Problem:
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∘ 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 OP2=nm, where m and n are relatively prime positive integers. Find m+n.
Solution:
Suppose the bug begins at the origin (0,0) and travels a units due east on the first day. On each subsequent day, the bug travels a distance equal to r times the distance traveled the previous day, where 0<r<1 is a fixed real number.
On day n, the bug travels along a vector vn with magnitude arn−1 and direction given by the unit vector ⟨cos(n⋅60∘),sin(n⋅60∘)⟩. The point P represents the terminal point of the infinite sum of these vectors:
v1+v2+v3+⋯
The point P represents the final position of the bug after infinitely many days of movement — that is, the endpoint of the infinite sum of its daily displacement vectors.
The x-coordinate of this total displacement is given by: