data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Linda is tasked with writing a report on extracurricular activities at her school. The school offers two clubs: Robotics and Science. Linda has a list of 30 students who are members of at least one club. She knows that 22 students are in the Robotics club and 24 students are in... | math | {
"style": "rule",
"ground_truth": "\\frac{392}{435}"
} | {
"split": "train",
"index": 8600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10. Let's think step... | math | {
"style": "rule",
"ground_truth": "1 - \\left( \\frac{8}{9} \\right)^n - \\left( \\frac{5}{9} \\right)^n + \\left( \\frac{4}{9} \\right)^n"
} | {
"split": "train",
"index": 8601,
"solution": "For the product to be divisible by 10, there must be a factor of 2 and a factor of 5 in there.\nThe probability that there is no 5 is $\\left( \\frac{8}{9}\\right)^n$ .\nThe probability that there is no 2 is $\\left( \\frac{5}{9}\\right)^n$ .\nThe probability that t... |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 8602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. Ho... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 8603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tetrahedron with each edge length equal to $\\sqrt{2}$ has all its vertices on the same sphere. Calculate the surface area of this sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 8604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$,... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 8605,
"solution": "Once we establish that $\\Delta ANO$ is equilateral, we have \\[[{\\rm Sector } BON] = 2[{\\rm Sector } AON], [BCT'U]=2[ADT'U]\\] \\[\\Rightarrow [\\overset{\\large\\frown}{NB} CT'UO]=2[\\overset{\\large\\frown}{NA} DT'UO]\\] On the other hand, \\[[\\overset{\\large... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \\_\\_\\_\\_\\_ Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 8606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\ge 2$ be an integer. Consider an $n \\times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each... | math | {
"style": "rule",
"ground_truth": "k = \\left\\lfloor \\sqrt{n - 1}\\right\\rfloor"
} | {
"split": "train",
"index": 8607,
"solution": "\nLet \\( n \\geq 2 \\) be an integer, and consider an \\( n \\times n \\) chessboard. We place \\( n \\) rooks on this board such that each row and each column contains exactly one rook. This is defined as a peaceful configuration of rooks. The objective is to find... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number \\( r \\) with the following property: For every choice of 2023 unit vectors \\( \\mathbf{v}_1, \\mathbf{v}_2, \\ldots, \\mathbf{v}_{2023} \\in \\mathbb{R}^2 \\), a point \\( \\mathbf{p} \\) can be found in the plane such that for each sub... | math | {
"style": "rule",
"ground_truth": "2023/2"
} | {
"split": "train",
"index": 8608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from two standard decks of 104 cards mixed together. What is the probability that the first card drawn is an ace and the second card drawn is also an ace? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{1339}"
} | {
"split": "train",
"index": 8609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_1: ax+2y+6=0$ and $l_2: x+(a-1)y+a^2-1=0$. When $a$ \\_\\_\\_\\_\\_\\_, $l_1$ intersects $l_2$; when $a$ \\_\\_\\_\\_\\_\\_, $l_1$ is perpendicular to $l_2$; when $a$ \\_\\_\\_\\_\\_\\_, $l_1$ coincides with $l_2$; when $a$ \\_\\_\\_\\_\\_\\_, $l_1$ is parall... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn a department store, they received 10 suitcases and 10 keys separately in an envelope. Each key opens only one suitcase, and every suitcase can be matched with a corresponding key.\n\nA worker in the department store, who received the suitcases, sighed:\n\n- So much hassle ... | math | {
"style": "rule",
"ground_truth": "29.62"
} | {
"split": "train",
"index": 8611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30\\%"
} | {
"split": "train",
"index": 8612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of positive real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) = f(xy) + 2005 \\left( \\frac{1}{x} + \\frac{1}{y} + 2004 \\right)\\]for all $x,$ $y > 0.$\n\nLet $n$ be the number of possible values of $f(2),$ and let $s$ be the ... | math | {
"style": "rule",
"ground_truth": "\\frac{4011}{2}"
} | {
"split": "train",
"index": 8613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the reciprocal of $\\frac{1}{2}+\\frac{2}{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 8614,
