Difference between revisions of "2020 AIME I Problems/Problem 8"
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-molocyxu | -molocyxu | ||
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+ | == Solution 2 (Complex) == | ||
+ | We put the ant in the complex plane, with its first move going in the positive real direction. | ||
+ | Take | ||
+ | <cmath>|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^i})|^2</cmath> | ||
+ | and this is an infinite geometric series. Summing using <math>\frac{a}{1-r}</math> gives <math>\boxed{103}.</math> ~awang11 | ||
==See Also== | ==See Also== |
Revision as of 16:20, 12 March 2020
Note: Please do not post problems here until after the AIME.
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 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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