually contains about $70\\%$ water, while the honey produced from it contains only $17\\%$ water. How many kilograms of nectar must bees process to obtain 1 kilogram of honey? Pleas... | {
"ground_truth": "2.77",
"style": "rule"
} | {
"index": "5024",
"split": "train"
} |
math | [
{
"content": "Given two real numbers $p>1$ and $q>1$ so that $\\frac{1}{p} + \\frac{1}{q} = 1$ and $pq = 4$, what is $q$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5025",
"split": "train"
} |
math | [
{
"content": "If real numbers \\( x \\) and \\( y \\) satisfy \\( x^{3} + y^{3} + 3xy = 1 \\), then the minimum value of \\( x^{2} + y^{2} \\) is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5026",
"split": "train"
} |
math | [
{
"content": "Find the integer $n$, $-90 \\le n \\le 90$, such that $\\sin n^\\circ = \\sin 782^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-62",
"style": "rule"
} | {
"index": "5027",
"split": "train"
} |
math | [
{
"content": "The area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inches. What is the perimeter of the non-shaded region?\n\n[asy]size(101);\nfilldraw(((0,0)--(0,8)--(10,8)--(10,-2)--(6,-2)--(6,0)--cycle^^(2.5,3)--(2.5,5)--(7.5,5)--(7.5,3)--cycle),gra... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5028",
"split": "train"
} |
math | [
{
"content": "The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5029",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=4\\cos ωx\\sin \\left(wx- \\frac{π}{6}\\right)$ $(ω > 0)$ with the smallest positive period of $π$.\n\n(1) Find $ω$;\n\n(2) In triangle $△ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that the acute angle $A$ is a zero point o... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "5030",
"split": "train"
} |
math | [
{
"content": "Call a positive integer $N \\geq 2$ \"special\" if for every $k$ such that $2 \\leq k \\leq N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$? Plea... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "5031",
"split": "train"
} |
math | [
{
"content": "Find $x$ such that $\\lceil x \\rceil \\cdot x = 210$. Express $x$ as a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5032",
"split": "train"
} |
math | [
{
"content": "Determine $\\sqrt[4]{105413504}$ without a calculator. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "5033",
"split": "train"
} |
math | [
{
"content": "Let $g$ be a function defined on the positive integers, such that\n\\[g(xy) = g(x) + g(y)\\]\nfor all positive integers $x$ and $y$. Given $g(8) = 21$ and $g(18) = 26$, find $g(432)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "5034",
"split": "train"
} |
math | [
{
"content": "On this monthly calendar, the date behind one of the letters is added to the date behind $\\text{C}$. If this sum equals the sum of the dates behind $\\text{A}$ and $\\text{B}$, then the letter is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "P",
"style": "rule"
} | {
"index": "5035",
"split": "train"
} |
math | [
{
"content": "Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are reciprocals to them, and printed the resulting decimal fractions on 20 infinite ribbons (for example, the last ribbon had printed the number $\\frac{1}{99!}=0, \\underbrace{00 \\ldots 00}_{155 \\text { ... | {
"ground_truth": "155",
"style": "rule"
} | {
"index": "5036",
"split": "train"
} |
math | [
{
"content": "Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "96\\%",
"style": "rule"
} | {
"index": "5037",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\angle C = 90^\\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12 + 12\\sqrt{2}",
"style": "rule"
} | {
"index": "5038",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\(ABCD\\), \\(AB\\) is parallel to \\(DC\\) and \\(\\angle DAF = 90^\\circ\\). Point \\(E\\) is on \\(DC\\) such that \\(EB = BC = CE\\). Point \\(F\\) is on \\(AB\\) such that \\(DF\\) is parallel to \\(EB\\). In degrees, what is the measure of \\(\\angle FDA\\)? Please output the f... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "5039",
"split": "train"
} |
math | [
{
"content": "All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "194",
"style": "rule"
} | {
"index": "5040",
"split": "train"
} |
math | [
{
"content": "Let $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a function such that for any integers $x, y$, we have $f\\left(x^{2}-3 y^{2}\\right)+f\\left(x^{2}+y^{2}\\right)=2(x+y) f(x-y)$. Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \\cdot f(2016)$ is a perfect square. Find the minimum possible val... | {
"ground_truth": "246",
"style": "rule"
} | {
"index": "5041",
"split": "train"
} |
math | [
{
"content": "Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$. \n$(1)$ If there are two tangents passing through point $A$, find the range of $m$. \n$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilate... | {
"ground_truth": "\\frac{7\\sqrt{15}}{5}",
"style": "rule"
} | {
"index": "5042",
"split": "train"
} |
math | [
{
"content": "Given a sequence ${a_n}$ with the sum of its first $n$ terms denoted as $T_n$, where $a_1 = 1$ and $a_1 + 2a_2 + 4a_3 + ... + 2^{n-1}a_n = 2n - 1$, find the value of $T_8 - 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{63}{64}",