"solution": "1. **Find the common denominator** for the fractions $\\frac{1}{2}$ and $\\frac{2}{3}$:\n \\[\n \\frac{1}{2} = \\frac{3}{6} \\quad \\text{and} \\quad \\frac{2}{3} = \\frac{4}{6}\n \\]\n Here, the least common multiple of 2 and 3 is 6.\n\n2. **Add the fractio... |
deepscaler | [
{
"role": "user",
"content": "Consider all prisms whose base is a convex 2015-gon.\n\nWhat is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 8615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube \\( A B C D A_{1} B_{1} C_{1} D_{1} \\), \\( M \\) is the center of the face \\( A B B_{1} A_{1} \\), \\( N \\) is a point on the edge \\( B_{1} C_{1} \\), \\( L \\) is the midpoint of \\( A_{1} B_{1} \\); \\( K \\) is the foot of the perpendicular dropped from \\(... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 8616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In convex hexagon $ABCDEF$, all six sides are congruent, $\\angle A$ and $\\angle D$ are right angles, and $\\angle B, \\angle C, \\angle E,$ and $\\angle F$ are congruent. The area of the hexagonal region is $2116(\\sqrt{2}+1).$ Find $AB$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 8617,
"solution": "Let the side length be called $x$, so $x=AB=BC=CD=DE=EF=AF$.\n The diagonal $BF=\\sqrt{AB^2+AF^2}=\\sqrt{x^2+x^2}=x\\sqrt{2}$. Then the areas of the triangles AFB and CDE in total are $\\frac{x^2}{2}\\cdot 2$, and the area of the rectangle BCEF equals $x\\cdot x\\sq... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\\angle A$. Let's think ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 8618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$ and $b$ are positive integers for which $8 a^{a} b^{b}=27 a^{b} b^{a}$. Find $a^{2}+b^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 8619,
"solution": "We have $$8 a^{a} b^{b}=27 a^{b} b^{a} \\Longleftrightarrow \\frac{a^{a} b^{b}}{a^{b} b^{a}}=\\frac{27}{8} \\Longleftrightarrow \\frac{a^{a-b}}{b^{a-b}}=\\frac{27}{8} \\Longleftrightarrow\\left(\\frac{a}{b}\\right)^{a-b}=\\frac{27}{8}$$ Since $27=3^{3}$ and $8=2^{3}... |
deepscaler | [
{
"role": "user",
"content": "There are a total of $400$ machine parts. If person A works alone for $1$ day, and then person A and person B work together for $2$ days, there will still be $60$ parts unfinished. If both work together for $3$ days, they can produce $20$ parts more than needed. How many parts ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 8620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 8621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos\\left(\\frac {\\pi}{4} + \\theta\\right) = -\\frac {3}{5}$, and $\\frac {11\\pi}{12} < \\theta < \\frac {5\\pi}{4}$, find the value of $\\frac {\\sin{2\\theta} + 2\\sin^{2}{\\theta}}{1 - \\tan{\\theta}}$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac {28}{75}"
} | {
"split": "train",
"index": 8622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\frac{x+4}{x^2+ax+b}$, and $f(x)$ has two vertical asymptotes at $x=1$ and $x=-2$, find the sum of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the inequality $\\ln (x+1)-(a+2)x\\leqslant b-2$ that always holds, find the minimum value of $\\frac {b-3}{a+2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1-e"
} | {
"split": "train",
"index": 8624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two sides of a triangle measure 4 cm and 9 cm. What is the greatest integer number of centimeters that could be the third side of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $l_{1}$: $x+my+6=0$ is parallel to the line $l_{2}$: $(m-2)x+3y+2m=0$. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8627,
"solution": "This is obtainable by construction. Consider two tetrahedrons glued along a face; this gives us 5 vertices, 9 edges, and 6 faces, for a total of 20 , and one readily checks that the required Eulerian path exists. Now, to see that we cannot do better, first notice th... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin \\omega x\\cos \\omega x+2 \\sqrt{3}\\sin ^{2}\\omega x- \\sqrt{3} (\\omega > 0)$ has the smallest positive period of $\\pi$. \n$(1)$ Find the intervals of increase for the function $f(x)$; \n$(2)$ Shift the graph of the function $f(x)$ to the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {59\\pi}{12}"
} | {
"split": "train",
"index": 8628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\\dfrac{m}{n}$, wher... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 8629,
"solution": "Assign mass points as follows: by Angle-Bisector Theorem, $BD / DC = 20/11$, so we assign $m(B) = 11, m(C) = 20, m(D) = 31$. Since $AM = MD$, then $m(A) = 31$, and $\\frac{CP}{PA} = \\frac{m(A) }{ m(C)} = \\frac{31}{20}$, so $m+n = \\boxed{51}$."
