2011 AIME II Problem 13

Point P lies on the diagonal AC of square ABCD with AP > CP. Let O_1 and O_2 be the circumcenters of triangles ABP and CDP, respectively. Given that AB = 12 and \angle O_1PO_2 = 120 ^{\circ}, then AP = \sqrt{a} + \sqrt{b}, where a and b are positive integers. Find a + b.

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.