Question 8
A bug walks all day and sleeps all night. On the first day, it starts at point $O,$ faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P.$ Then $OP^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$