data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A function $f(x)$ defined on $\\mathbb{R}$ satisfies $f(0)=0$, $f(x)+f(1-x)=1$, $f(\\frac{x}{5})=\\frac{1}{2}f(x)$, and $f(x_1)\\leqslant f(x_2)$ when $0\\leqslant x_1 < x_2\\leqslant 1$. Find the value of $f(\\frac{1}{2007})$.\n\nA) $\\frac{1}{2}$\nB) $\\frac{1}{16}$\nC) $\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"split": "train",
"index": 27549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven people are standing in two rows, with 3 people in the front row and 4 people in the back row. Three people, A, B, and C, are added to the queue, with one person added to the front row and two people added to the back row, while the positions of the other people remain unc... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 27550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "679 contestants participated in HMMT February 2017. Let \\(N\\) be the number of these contestants who performed at or above the median score in at least one of the three individual tests. Estimate \\(N\\). An estimate of \\(E\\) earns \\(\\left\\lfloor 20-\\frac{|E-N|}{2}\\rig... | math | {
"style": "rule",
"ground_truth": "516"
} | {
"split": "train",
"index": 27551,
"solution": "Out of the 679 total contestants at HMMT February 2017, 188 contestants scored at least the median on all three tests, 159 contestants scored at least the median on two tests, and 169 contestants scored at least the median on one test, giving a total of 516 contest... |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 294 distinct cards with numbers \\(7, 11, 7^{2}, 11^{2}, \\ldots, 7^{147}, 11^{147}\\) (each card has exactly one number, and each number appears exactly once). How many ways can two cards be selected so that the product of the numbers on the selected cards is a perfe... | math | {
"style": "rule",
"ground_truth": "15987"
} | {
"split": "train",
"index": 27553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solution of $8x+1\\equiv 5 \\pmod{12}$ is $x\\equiv a\\pmod{m}$ for some positive integers $m\\geq 2$ and $a<m$. Find $a+m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 27554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x, \\) \\( y, \\) and \\( k \\) are positive real numbers such that \n\\[\n4 = k^2\\left(\\frac{x^2}{y^2}+\\frac{y^2}{x^2}\\right)+k\\left(\\frac{x}{y}+\\frac{y}{x}\\right),\n\\]\nfind the maximum possible value of \\( k \\). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{-1+\\sqrt{17}}{4}"
} | {
"split": "train",
"index": 27555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum positive period of the function $f(x)=\\sin x$ is $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 27556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In some cases, it is not necessary to calculate the result to remove the absolute value. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7.\\left(1\\right)$ According to the above rule, express the following expressions in the form without absolute value symbols: ... | math | {
"style": "rule",
"ground_truth": "\\frac{505}{1011}"
} | {
"split": "train",
"index": 27557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the positive solutions to \\( 2x^2 - x \\lfloor x \\rfloor = 5 \\), where \\( \\lfloor x \\rfloor \\) is the largest integer less than or equal to \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{41} + 2\\sqrt{11}}{4}"
} | {
"split": "train",
"index": 27558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are two types of golden triangles, one with a vertex angle of $36^{\\circ}$ and the other with a vertex angle of $108^{\\circ}$, and the ratio of the side opposite the $36^{\\circ}$ angle to the side opposite the $72^{\\circ}$ angle in the golden triangle with ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 27559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive numbers \\(a\\), \\(b\\), and \\(c\\) satisfy the following equations:\n\n\\[ a^{2} + a b + b^{2} = 1 \\]\n\\[ b^{2} + b c + c^{2} = 3 \\]\n\\[ c^{2} + c a + a^{2} = 4 \\]\n\nFind \\(a + b + c\\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 27560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?\n$\\textbf{(I)}\\ x+y < a+b$\n$\\textbf{(II)}\\ x-y < a-b$\n$\\textbf{(III)}\\ xy < ab$\n$\\textbf{(IV)}\\ \\frac{x}{y} < \\frac{a}{b}$ Let's think step... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 27561,
