2019 AIME I Problem 15

Let \overline{AB} be a chord of a circle \omega, and let P be a point on the chord \overline{AB}. Circle \omega_1 passes through A and P and is internally tangent to \omega. Circle \omega_2 passes through B and P and is internally tangent to \omega. Circles \omega_1 and \omega_2 intersect at points P and Q. Line PQ intersects \omega at X and Y. Assume that AP=5, PB=3, XY=11, and PQ^2 = \tfrac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.