Question 11
The circumcircle of acute $\triangle ABC$ has center $O.$ The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q,$ respectively. Also $AB = 5,$ $BC = 4,$ $BQ = 4.5,$ and $BP = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$