Question 14
In right triangle $ABC$ with the right angle at $C,$ $\angle BAC \lt 45^\circ$ and $AB = 4.$ Point $P$ on $\overline{AB}$ has the properties 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 $p,$ $q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Find $p + q + r.$