data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given squares $ABCD$ and $EFGH$ are congruent, $AB=12$, and $H$ is located at vertex $D$ of square $ABCD$. Calculate the total area of the region in the plane covered by these squares."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 14400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition."
}
] | math | {
"style": "rule",
"ground_truth": "24068"
} | {
"index": 14401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given tetrahedron $P-ABC$, if one line is randomly selected from the lines connecting the midpoints of each edge, calculate the probability that this line intersects plane $ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\n\\[\n\\frac{1}{x} + \\frac{1}{y} = 4, \\quad x + y = 5,\n\\]\n\ncompute \\(x^2 + y^2\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{2}"
} | {
"index": 14403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\n\\begin{align*}\n\\frac{1}{x}+\\frac{1}{y}&=3,\\\\\nxy+x+y&=4,\n\\end{align*}\n\ncompute $x^2y+xy^2$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n$$\n\\begin{array}{l}\na + b + c = 5, \\\\\na^2 + b^2 + c^2 = 15, \\\\\na^3 + b^3 + c^3 = 47.\n\\end{array}\n$$\nFind the value of \\((a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 14405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n$\\frac 1{2!17!}+\\frac 1{3!16!}+\\frac 1{4!15!}+\\frac 1{5!14!}+\\frac 1{6!13!}+\\frac 1{7!12!}+\\frac 1{8!11!}+\\frac 1{9!10!}=\\frac N{1!18!}$\n find the greatest integer that is less than $\\frac N{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 14406,
"solution": "Let $f(x) = (1+x)^{19}.$ Applying the binomial theorem gives us $f(x) = \\binom{19}{19} x^{19} + \\binom{19}{18} x^{18} + \\binom{19}{17} x^{17}+ \\cdots + \\binom{19}{0}.$ Since $\\frac 1{2!17!}+\\frac 1{3!16!}+\\dots+\\frac 1{8!11!}+\\frac 1{9!10!} = \\frac{\\frac{f(1)}{2} - \\binom... |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n$\\frac 1{2!17!}+\\frac 1{3!16!}+\\frac 1{4!15!}+\\frac 1{5!14!}+\\frac 1{6!13!}+\\frac 1{7!12!}+\\frac 1{8!11!}+\\frac 1{9!10!}=\\frac N{1!18!}$\nfind the greatest integer that is less than $\\frac N{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 14407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n$\\frac{((3!)!)!}{3!} = k \\cdot n!,$\n where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n.$"
}
] | math | {
"style": "rule",
"ground_truth": "839"
} | {
"index": 14408,
"solution": "Note that\\[{{\\left((3!)!\\right)!}\\over{3!}}= {{(6!)!}\\over{6}}={{720!}\\over6}={{720\\cdot719!}\\over6}=120\\cdot719!.\\]Because $120\\cdot719!<720!$, conclude that $n$ must be less than 720, so the maximum value of $n$ is 719. The requested value of $k+n$ is therefore $120+719=\... |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\\[\n2^{-\\frac{5}{3} + \\sin 2\\theta} + 2 = 2^{\\frac{1}{3} + \\sin \\theta},\n\\]\ncompute \\(\\cos 2\\theta.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\\[2^{-\\frac{3}{2} + 2 \\cos \\theta} + 1 = 2^{\\frac{1}{4} + \\cos \\theta},\\]compute $\\cos 2 \\theta.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 14410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\\begin{align*}x_{1}&=211,\\\\ x_{2}&=375,\\\\ x_{3}&=420,\\\\ x_{4}&=523,\\ \\text{and}\\\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\\ \\text{when}\\ n\\geq5, \\end{align*}\n find the value of $x_{531}+x_{753}+x_{975}$."
