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deepscaler
[ { "role": "user", "content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with...
math
{ "style": "rule", "ground_truth": "20\\sqrt{5}" }
{ "split": "train", "index": 40049, "solution": "" }
deepscaler
[ { "role": "user", "content": "Let $f(x)=\\frac{\\cos 2x-(a+2)\\cos x+a+1}{\\sin x}.$ For constant $a$ such that $\\lim_{x\\rightarrow 0} \\frac{f(x)}{x}=\\frac 12$ , evaluate $\\int_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\frac{1}{f(x)}dx.$ Let's think step by step and put only the final answer within \...
math
{ "style": "rule", "ground_truth": "\\frac{1}{2}" }
{ "split": "train", "index": 40050, "solution": "" }