Problem of the Day #23 (Challenge)

POTD Challenge December 12, 2023
Author: awesomeming327

Consider a circle \omega of radius 3 with center O and a point P outside such that OP=5. Let the tangents from P to \omega meet \omega at A and B. Consider a point X on the minor arc AB, and let the circumcircle of POX meet \omega again at Y. Let XY intersect AB at Q. If XQ = \frac{4}{9}YQ, and XY = \frac{m}{n}, find the value of m+n.