}
] | math | {
"style": "rule",
"ground_truth": "898"
} | {
"index": 14411,
"solution": "Calculate the first few terms:\n\\[211,375,420,523,267,-211,-375,-420,-523,\\dots\\]\n At this point it is pretty clear that the sequence is periodic with period 10 (one may prove it quite easily like in solution 1) so our answer is obviously $211+420+267=\\boxed{898}$\n ~Dhillonr25\n... |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\\begin{eqnarray*}&(1)& x\\text{ and }y\\text{ are both integers between 100 and 999, inclusive;}\\qquad \\qquad \\qquad \\qquad \\qquad \\\\ &(2)& y\\text{ is the number formed by reversing the digits of }x\\text{; and}\\\\ &(3)& z=|x-y|. \\end{eqnarray*}\n... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 14412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that\n\\begin{eqnarray*}&(1)& x\\text{ and }y\\text{ are both integers between 100 and 999, inclusive;}\\qquad \\qquad \\qquad \\qquad \\qquad \\\\ &(2)& y\\text{ is the number formed by reversing the digits of }x\\text{; and}\\\\ &(3)& z=|x-y|. \\end{eqnarray*}\n How man... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 14413,
"solution": "We express the numbers as $x=100a+10b+c$ and $y=100c+10b+a$. From this, we have \\begin{eqnarray*}z&=&|100a+10b+c-100c-10b-a|\\\\&=&|99a-99c|\\\\&=&99|a-c|\\\\ \\end{eqnarray*} Because $a$ and $c$ are digits, and $a$ and $c$ are both between 1 and 9 (from condition 1), there are $\\bo... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $ 2^{2004}$ is a $ 604$ -digit number whose first digit is $ 1$ , how many elements of the set $ S \\equal{} \\{2^0,2^1,2^2, \\ldots,2^{2003}\\}$ have a first digit of $ 4$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 14414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $ a,b,c,d$ are rational numbers with $ a>0$ , find the minimal value of $ a$ such that the number $ an^{3} + bn^{2} + cn + d$ is an integer for all integers $ n \\ge 0$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 14415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\\leq n\\leq2017$ does $5^n$ start with $1$ ?\n\n*2018 CCA Math Bonanza Tiebreaker Round #3*"
}
] | math | {
"style": "rule",
"ground_truth": "607"
} | {
"index": 14416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\log_{10}2 \\approx 0.30103$ , find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 14417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan A=\\frac{12}{5}$ , $\\cos B=-\\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\\cos (A-B)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{63}{65}$"
} | {
"index": 14418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tbinom{n}{k}=\\tfrac{n!}{k!(n-k)!}$ , the value of $$ \\sum_{n=3}^{10}\\frac{\\binom{n}{2}}{\\binom{n}{3}\\binom{n+1}{3}} $$ can be written in the form $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "329"
} | {
"index": 14419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of $$ |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \\dots + |a_{99} - 99|. $$"
}
] | math | {
"style": "rule",
"ground_truth": "4900"
} | {
"index": 14420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $ \\sin \\left( \\frac{\\pi}{3} - \\frac{\\alpha}{2} \\right) = -\\frac{\\sqrt{3}}{2} $, find the value of $ \\cos \\left( \\frac{\\pi}{3} + \\alpha \\right) $."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 14422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 14423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$\\sqrt {0.1587}$$≈0.3984, $$\\sqrt {1.587}$$≈1.260, $$\\sqrt[3]{0.1587}$$≈0.5414, $$\\sqrt[3]{1.587}$$≈1.166, can you find without using a calculator:\n(1) $$\\sqrt {15.87}$$≈\\_\\_\\_\\_\\_\\_;\n(2) - $$\\sqrt {0.001587}$$≈\\_\\_\\_\\_\\_\\_;\n(3) $$\\sqrt[3]{1.58... | math | {
"style": "rule",
"ground_truth": "0.05414"
} | {
"index": 14424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$cos(α- \\frac {π}{6})-sinα= \\frac {2 \\sqrt {3}}{5}$$, find the value of $$cos(α+ \\frac {7π}{6})$$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {2 \\sqrt {3}}{5}"
} | {
"index": 14425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$x^{5}=a_{0}+a_{1}(2-x)+a_{2}(2-x)^{2}+…+a_{5}(2-x)^{5}$$, find the value of $$\\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}}$$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {61}{60}"
} | {
"index": 14426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $$α∈(0， \\frac {π}{3})$$ and vectors $$a=( \\sqrt {6}sinα， \\sqrt {2})$$, $$b=(1，cosα- \\frac { \\sqrt {6}}{2})$$ are orthogonal,\n\n(1) Find the value of $$tan(α+ \\frac {π}{6})$$;\n\n(2) Find the value of $$cos(2α+ \\frac {7π}{12})$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}- \\sqrt {30}}{8}"
