data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{1, a, \\frac{b}{a}\\} = \\{0, a^2, a+b\\}$, find the value of $a^{2015} + b^{2014}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{1, a, \\frac{b}{a}\\} = \\{0, a^2, a+b\\}$, find the value of $a^{2017} + b^{2017}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a\\_n\\}$ is a geometric sequence, $a\\_2=2$, $a\\_6=162$, find $a\\_{10}$ = $\\_\\_\\_\\_\\_\\_$ ."
}
] | math | {
"style": "rule",
"ground_truth": "13122"
} | {
"index": 14802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a geometric sequence and $S_n$ is the sum of the first $n$ terms, with $S_3=2$ and $S_6=6$, find the value of $a_{13}+a_{14}+a_{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.\n\n(Ⅰ) Find the value of $q$; \n(Ⅱ) Let the sum of the first $n$ terms of the sequence $\\{a_n\\}$ be $S_n$. Determine whether $S_m$, $S_{m+2... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 14804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.\n\n(Ⅰ) Find the value of $q$; \n(Ⅱ) Let the sum of the first $n$ terms of the sequence $\\{a_n\\}$ be $S_n$. Determine whether $S_m$, $S_{m+2... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 14805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a geometric sequence, and $S_n$ is the sum of its first $n$ terms. If $a_2 \\cdot a_3 = 2a_1$, and the arithmetic mean of $a_4$ and $2a_7$ is $\\frac{5}{4}$, then $S_6 = \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{2}"
} | {
"index": 14806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a sequence of positive numbers, and the sum of its first $n$ terms $2S_n = a_n^2 + a_n$ ($n \\in \\mathbb{N}^*$), the sequence $\\{b_n\\}$ satisfies $b_1 = \\frac{3}{2}$, $b_{n+1} = b_n + 3^{a_n}$ ($n \\in \\mathbb{N}^*$).\n\n(I) Find the general formu... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence with common difference $d$, and $S_n$ is the sum of the first $n$ terms, if only $S_4$ is the minimum term among $\\{S_n\\}$, then the correct conclusion(s) can be drawn is/are ______.\n\n$(1) d > 0$ $(2) a_4 < 0$ $(3) a_5 >... | math | {
"style": "rule",
"ground_truth": "(1)(2)(3)(4)"
} | {
"index": 14808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence with the first term $a_1 > 0$, and $a_{1007} + a_{1008} > 0$, $a_{1007} \\cdot a_{1008} < 0$, determine the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"index": 14809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \\cdot a_{24} < 0$, determine the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 14810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, if $\\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 14811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_5+a_6 > 0$, and $a_5a_6 < 0$, calculate the maximum natural number $n$ for which the sum of the first $n$ terms $S_n > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_{n}\\}$ is a geometric sequence, and $a_{3}$ and $a_{7}$ are two roots of the equation $x^{2}+4x+1=0$, calculate the value of $a_{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_{n}\\}$ is an arithmetic progression, $\\{b_{n}\\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$. \n$(1)$ Find the general formulas for $\\{a_{n}\\}$ and $\\{b_{n}\\}$. \n$(2)$ Arrange the terms of $\\{a_{n}\\}$ and $\\{b_{n}\\}$ in a... | math | {
"style": "rule",
"ground_truth": "2726"
} | {
"index": 14815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\{a_{n}\\}$ is an arithmetic sequence, with the sum of its first $n$ terms denoted as $S_{n}$, and $a_{4}=-3$, choose one of the following conditions as known: <br/>$(Ⅰ)$ The arithmetic sequence $\\{a_{n}\\}$'s general formula; <br/>$(Ⅱ)$ The minimum value of $S_{n... | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"index": 14816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a = \\sin (2015\\pi - \\frac {\\pi}{6})$ and the function $f(x) = \\begin{cases} a^{x}, & x > 0 \\\\ f(-x), & x < 0 \\end{cases}$, calculate the value of $f(\\log_{2} \\frac {1}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{6}"
} | {
"index": 14817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$ and $b > 0$, if the inequality $\\frac{3}{a} + \\frac{1}{b} \\geq \\frac{m}{a + 3b}$ always holds true, find the maximum value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$ and $b > 0$, they satisfy the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 14819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \\frac{a^{2}+1}{ab} - 2) \\cdot c + \\frac{\\sqrt{2}}{c - 1}$."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"index": 14820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $2a+b=1$, find the maximum value of $2 \\sqrt {ab}-4a^{2}-b^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {2}-1}{2}"
} | {
"index": 14821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $4a - b \\geq 2$, find the maximum value of $\\frac{1}{a} - \\frac{1}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $\\frac{1}{a} + \\frac{2}{b} = 2$.\n(1) Find the minimum value of $ab$;\n(2) Find the minimum value of $a + 2b$, and find the corresponding values of $a$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 14823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $\\frac{1}{a}$, $\\frac{1}{2}$, $\\frac{1}{b}$ form an arithmetic sequence, find the minimum value of $a+9b$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $a + b = 1$, find the maximum value of $(-\\frac{1}{2a} - \\frac{2}{b})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"index": 14825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $a + b = 1$, find the minimum value of $\\frac{2}{a} + \\frac{3}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "5 + 2\\sqrt{6}"