"solution": "We are given that $x < a$ and $y < b$ for nonzero real numbers $x$, $y$, $a$, and $b$. We need to determine how many of the given inequalities must be true under these conditions.\n\n1. **Analyzing $\\textbf{(I)}\\ x+y < a+b$:**\n - Since $x < a$ and $y < b$, by ... |
deepscaler | [
{
"role": "user",
"content": "The expression $\\circ \\ 1\\ \\circ \\ 2 \\ \\circ 3 \\ \\circ \\dots \\circ \\ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many p... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 27562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\,{\\mathbb{R}}\\,$ denote the set of all real numbers. Find all functions $\\,f: {\\mathbb{R}}\\rightarrow {\\mathbb{R}}\\,$ such that \\[ f\\left( x^{2}+f(y)\\right) =y+\\left( f(x)\\right) ^{2}\\hspace{0.2in}\\text{for all}\\,x,y\\in \\mathbb{R}. \\] Let's think step b... | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"split": "train",
"index": 27563,
"solution": "\nLet \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a function satisfying the functional equation:\n\n\\[\nf(x^2 + f(y)) = y + (f(x))^2 \\quad \\text{for all } x, y \\in \\mathbb{R}.\n\\]\n\n**Step 1**: Evaluate the functional equation at specific points.\n\nF... |
deepscaler | [
{
"role": "user",
"content": "Express $0.7\\overline{32}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1013}{990}"
} | {
"split": "train",
"index": 27564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the broadcast time of the \"Midday News\" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 27565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f_{}^{}$ has the property that, for each real number $x,\\,$\n$f(x)+f(x-1) = x^2.\\,$\n If $f(19)=94,\\,$ what is the remainder when $f(94)\\,$ is divided by $1000$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "561"
} | {
"split": "train",
"index": 27566,
"solution": "\\begin{align*}f(94)&=94^2-f(93)=94^2-93^2+f(92)=94^2-93^2+92^2-f(91)=\\cdots \\\\ &= (94^2-93^2) + (92^2-91^2) +\\cdots+ (22^2-21^2)+ 20^2-f(19) \\\\ &= 94+93+\\cdots+21+400-94 \\\\ &= 4561 \\end{align*}\n So, the remainder is $\\boxed{561}$."
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:\n$\\textbf{(A)}\\ s^2\\le8r^2\\qquad \\textbf{(B)}\\ s^... | math | {
"style": "rule",
"ground_truth": "s^2 \\le 8r^2"
} | {
"split": "train",
"index": 27567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given list of three numbers, the operation \"changesum\" replaces each number in the list with the sum of the other two. For example, applying \"changesum\" to \\(3,11,7\\) gives \\(18,10,14\\). Arav starts with the list \\(20,2,3\\) and applies the operation \"changesum\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c, d \\) be positive integers such that \\( \\gcd(a, b) = 24 \\), \\( \\gcd(b, c) = 36 \\), \\( \\gcd(c, d) = 54 \\), and \\( 70 < \\gcd(d, a) < 100 \\). Which of the following numbers is a factor of \\( a \\)? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1} \\) and \\( x_{2} \\) be the largest roots of the polynomials \\( f(x) = 1 - x - 4x^{2} + x^{4} \\) and \\( g(x) = 16 - 8x - 16x^{2} + x^{4} \\), respectively. Find \\( \\frac{x_{1}}{x_{2}} \\). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 27570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If in the expansion of the binomial \\((x- \\frac {2}{ \\sqrt {x}})^{n}\\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \\(x^{2}\\) is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 27571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $62^{2}+122^{2}=18728$, find positive integers $(n, m)$ such that $n^{2}+m^{2}=9364$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(30,92) \\text{ OR } (92,30)"
} | {
"split": "train",
"index": 27572,
"solution": "If $a^{2}+b^{2}=2 c$, then $\\left(\\frac{a+b}{2}\\right)^{2}+\\left(\\frac{a-b}{2}\\right)^{2}=\\frac{2 a^{2}+2 b^{2}}{4}=\\frac{a^{2}+b^{2}}{2}=c$. Thus, $n=\\frac{62+122}{2}=92$ and $m=\\frac{122-62}{2}=30$ works."