} | {
"index": 14427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(1+\\sin t)(1+\\cos t)=5/4$ and\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\n where $k, m,$ and $n_{}$ are positive integers with $m_{}$ and $n_{}$ relatively prime, find $k+m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 14428,
"solution": "We want $1+\\sin t \\cos t-\\sin t-\\cos t$. However, note that we only need to find $\\sin t+\\cos t$.\n Let $y = \\sin t+\\cos t \\rightarrow y^2 = \\sin^2 t + \\cos^2 t + 2\\sin t \\cos t = 1 + 2\\sin t \\cos x$\n From this we have $\\sin t \\cos t = \\frac{y^2-1}{2}$ and $\\sin t ... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(1+\\sin t)(1+\\cos t)=5/4$ and\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\nwhere $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 14429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(1-3x)^6 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6$, find the total sum of elements in all subsets containing 2 elements of the set $\\{a_1, a_2, a_3, a_4, a_5, a_6\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 14430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(2-x)^{5}=a\\_{0}+a\\_{1}x+a\\_{2}x^{2}+…+a\\_{5}x^{5}$, find the value of $\\frac{a\\_0+a\\_2+a\\_4}{a\\_1+a\\_3}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{61}{60}"
} | {
"index": 14431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:\n(1) $a_0+a_1+...+a_5$;\n(2) $|a_0|+|a_1|+...+|a_5|$;\n(3) $a_1+a_3+a_5$;\n(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"index": 14432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(a + 1)x^2 + (a^2 + 1) + 8x = 9$ is a quadratic equation in terms of $x$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(a + \\frac{1}{a})^3 = 3$, find the value of $a^4 + \\frac{1}{a^4}$.\nA) $9^{1/3} - 4 \\cdot 3^{1/3} + 2$\nB) $9^{1/3} - 2 \\cdot 3^{1/3} + 2$\nC) $9^{1/3} + 4 \\cdot 3^{1/3} + 2$\nD) $4 \\cdot 3^{1/3} - 9^{1/3}$"
}
] | math | {
"style": "rule",
"ground_truth": "9^{1/3} - 4 \\cdot 3^{1/3} + 2"
} | {
"index": 14434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(a+1)x - 1 - \\ln x \\leqslant 0$ holds for any $x \\in [\\frac{1}{2}, 2]$, find the maximum value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 2\\ln 2"
} | {
"index": 14435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $(a_n)_{n \\equal{} 1}^\\infty$ is defined on real numbers with $a_n \\not \\equal{} 0$, $a_na_{n \\plus{} 3} = a_{n \\plus{} 2}a_{n \\plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \\cdots + a_{41}a_{42}$."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 14436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"index": 14437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $({x}^{2}+1){(2x+1)}^{9}={a}_{0}+{a}_{1}(x+2)+{a}_{2}{(x+2)}^{2}+\\cdots +{a}_{11}{(x+2)}^{11}$, find the value of $({a}_{0}+{a}_{1}+{a}_{2}+\\cdots +{a}_{11})$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-1 - 4\\sqrt{2}$ is a root of the equation \\[x^3 + ax^2 + bx + 31 = 0\\]and that $a$ and $b$ are rational numbers, compute $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 14439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-4\\leq x\\leq-2$ and $2\\leq y\\leq4$, what is the largest possible value of $\\frac{x+y}{x}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-6 \\leq x \\leq -3$ and $1 \\leq y \\leq 5$, what is the largest possible value of $\\frac{x+y}{x}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 14441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-7$ is a solution to $x^2 + bx -28 = 0$, what is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-9, a_1, a_2, -1$ form an arithmetic sequence and $-9, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $b_2(a_2 - a_1)$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 14443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-\\frac{\\pi}{2} < x < 0$ and $\\sin x + \\cos x = \\frac{1}{5}$.\n\n(I) Find the value of $\\sin x - \\cos x$.\n\n(II) Find the value of $\\frac{3\\sin^2{\\frac{x}{2}} - 2\\sin{\\frac{x}{2}}\\cos{\\frac{x}{2}} + \\cos^2{\\frac{x}{2}}}{\\tan x + \\cot x}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{108}{125}"
} | {
