Question 164
Let $p \geq 3$ be a prime number and consider the right triangle with longer leg $p^{2}-1$ and shorter leg $2p$. We inscribe in the triangle a semicircle whose diameter lies on the longer leg of the triangle and which is tangent to the hypotenuse and to the shorter leg of the triangle. Find the values of $p$ for which the radius of the semicircle is an integer.