} |
deepscaler | [
{
"role": "user",
"content": "Of all positive integral solutions $(x,y,z)$ to the equation \\[x^3+y^3+z^3-3xyz=607,\\] compute the minimum possible value of $x+2y+3z.$ *Individual #7* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1215"
} | {
"split": "train",
"index": 8630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yangyang leaves home for school. If she walks 60 meters per minute, she arrives at school at 6:53. If she walks 75 meters per minute, she arrives at school at 6:45. What time does Yangyang leave home? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "6:13"
} | {
"split": "train",
"index": 8631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{13}{16}"
} | {
"split": "train",
"index": 8632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos (\\frac{\\pi}{4}-\\frac{\\theta}{2})=\\frac{2}{3}$, find the value of $\\sin \\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"split": "train",
"index": 8633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-4^2÷(-32)×(\\frac{2}{3})^2$;<br/>$(2)(-1)^{10}÷2+(-\\frac{1}{2})^3×16$;<br/>$(3)\\frac{12}{7}×(\\frac{1}{2}-\\frac{2}{3})÷\\frac{5}{14}×1\\frac{1}{4}$;<br/>$(4)1\\frac{1}{3}×[1-(-4)^2]-(-2)^3÷\\frac{4}{5}$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 8634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 8635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the moving points $P$, $M$, and $N$ are respectively on the $x$-axis, the circle $\\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and the circle $\\Gamma_{2}$: $(x-3)^{2}+(y-4)^{2}=3$, find the minimum value of $|PM| + |PN|$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10} - \\sqrt{3} - 1"
} | {
"split": "train",
"index": 8636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has five children. What is the probability that at least half of them are girls? (We can assume a boy is equally likely to be born as is a girl, and vice-versa.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 8637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\frac{1}{2} \\end{pmatrix}^n = \\mathbf{I}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 8638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe is studying a bacteria population. There are 20 bacteria present at 3:00 p.m. and the population doubles every 3 minutes. Assuming none of the bacteria die, how many bacteria are present at 3:15 p.m. the same day? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 8639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find, with proof, the smallest real number $C$ with the following property:\nFor every infinite sequence $\\{x_i\\}$ of positive real numbers such that $x_1 + x_2 +\\cdots + x_n \\leq x_{n+1}$ for $n = 1, 2, 3, \\cdots$, we have\n\\[\\sqrt{x_1}+\\sqrt{x_2}+\\cdots+\\sqrt{x_n} ... | math | {
"style": "rule",
"ground_truth": "C=1+\\sqrt{2}"
} | {
"split": "train",
"index": 8640,
"solution": "\nTo find the smallest real number \\( C \\) such that the inequality \n\n\\[\n\\sqrt{x_1} + \\sqrt{x_2} + \\cdots + \\sqrt{x_n} \\leq C \\sqrt{x_1 + x_2 + \\cdots + x_n}\n\\]\n\nholds for every infinite sequence \\(\\{x_i\\}\\) of positive real numbers satisfying \... |
deepscaler | [
{
"role": "user",
"content": "In the expression $(\\underline{\\qquad}\\times\\underline{\\qquad})+(\\underline{\\qquad}\\times\\underline{\\qquad})$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? Let's think ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 8641,
"solution": "To solve this problem, we need to determine how many distinct values can be obtained by filling in the blanks in the expression $\\left(a \\times b\\right) + \\left(c \\times d\\right)$, where $a, b, c, d$ are the digits $1, 2, 3, 4$, each used exactly once.\n\n1. *... |
deepscaler | [
{
"role": "user",
"content": "For any two positive integers m and n, define an operation \"※\" as follows: when m and n are both positive even numbers or positive odd numbers, m※n=m+n; when one of m and n is a positive even number and the other is a positive odd number, m※n=mn. Determine the number of eleme... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 8642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\diamondsuit$ and the second card dealt is a $\\spadesuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"split": "train",
"index": 8643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (5\\sqrt{3}\\cos x, \\cos x)$ and $\\overrightarrow{b} = (\\sin x, 2\\cos x)$, and the function $f(x) = \\overrightarrow{a} \\cdot \\overrightarrow{b} + |\\overrightarrow{b}|^2 + \\frac{3}{2}$.\n(I) Find the range of $f(x)$ when $x \\in [\\f... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2} + 7"
} | {
"split": "train",