"index": 14444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $-\\frac{\\pi}{2}<\\alpha<\\frac{\\pi}{2}, 2 \\tan \\beta=\\tan 2\\alpha, \\tan (\\beta-\\alpha)=-2 \\sqrt{2}$, find the value of $\\cos \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 14445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < \\alpha < \\frac{\\pi}{2}$ and $0 < \\beta < \\frac{\\pi}{2}$, if $\\sin\\left(\\frac{\\pi}{3}-\\alpha\\right) = \\frac{3}{5}$ and $\\cos\\left(\\frac{\\beta}{2} - \\frac{\\pi}{3}\\right) = \\frac{2\\sqrt{5}}{5}$, \n(I) find the value of $\\sin \\alpha$; \n(I... | math | {
"style": "rule",
"ground_truth": "\\frac{11\\sqrt{5}}{25}"
} | {
"index": 14446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < \\alpha < \\frac{\\pi}{2}$, $-\\frac{\\pi}{2} < \\beta < 0$, $\\cos\\left(\\frac{\\pi}{4}+\\alpha\\right) = \\frac{1}{3}$, $\\cos\\left(\\frac{\\pi}{4}-\\frac{\\beta}{2}\\right) = \\frac{\\sqrt{3}}{3}$, \nfind \n$(1)$ the value of $\\cos \\alpha$; \n$(2)$ the... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{9}"
} | {
"index": 14447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < \\alpha < \\pi$ and $\\cos{\\alpha} = -\\frac{3}{5}$, find the value of $\\tan{\\alpha}$ and the value of $\\cos{2\\alpha} - \\cos{(\\frac{\\pi}{2} + \\alpha)}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{25}"
} | {
"index": 14448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < a < \\pi, \\tan a=-2$,\n\n(1) Find the value of $\\cos a$;\n\n(2) Find the value of $2\\sin^{2}a - \\sin a \\cos a + \\cos^{2}a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{5}"
} | {
"index": 14449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < a \\leqslant \\frac{5}{4}$, find the range of real number $b$ such that all real numbers $x$ satisfying the inequality $|x - a| < b$ also satisfy the inequality $|x - a^2| < \\frac{1}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 14450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < x < \\frac{1}{2}$, find the minimum and maximum value of the function $x^{2}(1-2x)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{27}"
} | {
"index": 14451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < x < \\frac{\\pi}{2}$ and $\\sin(2x - \\frac{\\pi}{4}) = -\\frac{\\sqrt{2}}{10}$, find the value of $\\sin x + \\cos x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{10}}{5}"
} | {
"index": 14452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, $- \\dfrac {π}{2} < β < 0$, $\\cos ( \\dfrac {π}{4}+α)= \\dfrac {1}{3}$, and $\\cos ( \\dfrac {π}{4}-β)= \\dfrac { \\sqrt {3}}{3}$, find $\\cos (α+β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {5 \\sqrt {3}}{9}"
} | {
"index": 14453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, $\\sin α= \\dfrac {4}{5}$, and $\\tan (α-β)=- \\dfrac {1}{3}$, find the value of $\\tan β$ and compute the expression $\\dfrac {\\sin (2β- \\dfrac {π}{2})\\cdot \\sin (β+π)}{ \\sqrt {2}\\cos (β+ \\dfrac {π}{4})}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {6}{5}"
} | {
"index": 14454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, and $\\cos ( \\dfrac {π}{3}+α)= \\dfrac {1}{3}$, find the value of $\\cos α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {2 \\sqrt {6}+1}{6}"
} | {
"index": 14455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, and $\\cos (2π-α)-\\sin (π-α)=- \\dfrac { \\sqrt {5}}{5}$,\n(1) Find the value of $\\sin α+\\cos α$;\n(2) Find the value of $\\sin (2α- \\dfrac {π}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {7 \\sqrt {2}}{10}"
} | {
"index": 14456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac{\\pi }{2}$, $-\\dfrac{\\pi }{2} < β < 0$, $\\cos \\left( \\dfrac{\\pi }{4}+α \\right)=\\dfrac{1}{3}$, and $\\cos \\left( \\dfrac{\\pi }{4}-\\dfrac{\\beta }{2} \\right)=\\dfrac{\\sqrt{3}}{3}$, calculate the value of $\\cos \\left( α +\\dfrac{\\beta }{... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\sqrt{3}}{9}"
} | {
"index": 14457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\frac {π}{2}$, and $\\cos (2π-α)-\\sin (π-α)=- \\frac { \\sqrt {5}}{5}$.\n(1) Find the value of $\\sin α+\\cos α$\n(2) Find the value of $\\frac {2\\sin α\\cos α-\\sin ( \\frac {π}{2}+α)+1}{1-\\cot ( \\frac {3π}{2}-α)}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {5}-9}{5}"
} | {
"index": 14458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < α < \\frac{π}{2}$ and $\\frac{π}{2} < β < π$, with $\\cos(α + \\frac{π}{4}) = \\frac{1}{3}$ and $\\cos(\\frac{π}{4} - \\frac{β}{2}) = \\frac{\\sqrt{3}}{3}$,\n\n1. Find the value of $\\cos β$;\n2. Find the value of $\\cos(2α + β)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 < β < \\dfrac{π}{2} < α < π$, and $\\cos (α- \\dfrac{β}{2} )= \\dfrac{5}{13} $, $\\sin ( \\dfrac{α}{2}-β)= \\dfrac{3}{5} $. Find the values of:\n$(1) \\tan (α- \\dfrac{β}{2} )$\n$(2) \\cos ( \\dfrac{α+β}{2} )$"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{56}{65}"
