2014 AIME I Problem 7

Let w and z be complex numbers such that |w| = 1 and |z| = 10. Let \theta = \arg \left(\tfrac{w-z}{z}\right). The maximum possible value of \tan^2 \theta can be written as \tfrac{p}{q}, where p and q are relatively prime positive integers. Find p+q. (Note that \arg(w), for w \neq 0, denotes the measure of the angle that the ray from 0 to w makes with the positive real axis in the complex plane.)

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.