2010 AIME II Problem 14

Triangle ABC with right angle at C, \angle BAC < 45^\circ and AB = 4. Point P on \overline{AB} is chosen such that \angle APC = 2\angle ACP and CP = 1. The ratio \frac{AP}{BP} can be represented in the form p + q\sqrt{r}, where p, q, r are positive integers and r is not divisible by the square of any prime. Find p+q+r.

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.