} | {
"index": 14460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0 \\leqslant x \\leqslant 2$, find the minimum and maximum values of the function $f(x) = 4^{x - \\frac{1}{2}} - 3 \\cdot 2^x + 5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 14461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $0\\le x_3 \\le x_2 \\le x_1\\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\\frac{1}{4},$ find $x_1.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $1 \\leq x \\leq 2$, calculate the number of different integer values of the expression $10-10|2x-3|$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $11^{-1} \\equiv 3 \\pmod{31}$, find $20^{-1} \\pmod{31}$. Provide the result as a residue modulo 31 (a number between 0 and 30, inclusive)."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 14464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $13^{-1} \\equiv 29 \\pmod{47}$, find $34^{-1} \\pmod{47}$, as a residue modulo 47. (Give a number between 0 and 46, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 14465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $15^{-1} \\equiv 31 \\pmod{53}$, find $38^{-1} \\pmod{53}$, as a residue modulo 53."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 14466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 11 \\pmod{53}$, find $36^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 14467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 26 \\pmod{53}$, find $36^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 14468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 31 \\pmod{53}$, find $36^{-1} \\pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive)."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 14469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2 - 5\\sqrt{3}$ is a root of the equation \\[x^3 + ax^2 + bx - 48 = 0\\] and that $a$ and $b$ are rational numbers, compute $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{332}{71}"
} | {
"index": 14470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2$ boys and $4$ girls are lined up, calculate the probability that the boys are neither adjacent nor at the ends."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 14471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2+\\sqrt{3}$ is a root of the equation \\[x^3 + ax^2 + bx + 10 = 0\\]and that $a$ and $b$ are rational numbers, compute $b.$"
}
] | math | {
"style": "rule",
"ground_truth": "-39"
} | {
"index": 14472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $21^{-1} \\equiv 15 \\pmod{61}$, find $40^{-1} \\pmod{61}$, as a residue modulo 61. (Provide a number between 0 and 60, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 14473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $21^{-1} \\equiv 17 \\pmod{53}$, find $32^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2^x+ 2^x+ 2^x+ 2^x= 128$, what is the value of $(x + 1)(x - 1)$?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 14475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2^x+ 2^x+ 2^x+ 2^x= 512$, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 14476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2ax^3 - 3bx + 8 = 18$ when $x = -1$, determine the value of $9b - 6a + 2$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2x + 5y = 20$ and $5x + 2y = 26$, find $20x^2 + 60xy + 50y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59600}{49}"
} | {
"index": 14478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $2x + y = 4$ and $x + 2y = 5$, find $5x^2 + 8xy + 5y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 14479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $3\\sin \\alpha + 4\\cos \\alpha = 5$.\n(1) Find the value of $\\tan \\alpha$;\n(2) Find the value of $\\cot (\\frac{3\\pi}{2} - \\alpha) \\cdot \\sin^2 (\\frac{3\\pi}{2} + \\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"index": 14480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $3\\sin \\alpha - 2\\cos \\alpha = 0$, find the value of the following expressions:\n$$(1)\\ \\frac{\\cos \\alpha - \\sin \\alpha}{\\cos \\alpha + \\sin \\alpha} + \\frac{\\cos \\alpha + \\sin \\alpha}{\\cos \\alpha - \\sin \\alpha};$$\n$$(2)\\ \\sin^2\\alpha - 2\\si... | math | {
"style": "rule",
"ground_truth": "\\frac{28}{13}"
} | {
"index": 14481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $3^8 \\cdot 5^2 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "407"
} | {
"index": 14482,