"style": "rule"
} | {
"index": "5043",
"split": "train"
} |
math | [
{
"content": "The slope angle of the tangent line to the curve $f\\left(x\\right)=- \\frac{ \\sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\\tan^{-1}\\left( \\frac{f'\\left(1\\right)}{\\mid f'\\left(1\\right) \\mid} \\right)$, where $f'\\left(x\\right)$ is the derivative of $f\\left(x\\right)$. Please output the final an... | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "5044",
"split": "train"
} |
math | [
{
"content": "Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75 \\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25 \\%$. He decides not to make it rain... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "5045",
"split": "train"
} |
math | [
{
"content": "If $g(x)=1-x^2$ and $f(g(x))=\\frac{1-x^2}{x^2}$ when $x\\not=0$, then $f(1/2)$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5046",
"split": "train"
} |
math | [
{
"content": "Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\\qquad$. Please output the final answer with... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5047",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $\\sin A = \\frac{3}{5}$, $\\tan B = 2$. Find the value of $\\tan 2\\left(A+B\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{44}{117}",
"style": "rule"
} | {
"index": "5048",
"split": "train"
} |
math | [
{
"content": "$(1)$ State the operation laws or rules used in each step of the following calculation:<br/>$(-0.4)\\times \\left(-0.8\\right)\\times \\left(-1.25\\right)\\times 2.5$<br/>$=-\\left(0.4\\times 0.8\\times 1.25\\times 2.5\\right) \\text{(Step 1)}$<br/>$=-\\left(0.4\\times 2.5\\times 0.8\\times 1.25\\... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "5049",
"split": "train"
} |
math | [
{
"content": "Consider the polynomial \\( P(x)=x^{3}+x^{2}-x+2 \\). Determine all real numbers \\( r \\) for which there exists a complex number \\( z \\) not in the reals such that \\( P(z)=r \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "r>3, r<49/27",
"style": "rule"
} | {
"index": "5050",
"split": "train"
} |
math | [
{
"content": "The maximum value of the function \\( y = \\tan x - \\frac{2}{|\\cos x|} \\) is to be determined. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{3}",
"style": "rule"
} | {
"index": "5051",
"split": "train"
} |
math | [
{
"content": "Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\\frac {\\sin B}{\\sin A} = \\cos(A + B),$$ find the maximum value of $\\tan B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "5052",
"split": "train"
} |
math | [
{
"content": "Given that the odd function $f(x)$ is also a periodic function, and the smallest positive period of $f(x)$ is $\\pi$, when $x \\in \\left(0, \\frac{\\pi}{2}\\right)$, $f(x) = 2\\sin x$. Find the value of $f\\left(\\frac{11\\pi}{6}\\)$. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5053",
"split": "train"
} |
math | [
{
"content": "**\nConsider a set of numbers $\\{1, 2, 3, 4, 5, 6, 7\\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.\n\n** Please output the final answer ... | {
"ground_truth": "\\frac{17}{21}",
"style": "rule"
} | {
"index": "5054",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac{x}{3} = y^2$ and $\\frac{x}{6} = 3y$, solve for the value of $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "5055",
"split": "train"
} |
math | [
{
"content": "Given the quadratic function \\( f(x) = a x^{2} + b x + c \\) where \\( a, b, c \\in \\mathbf{R}_{+} \\), if the function has real roots, determine the maximum value of \\( \\min \\left\\{\\frac{b+c}{a}, \\frac{c+a}{b}, \\frac{a+b}{c}\\right\\} \\). Please output the final answer within \\boxed{}.... | {
"ground_truth": "5/4",
"style": "rule"
} | {
"index": "5056",
"split": "train"
} |
math | [
{
"content": "Find the smallest constant \\(C\\) such that\n\\[ x^2 + y^2 + z^2 + 1 \\ge C(x + y + z) \\]\nfor all real numbers \\(x, y,\\) and \\(z.\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{4}{3}}",
"style": "rule"
} | {
"index": "5057",
"split": "train"
} |
math | [
{
"content": "Compute the sum of the series:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "510",
"style": "rule"
} | {
"index": "5058",
"split": "train"
} |
math | [
{
"content": "There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty box... | {
"ground_truth": "28182",
"style": "rule"
} | {
"index": "5059",
"split": "train"
} |
math | [
{
"content": "How many solutions does the equation \n\\[\n\\frac{(x-1)(x-2)(x-3)\\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\\dotsm(x-10^2)(x-11^3)(x-12^3)\\dotsm(x-13^3)}\n\\]\nhave for \\(x\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "190",
"style": "rule"
} | {
"index": "5060",
"split": "train"
} |
math | [
{
"content": "Suppose \\(ABCD\\) is a rectangle whose diagonals meet at \\(E\\). The perimeter of triangle \\(ABE\\) is \\(10\\pi\\) and the perimeter of triangle \\(ADE\\) is \\(n\\). Compute the number of possible integer values of \\(n\\). Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "5061",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\\plus{}a \\cdot 32^n$ for some odd $ n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "436",