"index": 8644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class, no two boys were born on the same day of the week and no two girls were born in the same month. If another child were to join the class, this would no longer be true. How many children are there in the class? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 8645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $C = 2A$, $\\cos A = \\frac{3}{4}$, and $2 \\overrightarrow{BA} \\cdot \\overrightarrow{CB} = -27$.\n\n(I) Find the value of $\\cos B$;\n\n(II) Find the perimeter of $\\triangle ABC$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 8646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos\\alpha + \\cos\\beta = \\frac{1}{2}$, $\\sin\\alpha + \\sin\\beta = \\frac{\\sqrt{3}}{2}$, then $\\cos(\\alpha - \\beta) =$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 8647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What value of $x$ makes the equation below true: $$2x + 4 = |{-17 + 3}|$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 8648,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $\\cos(-\\pi - \\alpha)$ given a point $P(-3, 4)$ on the terminal side of angle $\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3}{5}"
} | {
"split": "train",
"index": 8649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 40^\\circ + 4 \\sin 40^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 8650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)= \\begin{cases} \\sin \\pi x & \\text{if } x\\geqslant 0\\\\ \\cos \\left( \\frac {\\pi x}{2}+ \\frac {\\pi}{3}\\right) & \\text{if } x < 0\\end{cases}$. Evaluate $f(f( \\frac {15}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 8651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh \"I know the sum of the numbers on your cards is... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, side $a$ is 2 units longer than side $b$, and side $b$ is 2 units longer than side $c$. If the sine of the largest angle is $\\frac {\\sqrt {3}}{2}$, then the area of triangle $\\triangle ABC$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac {15 \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 8653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the lengths of the sides of a triangle are positive integers not greater than 5, how many such distinct triangles exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 8654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "762"
} | {
"split": "train",
"index": 8655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Scatterbrained Scientist had a sore knee. The doctor prescribed 10 pills for the knee, to be taken one pill daily. These pills help in $90 \\%$ of cases, but in $2 \\%$ of cases, there is a side effect—it eliminates scatterbrainedness, if present.\n\nAnother doctor prescribed t... | math | {
"style": "rule",
"ground_truth": "0.69"
} | {
"split": "train",
"index": 8656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\( ABCD \\), angles \\( A \\) and \\( D \\) are right angles, \\( AB = 1 \\), \\( CD = 4 \\), \\( AD = 5 \\). Point \\( M \\) is taken on side \\( AD \\) such that \\(\\angle CMD = 2 \\angle BMA \\).\n\nIn what ratio does point \\( M \\) divide side \\( AD \\)... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 8657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the product $\\frac{1}{3} \\cdot \\frac{9}{1} \\cdot \\frac{1}{27} \\cdot \\frac{81}{1} \\dotsm \\frac{1}{6561} \\cdot \\frac{19683}{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 8658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\\tfrac mn$, where $m$ and $n$ are relatively p... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 8659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum ... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 8660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 8661,
"solution": "Given that $L$ has slope $\\frac{5}{12}$ and contains the point $A=(24,-1)$, we may write the point-slope equation for $L$ as $y+1=\\frac{5}{12}(x-24)$. Since $M$ is perpendicular to $L$ and contains the point $B=(5,6)$, we have that the slope of $M$ is $-\\frac{12}... |
deepscaler | [
{
"role": "user",
"content": "There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point o... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 8662,
"solution": "1. **Understanding the problem**: We need to find the smallest time $T > 0$ such that at least 5 of the horses are at the starting point. Each Horse $k$ returns to the starting point every $k$ minutes.\n\n2. **Divisibility condition**: For Horse $k$ to be at the sta... |
deepscaler | [
{
"role": "user",
"content": "Shuxin begins with 10 red candies, 7 yellow candies, and 3 blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 8663,
"solution": "For there to be equal numbers of each colour of candy, there must be at most 3 red candies and at most 3 yellow candies, since there are 3 blue candies to start. Thus, Shuxin ate at least 7 red candies and at least 4 yellow candies. This means that Shuxin ate at lea... |
deepscaler | [
{
"role": "user",
"content": "For every $a \\in \\mathbb N$ denote by $M(a)$ the number of elements of the set\n\\[ \\{ b \\in \\mathbb N | a + b \\text{ is a divisor of } ab \\}.\\]\nFind $\\max_{a\\leq 1983} M(a).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 8664,
"solution": "\nTo solve the problem, we need to analyze the set \\( S(a) = \\{ b \\in \\mathbb{N} \\mid a + b \\text{ is a divisor of } ab \\} \\) for a given \\( a \\) in the natural numbers, and we need to find the maximum number of elements \\( M(a) \\) in this set for \\( a ... |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 8665,
