Question 14
The perimeter of triangle $APM$ is $152,$ and angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}.$ Given that $OP = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$