Question 48
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. Suppose that $BC = 27$, $CD = 25$, and $AP = 10$. If $MP = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.