2020 AIME I Problems/Problem 8
Contents
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 place the ant at the origin of the complex plane with its first move being in the positive real direction. Then the ant's journey can be represented as the infinite series
Using the formula for an infinite geometric series, this is equal to
We are looking for the square of the modulus of this value:
so the answer is
.
Solution 3 (Solution 1 faster)
The ant goes in the opposite direction every moves, going
the distance backwards. Using geometric series, he travels
the distance of the first three moves over infinity moves. Now, we use coordinates meaning
or
. Multiplying these by
, we get
.
~Lcz
Video Solution
https://www.youtube.com/watch?v=BtMBSoZ3cMQ
-avn
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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