2020 AIME I Problems/Problem 8
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Problem
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
![\[|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^i})|^2\]](http://latex.artofproblemsolving.com/d/5/e/d5e987edde92caaf6b7c37bdde60a2aa0a6eafce.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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