} |
deepscaler | [
{
"role": "user",
"content": "A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting t... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 27573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 27574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exists a constant $c,$ so that among all chords $\\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$\n\\[t = \\frac{1}{AC} + \\frac{1}{BC}\\]is a fixed constant. Find the constant $t.$\n\n[asy]\nunitsize(1 cm);\n\nreal parab (real x) {\n return(x^2);\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1 is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4436"
} | {
"split": "train",
"index": 27576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 27577,
"solution": "1. **Identify the dimensions of the squares and rectangles**: \n Let the side length of the inner square be $s$. Assume the shorter side of each rectangle is $y$ and the longer side is $x$. The rectangles are congruent and placed around the inner square such that... |
deepscaler | [
{
"role": "user",
"content": "Define a function $f$ from nonnegative integers to real numbers, with $f(1) = 1$ and the functional equation:\n\\[ f(m+n) + f(m-n) = 3(f(m) + f(n)) \\]\nfor all nonnegative integers $m \\ge n$. Determine $f(10)$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 27578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 3x + 1$, what is $f(3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An array of integers is arranged in a grid of 7 rows and 1 column with eight additional squares forming a separate column to the right. The sequence of integers in the main column of squares and in each of the two rows form three distinct arithmetic sequences. Find the value of... | math | {
"style": "rule",
"ground_truth": "\\frac{-851}{3}"
} | {
"split": "train",
"index": 27580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $u_0 = \\frac{1}{3}$, and for $k \\ge 0$, let $u_{k+1} = \\frac{3}{2}u_k - \\frac{3}{2}u_k^2$. This sequence tends to a limit; call it $M$. Determine the least value of $k$ such that $|u_k - M| \\le \\frac{1}{2^{1000}}$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n$ for which $\\frac{n^{3}+8}{n^{2}-4}$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0,1,3,4,6"
} | {
"split": "train",
"index": 27582,
"solution": "We have $\\frac{n^{3}+8}{n^{2}-4}=\\frac{(n+2)(n^{2}-2n+4)}{(n+2)(n-2)}=\\frac{n^{2}-2n+4}{n-2}$ for all $n \\neq -2$. Then $\\frac{n^{2}-2n+4}{n-2}=n+\\frac{4}{n-2}$, which is an integer if and only if $\\frac{4}{n-2}$ is an integer. This happens when $n-2=-4,-2,-... |
deepscaler | [
{
"role": "user",
"content": "Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 27583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a rectangular sheet of paper, a picture is drawn in the shape of a \"cross\" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. L... | math | {
"style": "rule",
"ground_truth": "52.5"
} | {
"split": "train",
"index": 27584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $A$ if it is known that $A^{6}$ is an eight-digit number composed of the digits $0, 1, 2, 2, 2, 3, 4, 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\\cdots A_{2n \\plus{} 1}$ be a regular $ 2n\\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles. Let... | math | {
"style": "rule",
"ground_truth": "(2n + 1) \\left[ \\binom{n}{m - 1} + \\binom{n + 1}{m - 1} \\right]"
} | {
"split": "train",
"index": 27586,
"solution": "\nGiven two integers \\( m \\) and \\( n \\) satisfying \\( 4 < m < n \\), let \\( A_1A_2\\cdots A_{2n+1} \\) be a regular \\( 2n+1 \\) polygon. Denote by \\( P \\) the set of its vertices. We aim to find the number of convex \\( m \\)-gons whose vertices belong to... |
deepscaler | [
{
"role": "user",
"content": "Given a complex number $z$ that satisfies the following two conditions:\n\n① $1 < z + \\frac{2}{z} \\leqslant 4$.\n\n② The real part and the imaginary part of $z$ are both integers, and the corresponding point in the complex plane is located in the fourth quadrant.\n\n(I) Find ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{65}}{5}"
} | {
"split": "train",
"index": 27587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\\overrightarrow{m}=(\\sin B+\\sin C,\\sin A+\\sin B)$, $\\overrightarrow{n}=(\\sin B-\\sin C,\\sin A)$, and $\\overrightarrow{m}\\perp \\overrightarrow{n}$.\n... | math | {
"style": "rule",
"ground_truth": "2+\\sqrt{3}"
} | {
"split": "train",
