Question 55
Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM$. Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1$, $CS = 3$, $PQ = 8 \sqrt{\frac{7}{37}}$, and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.