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"value": "Replace each asterisk in the addition example of decimal fractions \n\n$$\n0, ** + 0, ** + 0, ** + 0, ** = 1\n$$ \n\nwith the digit 2 or the digit 3 so that the equation is correct. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "If $3\\sin A + 4\\cos B = \\sqrt{5}$, what is the maximum value of $4\\sin A + 3\\cos B$? Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nWe need to solve a problem: given that 3 sin A + 4 cos ... |
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"value": "Factorize the expression $a^3 + b^3 + c^3 + (a+b)(b+c)(c+a) - 2abc$. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nThe problem: Factorize expression\n\n\\(E = a^3 + b^3 + c^3 + (a+b)(b+c)(c+a... |
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"value": "Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn ultrasound imaging, consider a microbubble contrast agent with shell viscosity \\( \\eta \\) and initial acoustic impedance \\( Z_b \\), suspended in blood (\\( Z_{\\text{blood}} \\... |
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"value": "Evaluate $\\int \\frac{(x-1)!}{x!} \\, dx$. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nWe need to evaluate the indefinite integral $\\int \\frac{(x-1)!}{x!} \\, dx$.\n\nFirst, we need to i... |
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"value": "How many integers \\(n\\) are there such that \\(n^3 + 8\\) has at most 3 positive divisors? Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "<think>\nWe need to find integer n (presumably integer, could... |
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"value": "Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?\n\n$\\textbf{(A)}\\ 204 \\qquad \\textbf{(B)}\\ 280 \\qquad \\t... |
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"value": "Does there exist a function \\( f: \\mathbb{C} \\rightarrow \\mathbb{N} \\) such that for all \\( a, b \\in \\mathbb{C} \\) with \\( |a - b| = 1 \\), it holds that \\( f(a) \\neq f(b) \\)? If such a function exists, determine the least number of elements that the range of \\( f \... |
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"value": "Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhich of the following events or factors played a crucial role in the independence movements of Latin American countries but did not uniformly affect all countries in the same way?\nA:... |
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"value": "Compute the angle $\\theta = \\tan^{-1} \\left(\\frac{\\sin(\\pi/18) + \\sin(2\\pi/18)}{\\cos(\\pi/18) + \\cos(2\\pi/18)}\\right)$. Express your answer in degrees. Let's think step by step and output the final answer within \\boxed{}./think"
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"value": "Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn the context of legal reasoning and ethical considerations, which of the following scenarios accurately describes a situation where a judge must balance ethical dilemmas with the str... |
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"value": "Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider the integer linear program:\nMaximize \\( x + y \\)\nSubject to:\n\\( 2x + y \\leq 6 \\)\n\\( 2x + 3y \\leq 9 \\)\n\\( x, y \\geq 0 \\), integers.\nThe optimal solution to its... |
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