"index": 27588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with equation \\( y = 2x + b \\) passes through the point \\((-4, 0)\\). What is the value of \\(b\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27589,
"solution": "Since the line with equation \\( y = 2x + b \\) passes through the point \\((-4, 0)\\), the coordinates of the point must satisfy the equation of the line. Substituting \\(x = -4\\) and \\(y = 0\\) gives \\(0 = 2(-4) + b\\) and so \\(0 = -8 + b\\) which gives \\(b ... |
deepscaler | [
{
"role": "user",
"content": "Jenny wants to distribute 450 cookies among $p$ boxes such that each box contains an equal number of cookies. Each box must contain more than two cookies, and there must be more than one box. For how many values of $p$ can this distribution be made? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the maximum number of bishops that can be placed on the squares of a $6 \\times 6$ chessboard such that no two bishops are attacking each other. Let $k$ be the number of ways to put $n$ bishops on an $6 \\times 6$ chessboard such that no two bishops are attacking eac... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 27591,
"solution": "Color the square with coordinates $(i, j)$ black if $i+j$ is odd and white otherwise, for all $1 \\leq i, j \\leq 6$. Looking at the black squares only, we note that there are six distinct diagonals which run upward and to the right, but that two of them consist on... |
deepscaler | [
{
"role": "user",
"content": "On June 14, 2018, the 21st FIFA World Cup will kick off in Russia. A local sports channel organized fans to guess the outcomes of the matches for the top four popular teams: Germany, Spain, Argentina, and Brazil. Each fan can choose one team from the four, and currently, three ... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{15}"
} | {
"split": "train",
"index": 27592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two ferries leave simultaneously from opposite shores of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest shore. Upon reaching the shore, they immediately depart back. On the retur... | math | {
"style": "rule",
"ground_truth": "1280"
} | {
"split": "train",
"index": 27593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \n$$\nf(x)= \n\\begin{cases} \n1 & (1 \\leq x \\leq 2) \\\\ \n\\frac{1}{2}x^2 - 1 & (2 < x \\leq 3)\n\\end{cases}\n$$\ndefine $h(a) = \\max\\{f(x) - ax \\mid x \\in [1, 3]\\} - \\min\\{f(x) - ax \\mid x \\in [1, 3]\\}$ for any real number $a$. \n1. Find the v... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 27594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}-\\pi"
} | {
"split": "train",
"index": 27595,
"solution": "1. **Identify the Geometry of the Problem:**\n A regular hexagon with side length $s = 1$ is inscribed in a circle. This means that each side of the hexagon subtends a $60^\\circ$ angle at the center of the circle.\n\n2. **Calculate the Radius of the Circle:**\n ... |
deepscaler | [
{
"role": "user",
"content": "A line segment $AB$ with a fixed length of $4$ has its endpoints moving along the positive $x$-axis and the positive $y$-axis, respectively, and $P(x,y)$ is a point on the circumcircle of triangle $OAB$. Find the maximum value of $x+y$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 27596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first two digits of a natural four-digit number are either both less than 5 or both greater than 5. The same condition applies to the last two digits. How many such numbers are there in total? Justify your answer. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1476"
} | {
"split": "train",
"index": 27597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write the digits from 0 to 9 in a line, in any order you choose. On the line below, combine the neighboring digits to form nine new numbers, and sum these numbers as in the example below:\n\n| 2 | | 1 | | 3 | | 7 | | 4 | | 9 | | 5 | | 8 | | 0 | | 6 |\n| :--- | :--- | :... | math | {
"style": "rule",
"ground_truth": "494"
} | {
"split": "train",
"index": 27598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the radius of the inscribed circle of a triangle with sides 15,16 , and 17 . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 27599,
