2015 AIME II Problem 11

The circumcircle of acute \triangle A B C has center O. The line passing through point O perpendicular to \overline{O B} intersects lines A B and B C at P and Q, respectively. Also A B=5, B C=4, B Q=4.5, and B P=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.