2020 AIME I Problems/Problem 8
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Problem
A bug walks all day and sleeps all night. On the first day, it starts at point 
faces east, and walks a distance of 
units due east. Each night the bug rotates 
counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point 
Then 
where 
and 
are relatively prime positive integers. Find 
Solution 1 (Coordinates)
We plot this on the coordinate grid with point 
as the origin. We will keep a tally of the x-coordinate and y-coordinate separately.
First move: The ant moves right .
Second move: We use properties of a 
triangle to get 
right, 
up.
Third move: 
left, 
up.
Fourth move: left.
Fifth move: 
left, 
down.
Sixth move: 
right, 
down.
Total of x-coordinate: 
.
Total of y-coordinate: 
.
After this cycle of six moves, all moves repeat with a factor of 
. Using the formula for a geometric series, multiplying each sequence by 
will give us the point 
.
, 
.
Therefore, the coordinates of point are 
, so using the Pythagorean Theorem, 
, for an answer of 
.
-molocyxu
Solution 2 (Complex)
We put the ant in the complex plane, with its first move going in the positive real direction.
Take
![\[|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^k})|^2\]](http://latex.artofproblemsolving.com/0/9/7/097331b13ec0b489a208c3d7fc579bd9dbb9abcd.png)
and this is an infinite geometric series. Summing using 
gives 
~awang11
See Also
| 2020 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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