Mock AIME 1 2006-2007/Problem 14

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Problem

Three points $A$ , $B$ , and $T$ are fixed such that $T$ lies on segment $AB$ , closer to point $A$ . Let $AT=m$ and $BT=n$ where $m$ and $n$ are positive integers. Construct circle $O$ with a variable radius that is tangent to $AB$ at $T$ . Let $P$ be the point such that circle $O$ is the incircle of $\triangle APB$ . Construct $M$ as the midpoint of $AB$ . Let $f(m,n)$ denote the maximum value $\tan^{2}\angle AMP$ for fixed $m$ and $n$ where $n>m$ . If $f(m,49)$ is an integer, find the sum of all possible values of $m$ .