"solution": "Hero's formula gives that the area is $\\sqrt{24 \\cdot 9 \\cdot 8 \\cdot 7}=24 \\sqrt{21}$. Then, using the result that the area of a triangle equals the inradius times half the perimeter, we see that the radius is $\\sqrt{21}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin 2\\alpha = 3\\sin 2\\beta$, calculate the value of $\\frac {\\tan(\\alpha-\\beta)}{\\tan(\\alpha +\\beta )}$. 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": 27600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be positive integers with $a\\ge$ $b\\ge$ $c$ such that\n$a^2-b^2-c^2+ab=2011$ and\n$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.\nWhat is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "253"
} | {
"split": "train",
"index": 27601,
"solution": "1. **Combine the given equations**:\n \\[\n a^2 - b^2 - c^2 + ab + a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = 2011 - 1997\n \\]\n Simplifying this, we get:\n \\[\n 2a^2 + 2b^2 + 2c^2 - 2ab - 2ac - 2bc = 14\n \\]\n\n2. **Factor and simplify**:\n \\[\n (... |
deepscaler | [
{
"role": "user",
"content": "A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 27602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of all fractions $\\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\\%$? Let's think ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 27603,
"solution": "1. **Formulate the equation**: Given that increasing both the numerator and denominator by 1 increases the fraction by 10%, we can write:\n \\[\n \\frac{x+1}{y+1} = 1.1 \\cdot \\frac{x}{y}\n \\]\n Simplifying the right side, we get:\n \\[\n \\frac{x+1}{... |
deepscaler | [
{
"role": "user",
"content": "The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangemen... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 27604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the vertex of angle $\\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \\frac {1}{2}x$ ($x\\leq0$). \n(Ⅰ) Find the value of $\\cos\\left( \\frac {\\pi}{2}+... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {2}}{10}"
} | {
"split": "train",
"index": 27605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have a bag with $6$ marbles numbered from $1$ to $6.$ Mathew has a bag with $12$ marbles numbered from $1$ to $12.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 27606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos \\left( 2x+\\dfrac{\\pi}{3} \\right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\\cos 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{6}"
} | {
"split": "train",
"index": 27607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular field has a length of 20 metres and a width of 5 metres. If its length is increased by 10 m, by how many square metres will its area be increased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 27608,
"solution": "Since the field originally has length 20 m and width 5 m, then its area is $20 \\times 5=100 \\mathrm{~m}^{2}$. The new length of the field is $20+10=30 \\mathrm{~m}$, so the new area is $30 \\times 5=150 \\mathrm{~m}^{2}$. The increase in area is $150-100=50 \\mat... |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=-x+1$ and the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, which intersect at points $A$ and $B$.\n\n(1) If the eccentricity of the ellipse is $\\frac{\\sqrt{3}}{3}$ and the focal length is $2$, find the length of the line segment $AB$.\n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 27609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 27610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the moving point $P$ is on the line $y=x+1$, and the moving point $Q$ is on the curve $x^{2}=-2y$, calculate the minimum value of $|PQ|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 27611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is randomly thrown on the segment [3, 8] and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "4/15"
} | {
"split": "train",
"index": 27612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Solve the inequality $\\log_{\\frac{1}{2}}(x+2) > -3$\n(2) Calculate: $(\\frac{1}{8})^{\\frac{1}{3}} \\times (-\\frac{7}{6})^{0} + 8^{0.25} \\times \\sqrt[4]{2} + (\\sqrt[3]{2} \\times \\sqrt{3})^{6}$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{221}{2}"
} | {
"split": "train",
"index": 27613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sector with a central angle of $\\alpha$ and a radius of $r$. \n$(1)$ If $\\alpha = 60^{\\circ}$ and $r = 3$, find the arc length of the sector. \n$(2)$ If the perimeter of the sector is $16$, at what angle $\\alpha$ will the area of the sector be maximized? Also, fin... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27615,