} | {
"index": 14826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, if $f(g(a)) = 18$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2\\sqrt{2} + 6}"
} | {
"index": 14827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 14828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0$, if the area enclosed by the curve $y=\\sqrt{x}$ and the lines $x=a$, $y=0$ is equal to $a$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 14829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 0, b > 0$ and $a+b=1$, find the minimum value of $\\frac{1}{a} + \\frac{2}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 14830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\\frac{2}{a - 1} + \\frac{a}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "4(1 + \\sqrt{2})"
} | {
"index": 14831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > 2b$ ($a, b \\in \\mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\\infty)$. Determine the minimum value of $$\\frac{a^2 + 4b^2}{a - 2b}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a > b > 0$, and $a + b = 2$, find the minimum value of $$\\frac {3a-b}{a^{2}+2ab-3b^{2}}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {5}}{4}"
} | {
"index": 14833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a \\in \\mathbb{R}$, if the real part and the imaginary part of the complex number $\\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\\_\\_\\_\\_\\_\\_$, $| \\overline{z}| = \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a \\in \\{0,1,2\\}$ and $b \\in \\{-1,1,3,5\\}$, the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function in the interval $(1, +\\infty)$ is $(\\quad\\quad)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b \\in R$, the function $f(x) = \\ln(x + 1) - 2$ is tangent to the line $y = ax + b - \\ln2$ at $x = -\\frac{1}{2}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \\leqslant g(x) \\leqslant m^2 - 2$ holds true in the interval $[1, 2]$, determine the real... | math | {
"style": "rule",
"ground_truth": "e + 1"
} | {
"index": 14836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are both positive real numbers, and $\\frac{1}{a} + \\frac{1}{b} = 2$, find the maximum value of $\\frac{1}{b}(\\frac{2}{a} + 1)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"index": 14837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are coefficients, and the difference between $ax^2+2xy-x$ and $3x^2-2bxy+3y$ does not contain a quadratic term, find the value of $a^2-4b$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 14838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are constants, and $a \\neq 0$, $f(x) = ax^2 + bx$, $f(2) = 0$.\n\n$(1)$ If the equation $f(x) - x = 0$ has a unique real root, find the expression for the function $f(x)$;\n\n$(2)$ When $a = 1$, find the maximum and minimum values of the function $f(x)$ ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 14840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the minimum value of $\\frac{1}{2a} + \\frac{1}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\frac{3}{2}"
} | {
"index": 14841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the value of $a$ when $a=$____, such that the minimum value of the algebraic expression $\\frac{{2{a^2}+1}}{{ab}}-2$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 14842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive real numbers satisfying $9a^{2}+b^{2}=1$, find the maximum value of $\\frac{ab}{3a+b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"index": 14843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive real numbers satisfying $a + 2b = 1$, find the minimum value of $a^2 + 4b^2 + \\frac{1}{ab}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"index": 14844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are rational numbers, a new operation is defined as follows: $a$☼$b=a^{3}-2ab+4$. For example, $2$☼$5=2^{3}-2\\times 2\\times 5+4=-8$. Find $4$☼$\\left(-9\\right)=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 14845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ and $b$ are real numbers, and $\\frac{a}{1-i} + \\frac{b}{2-i} = \\frac{1}{3-i}$, find the sum of the first 100 terms of the arithmetic sequence ${an + b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-910"
} | {
"index": 14846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is a multiple of $1428$, find the greatest common divisor of $a^2+9a+24$ and $a+4$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 14847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is a multiple of $456$, find the greatest common divisor of $3a^3+a^2+4a+57$ and $a$."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 14848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is a positive integer and $a = b - 2005$, if the equation $x^2 - ax + b = 0$ has a positive integer solution, what is the minimum value of $a$? \n(Hint: First, assume the two roots of the equation are $x_1$ and $x_2$, then…)"
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 14849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is a positive real number and $b$ is an integer between $1$ and $201$, inclusive, find the number of ordered pairs $(a,b)$ such that $(\\log_b a)^{2023}=\\log_b(a^{2023})$."
