2020 AIME I Problem 8

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.

Answer

Use complex numbers: the total displacement is

5\left(1 + z + z^2 + \cdots\right)

with

z = \frac{1}{2}e^{\frac{i\pi}{3}}.

The series sums to

\frac{5}{1 - z};

compute

|1 - z|^2

from

z = \frac{1}{4} + \frac{\sqrt{3}}{4}i.

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^{\frac{i\pi}{3}},

so

\begin{aligned} P &= 5\left(1 + z + z^2 + \cdots\right) \\ &= \frac{5}{1 - z}. \end{aligned}

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}{\frac{3}{4}} = \frac{100}{3},

and

m + n = 100 + 3 = 103.

Problem 8 in other years