"style": "rule"
} | {
"index": "5062",
"split": "train"
} |
math | [
{
"content": "Ottó decided to assign a number to each pair \\((x, y)\\) and denote it as \\((x \\circ y)\\). He wants the following relationships to hold:\na) \\(x \\circ y = y \\circ x\\)\nb) \\((x \\circ y) \\circ z = (x \\circ z) \\circ (y \\circ z)\\)\nc) \\((x \\circ y) + z = (x + z) \\circ (y + z)\\).\n\n... | {
"ground_truth": "1975.5",
"style": "rule"
} | {
"index": "5063",
"split": "train"
} |
math | [
{
"content": "Let $F=\\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 and 4",
"style": "rule"
} | {
"index": "5064",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \\in \\mathbb{R}$).\n\n$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.\n\n$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\dfrac{2}{3}",
"style": "rule"
} | {
"index": "5065",
"split": "train"
} |
math | [
{
"content": "When \\( N \\) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \\( 3^n + n^3 \\) are divisible by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "5066",
"split": "train"
} |
math | [
{
"content": "In $\\triangle A B C$, the incircle centered at $I$ touches sides $A B$ and $B C$ at $X$ and $Y$, respectively. Additionally, the area of quadrilateral $B X I Y$ is $\\frac{2}{5}$ of the area of $A B C$. Let $p$ be the smallest possible perimeter of a $\\triangle A B C$ that meets these conditions... | {
"ground_truth": "2 \\sqrt{5}",
"style": "rule"
} | {
"index": "5067",
"split": "train"
} |
math | [
{
"content": "What is the sum of the ten terms in the arithmetic sequence $-3, 4, \\dots, 40$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "285",
"style": "rule"
} | {
"index": "5068",
"split": "train"
} |
math | [
{
"content": "A taxi has a starting fare of 10 yuan. After exceeding 10 kilometers, for every additional kilometer, the fare increases by 1.50 yuan (if the increase is less than 1 kilometer, it is rounded up to 1 kilometer; if the increase is more than 1 kilometer but less than 2 kilometers, it is rounded up to... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "5069",
"split": "train"
} |
math | [
{
"content": "The minimum positive period of the function $f(x)=\\sin x$ is $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "5070",
"split": "train"
} |
math | [
{
"content": "An aluminum cube with an edge length of \\( l = 10 \\) cm is heated to a temperature of \\( t_{1} = 100^{\\circ} \\mathrm{C} \\). After this, it is placed on ice, which has a temperature of \\( t_{2} = 0^{\\circ} \\mathrm{C} \\). Determine the maximum depth to which the cube can sink. The specific... | {
"ground_truth": "0.0818",
"style": "rule"
} | {
"index": "5071",
"split": "train"
} |
math | [
{
"content": "How many odd whole numbers are factors of 180? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5072",
"split": "train"
} |
math | [
{
"content": "In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe, and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes ex... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5073",
"split": "train"
} |
math | [
{
"content": "Cybil and Ronda are sisters. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the 10 cards. What is the probability that one letter is from each sister's name? Express your answer ... | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "5074",
"split": "train"
} |
math | [
{
"content": "Given point P(-2,0) and the parabola C: y^2=4x, let A and B be the intersection points of the line passing through P and the parabola. If |PA|= 1/2|AB|, find the distance from point A to the focus of parabola C. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "5075",
"split": "train"
} |
math | [
{
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\\begin{cases} x=2+t \\\\ y=kt \\end{cases}$ ($t$ is the parameter), and the parametric equation of line $l_2$ is $\\begin{cases} x=-2+m \\\\ y=\\frac{m}{... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "5076",
"split": "train"
} |
math | [
{
"content": "If the inequality \n\\[\n((x+y)^2+4)((x+y)^2-2)\\geq A\\cdot (x-y)^2\n\\]\nis hold for every real numbers $x,y$ such that $xy=1$ , what is the largest value of $A$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5077",
"split": "train"
} |
math | [
{
"content": "Let \\( A, B, C, D, E, F, G, H \\) be distinct digits from 0 to 7 that satisfy the following equation:\n\n\\[ \\overrightarrow{A B C} + \\overline{D E} = \\overline{F G H} \\]\n\nFind \\(\\overline{D E}\\) if \\(\\overline{A B C} = 146\\).\n\n(Note: \\(\\overline{A B C}\\) denotes a three-digit nu... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "5078",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number with the following properties:\na) It ends in digit 6 in decimal notation;\nb) If you remove the last digit 6 and place this digit 6 at the beginning of the remaining number, the result is four times the original number. Please output the final answer within \\boxe... | {
