Question 22
Circles $\omega$ and $\gamma,$ both centered at $O,$ have radii $20$ and $17,$ respectively. Equilateral triangle $ABC,$ whose interior lies in the interior of $\omega$ but in the exterior of $\gamma,$ has vertex $A$ on $\omega,$ and the line containing side $BC$ is tangent to $\gamma.$ Segments $AO$ and $BC$ intersect at $P,$ and $\dfrac{BP}{CP} = 3.$ Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m,$ $n,$ $p,$ $q$ with $\gcd(m, n) = \gcd(p, q) = 1.$ What is $m + n + p + q?$