}
] | math | {
"style": "rule",
"ground_truth": "603"
} | {
"index": 14850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 14851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 14852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c \\in R^{+}$ and $a + b + c = 1$, find the maximum value of $\\sqrt{4a + 1} + \\sqrt{4b + 1} + \\sqrt{4c + 1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"index": 14853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c \\in \\mathbb{R}$, $a + b + c = 3$, $a \\geqslant b \\geqslant c$, the equation $ax^{2}+bx+c=0$ has real roots. Find the minimum value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 14854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are all non-zero, and the maximum value of $\\dfrac{a}{|a|} + \\dfrac{b}{|b|} + \\dfrac{c}{|c|} - \\dfrac{abc}{|abc|}$ is $m$, and the minimum value is $n$, find the value of $\\dfrac{n^{m}}{mn}$."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 14855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $ABC$, and $\\frac{a-c}{b-\\sqrt{2}c}=\\frac{sin(A+C)}{sinA+sinC}$. \n$(Ⅰ)$ Find the measure of angle $A$; \n$(Ⅱ)$ If $a=\\sqrt{2}$, $O$ is the circumcenter of triangle $ABC$, find the minimum... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 14856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $\\triangle ABC$, if $\\triangle ABC$ simultaneously satisfies three of the following four conditions:①$a=\\sqrt{3}$; ②$b=2$; ③$\\frac{{sinB+sinC}}{{sinA}}=\\frac{{a+c}}{{b-c}}$; ④${cos^2}({\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 14857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the opposite sides of the acute angles $A$, $B$, $C$ of triangle $\\triangle ABC$, $\\overrightarrow{m}=(3a,3)$, $\\overrightarrow{n}=(-2\\sin B,b)$, and $\\overrightarrow{m} \\cdot \\overrightarrow{n}=0$.\n$(1)$ Find $A$;\n$(2)$ If $a=2$ and the pe... | math | {
"style": "rule",
"ground_truth": "6 - 3\\sqrt{3}"
} | {
"index": 14858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and $2\\sin \\frac {7\\pi}{6}\\sin \\left( \\frac {\\pi}{6}+C\\right)+\\cos C= - \\frac {1}{2}$. \n$(1)$ Find $C$; \n$(2)$ If $c= \\sqrt {13}$ and the area of $\\triangl... | math | {
"style": "rule",
"ground_truth": "\\frac {7 \\sqrt {39}}{26}"
} | {
"index": 14859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and it satisfies $(2b-a) \\cdot \\cos C = c \\cdot \\cos A$.\n$(1)$ Find the size of angle $C$;\n$(2)$ Let $y = -4\\sqrt{3}\\sin^2\\frac{A}{2} + 2\\sin(C-B)$, find the max... | math | {
"style": "rule",
"ground_truth": "2-2 \\sqrt {3}"
} | {
"index": 14860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\\triangle ABC$, with $a=4$ and $(4+b)(\\sin A-\\sin B)=(c-b)\\sin C$, find the maximum value of the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 14861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are three positive real numbers, and $a(a+b+c)=bc$, find the maximum value of $\\frac{a}{b+c}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - 1}{2}"
} | {
"index": 14862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\\angle B=30^{\\circ}$, and the area of $\\triangle ABC$ is $\\frac{1}{2}$, determine the value of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+ \\sqrt{3}}{3}"
} | {
"index": 14863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, $d$ are the thousands, hundreds, tens, and units digits of a four-digit number, respectively, and the digits in lower positions are not less than those in higher positions. When $|a-b|+|b-c|+|c-d|+|d-a|$ reaches its maximum value, the minimum value of ... | math | {
"style": "rule",
"ground_truth": "1119"
} | {
"index": 14864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, $d$, $e$, and $f$ are all positive numbers, and $\\frac{bcdef}{a}=\\frac{1}{2}$, $\\frac{acdef}{b}=\\frac{1}{4}$, $\\frac{abdef}{c}=\\frac{1}{8}$, $\\frac{abcef}{d}=2$, $\\frac{abcdf}{e}=4$, $\\frac{abcde}{f}=8$, find $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{... | math | {
"style": "rule",
"ground_truth": "\\frac{119}{8}"
} | {
"index": 14865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, and $d$ are distinct positive integers, and $abcd = 441$, calculate the value of $a+b+c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are integers, and $a-2b=4$, $ab+c^2-1=0$, find the value of $a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are nonzero real numbers, find all possible values of the expression\n\\[\\frac{a}{|a|} + \\frac{b}{|b|} + \\frac{c}{|c|} + \\frac{abc}{|abc|}.\\]Enter all possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "4, 0, -4"