"ground_truth": "153846",
"style": "rule"
} | {
"index": "5079",
"split": "train"
} |
math | [
{
"content": "Let $P_{1}, P_{2}, \\ldots, P_{8}$ be 8 distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these 8 points, such that: (1) each $P_{i}$ is the endpoint of at most one segment and (2) two no segments intersect. (The... | {
"ground_truth": "323",
"style": "rule"
} | {
"index": "5080",
"split": "train"
} |
math | [
{
"content": "Given the distribution list of the random variable $X$, $P(X=\\frac{k}{5})=ak$, where $k=1,2,3,4,5$.\n\n1. Find the value of the constant $a$.\n2. Find $P(X\\geqslant\\frac{3}{5})$.\n3. Find $P(\\frac{1}{10}<X<\\frac{7}{10})$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "5081",
"split": "train"
} |
math | [
{
"content": "Inside a square of side length 1, four quarter-circle arcs are traced with the edges of the square serving as the radii. These arcs intersect pairwise at four distinct points, forming the vertices of a smaller square. This process is repeated for the smaller square, and continuously for each subse... | {
"ground_truth": "\\frac{2}{1 - (2 - \\sqrt{3})}",
"style": "rule"
} | {
"index": "5082",
"split": "train"
} |
math | [
{
"content": "What is the smallest three-digit number in Pascal's triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "5083",
"split": "train"
} |
math | [
{
"content": "What is the greatest common factor of 120, 180, and 300? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "5084",
"split": "train"
} |
math | [
{
"content": "Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on the Chapter 5 test? E... | {
"ground_truth": "76.17\\%",
"style": "rule"
} | {
"index": "5085",
"split": "train"
} |
math | [
{
"content": "What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{123456789}{2^{26}\\cdot 5^4}$ as a decimal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "5086",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be real numbers such that \\(9a^2 + 4b^2 + 25c^2 = 4\\). Find the maximum value of\n\\[6a + 3b + 10c.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{41}",
"style": "rule"
} | {
"index": "5087",
"split": "train"
} |
math | [
{
"content": "If the sum of all binomial coefficients in the expansion of $(2x- \\frac {1}{x^{2}})^{n}$ is $64$, then $n=$ \\_\\_\\_\\_\\_\\_; the constant term in the expansion is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "5088",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 10x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "5089",
"split": "train"
} |
math | [
{
"content": "Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2020",
"style": "rule"
} | {
"index": "5090",
"split": "train"
} |
math | [
{
"content": "Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "146",
"style": "rule"
} | {
"index": "5091",
"split": "train"
} |
math | [
{
"content": "What is the product of the digits in the base 8 representation of $9876_{10}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "5092",
"split": "train"
} |
math | [
{
"content": "If six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3888",
"style": "rule"
} | {
"index": "5093",
"split": "train"
} |
math | [
{
"content": "Let $\\bold{v} = \\begin{pmatrix} 5 \\\\ -3 \\end{pmatrix}$ and $\\bold{w} = \\begin{pmatrix} 11 \\\\ -2 \\end{pmatrix}$. Find the area of the parallelogram with vertices $\\bold{0}$, $\\bold{v}$, $\\bold{w}$, and $\\bold{v} + \\bold{w}$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "5094",
"split": "train"
} |
math | [
{
"content": "The radius of a sphere is \\( r = 10 \\text{ cm} \\). Determine the volume of the spherical segment whose surface area is in the ratio 10:7 compared to the area of its base. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288 \\pi",
"style": "rule"
} | {
"index": "5095",
"split": "train"
} |
math | [
{
"content": "In a right triangle, the length of one leg $XY$ is $30$ units, and the hypotenuse $YZ$ is $34$ units. Find the length of the second leg $XZ$, and then calculate $\\tan Y$ and $\\sin Y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{17}",
"style": "rule"
} | {
"index": "5096",
"split": "train"
} |
math | [
{
"content": "The taxicab distance between points $\\left(x_{1}, y_{1}\\right)$ and $\\left(x_{2}, y_{2}\\right)$ is $\\left|x_{2}-x_{1}\\right|+\\left|y_{2}-y_{1}\\right|$. A regular octagon is positioned in the $x y$ plane so that one of its sides has endpoints $(0,0)$ and $(1,0)$. Let $S$ be the set of all p... | {
"ground_truth": "2309",
"style": "rule"
} | {
"index": "5097",
"split": "train"
} |
math | [
{
"content": "What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits (leading zeros are not allowed)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10745",
"style": "rule"
} | {
"index": "5098",
"split": "train"
} |
math | [
{
"content": "Suppose $a_{1} < a_{2}< \\cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \\cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the t... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5099",
"split": "train"
} |
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