"solution": "It is actually reasonably easy to compute $m$ and $n$ exactly.\n First, note that if $4x+3y+2z=2009$, then $y$ must be odd. Let $y=2y'-1$. We get $4x + 6y' - 3 + 2z = 2009$, which simplifies to $2x + 3y' + z = 1006$. For any pair of positive integers $(x,y')$ such t... |
deepscaler | [
{
"role": "user",
"content": "There are 15 different-colored crayons in a box. Karl wants to first select three crayons for his art project and then select four crayons for his friend's project. How many ways can Karl select these seven crayons if the order of selection does not matter for each set? Let's t... | math | {
"style": "rule",
"ground_truth": "225225"
} | {
"split": "train",
"index": 8666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha =\\dfrac{\\sqrt{5}}{5}$ and $\\sin (\\alpha -\\beta )=\\dfrac{\\sqrt{10}}{10}$, calculate the value of $\\cos \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 8667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular tetrahedron \\(P-ABC\\) with edge length 1, let \\(E\\), \\(F\\), \\(G\\), and \\(H\\) be points on edges \\(PA\\), \\(PB\\), \\(CA\\), and \\(CB\\) respectively. If \\(\\overrightarrow{EF}+\\overrightarrow{GH}=\\overrightarrow{AB}\\) and \\(\\overrightarrow{EH}... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 8668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following expressions is not equivalent to $3x + 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(3x) + \\frac{2}{3}(9)"
} | {
"split": "train",
"index": 8669,
"solution": "We look at each of the five choices: (A) $3(x + 2) = 3x + 6$ (B) $\\frac{-9x - 18}{-3} = \\frac{-9x}{-3} + \\frac{-18}{-3} = 3x + 6$ (C) $\\frac{1}{3}(3x) + \\frac{2}{3}(9) = x + 6$ (D) $\\frac{1}{3}(9x + 18) = 3x + 6$ (E) $3x - 2(-3) = 3x + (-2)(-3) = 3x + 6$ The e... |
deepscaler | [
{
"role": "user",
"content": "Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 8670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the plane unit vectors $\\overrightarrow{{e}_{1}}$ and $\\overrightarrow{{e}_{2}}$ satisfy $|2\\overrightarrow{{e}_{1}}-\\overrightarrow{{e}_{2}}|\\leqslant \\sqrt{2}$. Let $\\overrightarrow{a}=\\overrightarrow{{e}_{1}}+\\overrightarrow{{e}_{2}}$, $\\overrightarrow{b... | math | {
"style": "rule",
"ground_truth": "\\frac{28}{29}"
} | {
"split": "train",
"index": 8671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of the quadratic polynomial $a(x^3 - x^2 + 3x) + b(2x^2 + x) + x^3 - 5$ when $x = 2$ is $-17$. What is the value of this polynomial when $x = -2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\left\\{ a_n \\right\\}$, $a_{15}+a_{16}+a_{17}=-45$, $a_{9}=-36$, and $S_n$ is the sum of the first $n$ terms.\n\n(1) Find the minimum value of $S_n$ and the corresponding value of $n$;\n\n(2) Calculate $T_n = \\left| a_1 \\right| + \\left| a_2 \\r... | math | {
"style": "rule",
"ground_truth": "-630"
} | {
"split": "train",
"index": 8673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3x + 8"
} | {
"split": "train",
"index": 8674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\\angle{BAC}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 8675,
"solution": "1. **Identify the Geometry and Key Properties**: \n Let the center of the circle be $O$. The lines $AB$ and $AC$ are tangents to the circle at points $B$ and $C$, respectively. By the property of tangents, the radii $OB$ and $OC$ are perpendicular to $AB$ and $AC... |
deepscaler | [
{
"role": "user",
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial\n$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$\n are all real can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime pos... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 8676,
"solution": "The polynomial we are given is rather complicated, so we could use Rational Root Theorem to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, $x = 1, -1, 2, -2$ are all possible rational roots. Upon plugging these roots into the polyn... |
deepscaler | [
{
"role": "user",
"content": "In the figure, triangles $ABC$ and $BCD$ are equilateral triangles. What is the value of $AD \\div BC$ when expressed in simplest radical form?\n\n[asy]\ndraw((0,0)--(5,8.7)--(10,0)--cycle);\ndraw((10,0)--(15,8.7)--(5,8.7));\nlabel(\"$A$\",(0,0),SW);\nlabel(\"$B$\",(5,8.7),N);\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 8677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{13500, 13501, 13502, \\ldots\\}"
} | {
"split": "train",
"index": 8678,
"solution": "\nTo determine which positive integers \\( N \\) are contagious, we consider 1000 consecutive non-negative integers and the sum of all their digits equating to \\( N \\).\n\nLet the consecutive integers be \\( x, x+1, x+2, \\ldots, x+999 \\). We need to calculate th... |
deepscaler | [
{
"role": "user",