} | {
"index": 14868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are positive real numbers satisfying $a + 2b + 3c = 6$, find the maximum value of $\\sqrt{a + 1} + \\sqrt{2b + 1} + \\sqrt{3c + 1}$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 14869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the lengths of the sides opposite the angles $A$, $B$, and $C$ in $\\triangle ABC$ respectively, with $a=2$, and $$\\frac{\\sin A - \\sin B}{\\sin C} = \\frac{c - b}{2 + b}.$$ Find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\\triangle ABC$, respectively, and $c\\cos A=5$, $a\\sin C=4$.\n\n(1) Find the length of side $c$; \n(2) If the area of $\\triangle ABC$, $S=16$, find the perimeter of $\\triangl... | math | {
"style": "rule",
"ground_truth": "13+ \\sqrt {41}"
} | {
"index": 14871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite the angles $A$, $B$, and $C$ in $\\triangle ABC$ respectively, and the equation $\\sqrt{3}b\\sin A - a\\cos B - 2a = 0$ holds, then the measure of $\\angle B$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 14873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\\triangle ABC$, respectively, and $\\sqrt{3}c\\sin A = a\\cos C$.\n\n$(I)$ Find the value of $C$;\n\n$(II)$ If $c=2a$ and $b=2\\sqrt{3}$, find the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15} - \\sqrt{3}}{2}"
} | {
"index": 14874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and $(\\sin A + \\sin B)(a-b) = (\\sin C - \\sin B)c$.\n\n1. Find the measure of angle $A$.\n2. If $a=4$, find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 14875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and $\\sin A$, $\\sin B$, and $\\sin C$ form a geometric sequence. When $B$ takes the maximum value, the maximum value of $\\sin A + \\sin C$ is _____."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and vectors $\\overrightarrow{m}=(1-\\cos (A+B),\\cos \\frac {A-B}{2})$ and $\\overrightarrow{n}=( \\frac {5}{8},\\cos \\frac {A-B}{2})$ with $\\overrightarrow{m}\\... | math | {
"style": "rule",
"ground_truth": "-\\frac {3}{8}"
} | {
"index": 14877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, if $\\cos B= \\frac {4}{5}$, $a=5$, and the area of $\\triangle ABC$ is $12$, find the value of $\\frac {a+c}{\\sin A+\\sin C}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {25}{3}"
} | {
"index": 14878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\\triangle ABC$ respectively, and $2\\sin \\frac{7\\pi }{6}\\sin (\\frac{\\pi }{6}+C)+ \\cos C=-\\frac{1}{2}$.\n\n(1) Find $C$;\n\n(2) If $c=2\\sqrt{3}$, find the maximum area of $\\triang... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 14879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\\triangle ABC$, respectively, and $\\overrightarrow{m}=(a,−\\sqrt {3}b)$, $\\overrightarrow{n}=(\\sin B,\\cos A)$, if $a= \\sqrt {7}$, $b=2$, and $\\overrightarrow{m} \\perp \\overrightar... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"index": 14880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and the altitude on side $BC$ is $\\frac{a}{2}$. Determine the maximum value of $\\frac{c}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 14881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a+b+c=0$, calculate the value of $\\frac{|a|}{a}+\\frac{|b|}{b}+\\frac{|c|}{c}+\\frac{|ab|}{ab}+\\frac{|ac|}{ac}+\\frac{|bc|}{bc}+\\frac{|abc|}{abc}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a+b=3$ and $a^3+b^3=81$, find $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 14883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 14884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b \\in R^{+}$ and $a + b = 1$, find the supremum of $- \\frac{1}{2a} - \\frac{2}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"index": 14885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b \\in R^{+}$, and $a + b = 1$, find the maximum value of $- \\frac{1}{2a} - \\frac{2}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"index": 14886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b \\in \\mathbb{R}$, and $a^2 + 2ab - 3b^2 = 1$, find the minimum value of $a^2 + b^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{4}"
} | {