"solution": "1. **Assign Variables:**\n Let the side length of the second square be $s_2$. Then, the diagonal of this square, which is also the side length of the first square, is denoted as $s_1$.\n\n2. **Relate Diagonal to Side Length:**\n The diagonal of a square can be ... |
deepscaler | [
{
"role": "user",
"content": "Convert -630° to radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\pi}{2}"
} | {
"split": "train",
"index": 27616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the quadrilateral formed by the points of intersection of the circle \\(x^2 + y^2 = 16\\) and the ellipse \\((x-3)^2 + 4y^2 = 36\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then find the number of different arrangements. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 27619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 27620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite sum $$\\sum_{n=2}^{\\infty} \\log _{2}\\left(\\frac{1-\\frac{1}{n}}{1-\\frac{1}{n+1}}\\right)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27621,
"solution": "Using the identity $\\log _{2}\\left(\\frac{a}{b}\\right)=\\log _{2} a-\\log _{2} b$, the sum becomes $$\\sum_{n=2}^{\\infty} \\log _{2}\\left(\\frac{n-1}{n}\\right)-\\sum_{n=2}^{\\infty} \\log _{2}\\left(\\frac{n}{n+1}\\right)$$ Most of the terms cancel out, excep... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{229}{8}"
} | {
"split": "train",
"index": 27622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the constant $t$ such that \\[(5x^2 - 6x + 7)(4x^2 +tx + 10) = 20x^4 -54x^3 +114x^2 -102x +70.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 27623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tower with a height of $8$ cubes, where a blue cube must always be at the top, determine the number of different towers the child can build using $2$ red cubes, $4$ blue cubes, and $3$ green cubes. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 27624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( T \\) be the set of all positive divisors of \\( 60^{100} \\). \\( S \\) is a subset of \\( T \\) such that no number in \\( S \\) is a multiple of another number in \\( S \\). Find the maximum value of \\( |S| \\). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "10201"
} | {
"split": "train",
"index": 27625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number in which each digit, starting from the third, is equal to the sum of all previous digits of the number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101248"
} | {
"split": "train",
"index": 27626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty mice were walking, carrying forty coins. Two less-capable mice carried two coins each. Several mice did not carry any coins at all. The larger ones carried seven coins each. The rest carried four coins each. How many mice were walking without any coins? Let's think step b... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 18 bahs are equal to 30 rahs, and 6 rahs are equivalent to 10 yahs, how many bahs are equal to 1200 yahs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 27628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 27629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers with $x+y=31$, what is the largest possible value of $x y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 27630,
"solution": "First, we note that the values of $x$ and $y$ cannot be equal since they are integers and $x+y$ is odd. Next, we look at the case when $x>y$. We list the fifteen possible pairs of values for $x$ and $y$ and the corresponding values of $x y$. Therefore, the largest ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin^{2}x+\\sqrt{3}\\sin x\\sin (x+\\frac{\\pi}{2})$.\n\n1. Find the value of $f(\\frac{\\pi}{12})$;\n2. Find the maximum and minimum values of the function $f(x)$ when $x\\in[0,\\frac{\\pi}{2}]$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 27631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x)$ be a third-degree polynomial with real coefficients satisfying \\[|g(0)|=|g(1)|=|g(3)|=|g(4)|=|g(5)|=|g(8)|=10.\\] Find $|g(2)|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $1010101_2 + 1001001_2$? Write your answer in base $10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 27633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "-x + 118"
} | {
"split": "train",
"index": 27634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the vo... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2} - 7"
} | {
"split": "train",
