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"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the problem, we need to find all pairs of integers \\((p, q)\\) such that the roots of bo(...TRUNCATED) | (6, 5) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To find the greatest common divisor (GCD) of two integers 315 and 2016, we can use the Euclidean a(...TRUNCATED) | 63 | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the problem, we need to use the formula for the projection of vector $\\overrightarrow{a}(...TRUNCATED) | \frac{9}{2} | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve this problem, we need to set up the equations based on the given information and solve fo(...TRUNCATED) | 30 | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the given problem, we need to address each part step by step.\n\n### Part (1): Intervals (...TRUNCATED) | n > 1 | [0.0,0.0,0.0,0.0,0.0,1.0,1.0,1.0,0.0,0.0,1.0,0.0,1.0,1.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve this problem, we need to use the given information to set up equations and solve for the (...TRUNCATED) | 4\sqrt{5} | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the problem, we need to analyze the functional equation \\( f(x+1) = \\frac{1}{2} + \\sqr(...TRUNCATED) | 1+\frac{\sqrt{2}}{2} | [0.0,0.0,0.0,1.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1.0,1.0,0.0,0.0,1.0,1.0,1.0,0.0,0.0,0.0,1.0,0.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the given expressions, we will break them down step by step and use Python to ensure accu(...TRUNCATED) | -25 | [0.0,1.0,0.0,1.0,1.0,0.0,0.0,0.0,0.0,0.0,1.0,0.0,1.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the problem, we need to find the maximum value of \\( | \\overrightarrow{a} + t \\overrig(...TRUNCATED) | \sqrt{3} | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) |
"<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the problem, we need to follow these steps:\n\n1. Calculate \\( M \\) by dividing 1764 by(...TRUNCATED) | 330.75 | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) |
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