"content": "Given that the left focus of the ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is $F$, the eccentricity is $\\frac{\\sqrt{2}}{2}$, and the distance between the left intersection point of the ellipse with the $x$-axis and point $F$ is $\\sqrt{2} - 1$.\n(I... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 8679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{r}$ and $\\mathbf{s}$ be two three-dimensional unit vectors such that the angle between them is $45^\\circ$. Find the area of the parallelogram whose diagonals correspond to $\\mathbf{r} + 3\\mathbf{s}$ and $3\\mathbf{r} + \\mathbf{s}$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 8680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with a common ratio $q \\in (1,2)$, and that $a_{n}$ is a positive integer for $1 \\leq n \\leq 6$, find the minimum value of $a_{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 8681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\\left[\\frac{5 x}{3}\\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \\leq x \\leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "734"
} | {
"split": "train",
"index": 8682,
"solution": "Note that, since $[x+n]=[x]+n$ for any integer $n$, $$f(x+3)=[x+3]+[2(x+3)]+\\left[\\frac{5(x+3)}{3}\\right]+[3(x+3)]+[4(x+3)]=f(x)+35$$ one only needs to investigate the interval $[0,3)$. The numbers in this interval at which at least one of the real numbers $x, 2 ... |
deepscaler | [
{
"role": "user",
"content": "A circle with diameter $\\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The leng... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 8683,
"solution": "Call the center of the larger circle $O$. Extend the diameter $\\overline{PQ}$ to the other side of the square (at point $E$), and draw $\\overline{AO}$. We now have a right triangle, with hypotenuse of length $20$. Since $OQ = OP - PQ = 20 - 10 = 10$, we know that ... |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"split": "train",
"index": 8684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "10 people attend a meeting. Everyone at the meeting exchanges business cards with everyone else. How many exchanges of business cards occur? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 8685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six senior students (including 4 boys and 2 girls) are arranged to intern at three schools, A, B, and C, with two students at each school, and the two girls cannot be at the same school or at school C, and boy A cannot go to school A, calculate the total number of differe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some middle school students in a city participated in a mathematics invitational competition, which consisted of 6 problems. It is known that each problem was solved by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. Wh... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 8687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\cdots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\), circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\), and so on, with circle \\( C_{1}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "3745"
} | {
"split": "train",
"index": 8689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is the geometric mean of $4$ and $16$, find the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 8690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is an arithmetic sequence with a common difference of $2$, and $a_1$, $a_2$, $a_5$ form a geometric sequence, find the value of $s_8$ (the sum of the first 8 terms). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 8691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = \\begin{cases} x^{2}+1 & (x>0) \\\\ 2f(x+1) & (x\\leq 0) \\end{cases}$, find $f(2)$ and $f(-2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 8692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Into how many regions do the x-axis and the graphs of \\( y = 2 - x^2 \\) and \\( y = x^2 - 1 \\) split the plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_{n}\\}$ is an increasing sequence of integers, and $a_{1}\\geqslant 3$, $a_{1}+a_{2}+a_{3}+\\ldots +a_{n}=100$. Determine the maximum value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 5$, $BC = 8$, and the length of median $AM$ is 4. Find $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{39}"
} | {
"split": "train",
"index": 8695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $y$ which satisfies\n\\[\\log_7 (y - 3) + \\log_{\\sqrt{7}} (y^2 - 3) + \\log_{\\frac{1}{7}} (y - 3) = 3.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\sqrt{343} + 3}"
} | {
"split": "train",
"index": 8696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 8697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $$log_{4}(a+4b)=log_{2}2 \\sqrt {ab}$$, find the minimum value of $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 8698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{35}"
} | {
"split": "train",
"index": 8699,
"solution": "We interpret the problem with geometric probability. Let the three segments have lengths $x, y, 1-x-y$ and assume WLOG that $x \\geq y \\geq 1-x-y$. The every possible $(x, y)$ can be found in the triangle determined by the points $\\left(\\frac{1}{3}, \\frac{1}{3}\... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.