"index": 27635,
"solution": "1. **Identify the Geometry of the Pyramid and Cube**: The pyramid has a square base with side length 1 and lateral faces that are equilateral triangles. A cube is placed inside such that one face is on the base of the pyramid and its opposite face has all its edges... |
deepscaler | [
{
"role": "user",
"content": "Given point \\( A(2,0) \\), point \\( B \\) lies on the curve \\( y = \\sqrt{1 - x^2} \\), and the triangle \\( \\triangle ABC \\) is an isosceles right triangle with \\( A \\) as the right angle vertex. Determine the maximum value of \\( |OC| \\). Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 1"
} | {
"split": "train",
"index": 27636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After a fair die with faces numbered 1 to 6 is rolled, the number on the top face is $x$. What is the most likely outcome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x > 2"
} | {
"split": "train",
"index": 27637,
"solution": "With a fair die that has faces numbered from 1 to 6, the probability of rolling each of 1 to 6 is $\\frac{1}{6}$. We calculate the probability for each of the five choices. There are 4 values of $x$ that satisfy $x>2$, so the probability is $\\frac{4}{6}=\\frac{2}{... |
deepscaler | [
{
"role": "user",
"content": "How can you weigh 1 kg of grain on a balance scale using two weights, one weighing 300 g and the other 650 g? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 27638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic polynomial $P(x),$ with real coefficients, satisfies\n\\[P(x^3 + x) \\ge P(x^2 + 1)\\]for all real numbers $x.$ Find the sum of the roots of $P(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In daily life, specific times are usually expressed using the 24-hour clock system. There are a total of 24 time zones globally, with adjacent time zones differing by 1 hour. With the Prime Meridian located in Greenwich, England as the reference point, in areas east of Greenwic... | math | {
"style": "rule",
"ground_truth": "10:00"
} | {
"split": "train",
"index": 27640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \\[(*)\\quad\\frac{S_{m+n}}{m+n}=\\frac{S_m}{m}\\frac{S_n}{n}\\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real number... | math | {
"style": "rule",
"ground_truth": "(2, 3), (3, 2), (2, 5), (5, 2)"
} | {
"split": "train",
"index": 27641,
"solution": "\nLet's start by understanding the problem statement correctly. We have a sequence defined by \n\\[ S_r = x^r + y^r + z^r \\]\nwhere \\( x, y, \\) and \\( z \\) are real numbers. We are informed that if \\( S_1 = x + y + z = 0 \\), then the following relationship h... |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n \\geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \\hdots, a_n$ with $\\text{max}(a_1,a_2,\\hdots,a_n) \\leq n \\cdot \\text{min}(a_1,a_2,\\hdots,a_n)$, there exist three that are the side lengths of an acute triangle. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "n \\geq 13"
} | {
"split": "train",
"index": 27642,
"solution": "\nTo solve this problem, we need to determine for which integers \\( n \\geq 3 \\), any set of \\( n \\) positive real numbers \\( a_1, a_2, \\ldots, a_n \\), under the condition \\( \\max(a_1, a_2, \\ldots, a_n) \\leq n \\cdot \\min(a_1, a_2, \\ldots, a_n) \\), co... |
deepscaler | [
{
"role": "user",
"content": "A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\\pi$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"split": "train",
"index": 27643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a new infinite geometric series: $$\\frac{7}{4} + \\frac{28}{9} + \\frac{112}{27} + \\dots$$\nDetermine the common ratio of this series. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 27644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.\nTom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding... | math | {
"style": "rule",
"ground_truth": "1866"
} | {
"split": "train",
"index": 27645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\\frac{4}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 27646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 27647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\\cos B=b\\cos C$. \n(Ⅰ) Find the magnitude of angle $B$; \n(Ⅱ) If $a=2$ and $c=3$, find the value of $\\sin C$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {21}}{14}"
} | {
"split": "train",
"index": 27648,
"solution": ""
} |
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