Question

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 $m + n.$

答案

Solution:

Work in the complex plane with $O$ at the origin and east along the positive real axis. Each day's displacement is the previous one multiplied by $z = \frac{1}{2}e^{i\pi/3},$ so $P = 5\left(1 + z + z^2 + \cdots\right) = \frac{5}{1 - z}.$

Since $z = \frac{1}{4} + \frac{\sqrt{3}}{4}i,$ we get $1 - z = \frac{3}{4} - \frac{\sqrt{3}}{4}i,$ whose squared magnitude is $\frac{9}{16} + \frac{3}{16} = \frac{3}{4}.$ Therefore $OP^2 = \frac{25}{3/4} = \frac{100}{3},$ and $m + n = 100 + 3 = 103.$

2020 AIME I Problem 8

Solution:

Problem 8 in Other Years