"index": 14887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b$, and $c$ are complex numbers satisfying $$\\begin{aligned} a^{2}+a b+b^{2} & =1+i \\\\ b^{2}+b c+c^{2} & =-2 \\\\ c^{2}+c a+a^{2} & =1 \\end{aligned}$$ compute $(a b+b c+c a)^{2}$. (Here, $\\left.i=\\sqrt{-1}.\\right)$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-11-4 i}{3}"
} | {
"index": 14888,
"solution": "More generally, suppose $a^{2}+a b+b^{2}=z, b^{2}+b c+c^{2}=x$, $c^{2}+c a+a^{2}=y$ for some complex numbers $a, b, c, x, y, z$. We show that $$f(a, b, c, x, y, z)=\\left(\\frac{1}{2}(a b+b c+c a) \\sin 120^{\\circ}\\right)^{2}-\\left(\\frac{1}{4}\\right)^{2}\\left[(x+y+z)^{2}-2\\left... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b, c$ are integers with $a b c=60$, and that complex number $\\omega \\neq 1$ satisfies $\\omega^{3}=1$, find the minimum possible value of $\\left|a+b \\omega+c \\omega^{2}\\right|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14889,
"solution": "Since $\\omega^{3}=1$, and $\\omega \\neq 1, \\omega$ is a third root of unity. For any complex number $z,|z|^{2}=z \\cdot \\bar{z}$. Letting $z=a+b \\omega+c \\omega^{2}$, we find that $\\bar{z}=a+c \\omega+b \\omega^{2}$, and $$|z|^{2} =a^{2}+a b \\omega+a c \\omega^{2}+a b \\omega^... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b, c$ are positive integers satisfying $$a+b+c=\\operatorname{gcd}(a, b)+\\operatorname{gcd}(b, c)+\\operatorname{gcd}(c, a)+120$$ determine the maximum possible value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 14890,
"solution": "240. Notice that $(a, b, c)=(240,120,120)$ achieves a value of 240. To see that this is maximal, first suppose that $a>b$. Notice that $a+b+c=\\operatorname{gcd}(a, b)+\\operatorname{gcd}(b, c)+$ $\\operatorname{gcd}(c, a)+120 \\leq \\operatorname{gcd}(a, b)+b+c+120$, or $a \\leq \\op... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b, c$ are positive real numbers and $\\log _{a} b+\\log _{b} c+\\log _{c} a=0$, find the value of $\\left(\\log _{a} b\\right)^{3}+\\left(\\log _{b} c\\right)^{3}+\\left(\\log _{c} a\\right)^{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14891,
"solution": "3. Let $x=\\log _{a} b$ and $y=\\log _{b} c$; then $\\log _{c} a=-(x+y)$. Thus we want to compute the value of $x^{3}+y^{3}-(x+y)^{3}=-3 x^{2} y-3 x y^{2}=-3 x y(x+y)$. On the other hand, $-x y(x+y)=\\left(\\log _{a} b\\right)\\left(\\log _{b} c\\right)\\left(\\log _{c} a\\right)=1$, ... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b, c$ are the sides opposite to angles $A, B, C$ respectively in $\\triangle ABC$, and $\\cos A \\sin B \\sin C + \\cos B \\sin A \\sin C = 2 \\cos C \\sin A \\sin B$, find the maximum value of $C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\\overline{c d a b}$ and $\\overline{a b c d}$ is a four-digit number of the form $\\overline{p q e f}$. If $\\overline{e f}$ is a perfect square and $\\o... | math | {
"style": "rule",
"ground_truth": "1983"
} | {
"index": 14893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a,b$ are constants, and $a \\neq 0$, $f\\left( x \\right)=ax^{2}+bx$, $f\\left( 2 \\right)=0$.\n\n(1) If the equation $f\\left( x \\right)-x=0$ has a unique real root, find the analytic expression of the function $f\\left( x \\right)$;\n\n(2) When $a=1$, find the ma... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a,b$ are positive real numbers, and $({(a-b)}^{2}=4{{(ab)}^{3}})$, find the minimum value of $\\dfrac{1}{a}+\\dfrac{1}{b}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a,b,c,d,e$ are real numbers such that\n$a+b+c+d+e=8$ ,\n$a^2+b^2+c^2+d^2+e^2=16$ .\nDetermine the maximum value of $e$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\frac{16}{5}\n\\]"
} | {
"index": 14896,
"solution": "By Cauchy Schwarz, we can see that $(1+1+1+1)(a^2+b^2+c^2+d^2)\\geq (a+b+c+d)^2$ thus $4(16-e^2)\\geq (8-e)^2$ Finally, $e(5e-16) \\geq 0$ which means $\\frac{16}{5} \\geq e \\geq 0$ so the maximum value of $e$ is $\\frac{16}{5}$ .\nfrom: Image from Gon Mathcenter.net "
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a-b=3$, find the value of $1+2b-(a+b)$.\nGiven that $2^x=3$, find the value of $2^{2x-3}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"index": 14897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 14898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a>0$, $b>1$, and $a+b=2$, find the minimum value of $$\\frac{1}{2a}+\\frac{2}{b-1}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 14899,
"solution": ""
} |
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