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"value": "Find the sum of the first six terms of an arithmetic sequence in which the sum of any number of terms is equal to four times the square of that number. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<t... |
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"value": "Calculate the value of the series\n\\[ S = \\sum_{n=1}^{+\\infty} \\frac{1}{(n \\cdot \\sinh(n\\pi))^2} \\]\nand express it in the form \\( S = \\frac{2}{3}A - \\frac{11\\pi^2}{180} \\), where \\( A = \\sum_{n=1}^{+\\infty} \\frac{(-1)^{n-1}}{(2n-1)^2} \\). Find the value of \\( ... |
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"value": "For positive real numbers \\(a, b, c\\), calculate the maximum value of the expression\n\\[\n\\frac{a}{3a^2 + b^2 + 2ac} + \\frac{b}{3b^2 + c^2 + 2ab} + \\frac{c}{3c^2 + a^2 + 2bc}.\n\\] Let's think step by step and output the final answer within \\boxed{}./think"
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"f... |
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"value": "Let \\( f \\) be a continuous function everywhere and \\( f(x) = \\frac{x^2 + a}{x^2 + x - 1} \\) for all \\( x \\neq 1 \\). Find the value of \\( a \\) such that \\( f \\) is continuous at \\( x = 1 \\). Let's think step by step and output the final answer within \\boxed{}./thin... |
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"value": "Sketch the curve\n\\[ y = \\frac{x}{1 + x^6 (\\sin x)^2}. \\] Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nWe need to \"Sketch the curve y = x / (1 + x^6 (sin x)^2)\". This is a request for ... |
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"value": "Solve the equation $\\left[\\frac{2x^2}{x^2 + 1}\\right] = x$, where $[a]$ denotes the largest integer less than or equal to $a$. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nWe need to solv... |
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"value": "Prove that if\n\n\\[\n\\begin{aligned}\nx_{0}^{4}+a_{1} x_{0}^{3}+a_{2} x_{0}^{2}+a_{3} x_{0}+a_{4} & =0 \\\\\n4 x_{0}^{3}+3 a_{1} x_{0}^{2}+2 a_{2} x_{0}+a_{3} & =0\n\\end{aligned}\n\\]\n\nthen \\(x^{4}+a_{1} x^{3}+a_{2} x^{2}+a_{3} x+a_{4}\\) is divisible by \\(\\left(x-x_{0}\\... |
[{"from":"user","value":"For which values of $n$ is the following implication true? For every $X, Y (...TRUNCATED) |
[{"from":"user","value":"Let \\( a, b \\), and \\( c \\) be positive real numbers. Determine the lar(...TRUNCATED) |
[{"from":"user","value":"Given points \\( A = (4,0,0) \\), \\( B = (0,3,0) \\), and \\( C = (0,0,5) (...TRUNCATED) |
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