"solution": "1. **Express the given equation in terms of $a$ and $b$:**\n Given the equation $3^8 \\cdot 5^2 = a^b$, we need to find $a$ and $b$ such that $a^b$ equals $3^8 \\cdot 5^2$.\n\n2. **Simplify the expression:**\n \\[\n 3^8 \\cdot 5^2 = (3^4)^2 \\cdot 5^2 = (81)^2 \\cdot 25 = 81 \\c... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $3x + y = 10$ and $x + 3y = 14$, find $10x^2 + 12xy + 10y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"index": 14483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $40\\%$ of students initially answered \"Yes\", $40\\%$ answered \"No\", and $20\\%$ were \"Undecided\", and $60\\%$ answered \"Yes\" after a semester, $30\\%$ answered \"No\", and $10\\%$ remained \"Undecided\", determine the difference between the maximum and minim... | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"index": 14484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $40\\%$ of the birds were geese, $20\\%$ were swans, $15\\%$ were herons, and $25\\%$ were ducks, calculate the percentage of the birds that were not ducks and were geese."
}
] | math | {
"style": "rule",
"ground_truth": "53.33\\%"
} | {
"index": 14485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $40\\%$ of the birds were pigeons, $20\\%$ were sparrows, $15\\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows."
}
] | math | {
"style": "rule",
"ground_truth": "18.75\\%"
} | {
"index": 14486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $4:5 = 20 \\div \\_\\_\\_\\_\\_\\_ = \\frac{()}{20} = \\_\\_\\_\\_\\_\\_ \\%$, find the missing values."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 14487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $4^{-1} \\equiv 57 \\pmod{119}$, find $64^{-1} \\pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 14488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 14489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $60\\%$ of the high school students like ice skating, $50\\%$ like skiing, and $70\\%$ like either ice skating or skiing, calculate the probability that a randomly selected student who likes skiing also likes ice skating."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"index": 14490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $62^{2}+122^{2}=18728$, find positive integers $(n, m)$ such that $n^{2}+m^{2}=9364$."
}
] | math | {
"style": "rule",
"ground_truth": "(30,92) \\text{ OR } (92,30)"
} | {
"index": 14491,
"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": "Given that $689\\Box\\Box\\Box20312 \\approx 69$ billion (rounded), find the number of ways to fill in the three-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 14492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $7^{-1} \\equiv 55 \\pmod{101}$, find $49^{-1} \\pmod{101}$, as a residue modulo 101 (Give an answer between 0 and 100, inclusive)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 14493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $7^{-1} \\equiv 55 \\pmod{78}$, find $49^{-1} \\pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 14494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $8!=40320$, what is the value of $8!\\div3!$?"
}
] | math | {
"style": "rule",
"ground_truth": "6720"
} | {
"index": 14495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $8^{-1} \\equiv 85 \\pmod{97}$, find $64^{-1} \\pmod{97}$, as a residue modulo 97. (Give an answer between 0 and 96, inclusive.)"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 14496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A = (\\sqrt{2008}+\\sqrt{2009}),$ $B = (-\\sqrt{2008}-\\sqrt{2009}),$ $C = (\\sqrt{2008}-\\sqrt{2009}),$ and $D = (\\sqrt{2009}-\\sqrt{2008}),$ find $ABCD.$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 14497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$ is an interior angle of $\\triangle ABC$, when $x= \\frac {5\\pi}{12}$, the function $f(x)=2\\cos x\\sin (x-A)+\\sin A$ attains its maximum value. The sides opposite to the angles $A$, $B$, $C$ of $\\triangle ABC$ are $a$, $b$, $c$ respectively. \n$(1)$ Find the... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"index": 14498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, $C$ are the three internal angles of $\\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\\cos ^{2} \\frac {A}{2}+\\cos A=0$.\n(1) Find the value of angle $A$;\n(2) If $a=2 \\sqrt {3},b+c=4$, find the area of $\\triangle ABC$."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 14499,
"solution": ""
} |
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