Difference between revisions of "2020 AIME I Problems/Problem 8"
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== Problem == | == Problem == | ||
− | == Solution == | + | == Solution 1 (Coordinates) == |
+ | We plot this on the coordinate grid with point <math>O</math> as the origin. We will keep a tally of the x-coordinate and y-coordinate separately. | ||
+ | |||
+ | First move: The ant moves right <math>5</math>. | ||
+ | Second move: We use properties of a <math>30-60-90</math> triangle to get <math>\frac{5}{4}</math> right, <math>\frac{5\sqrt{3}}{4}</math> up. | ||
+ | Third move: <math>\frac{5}{8}</math> left, <math>\frac{5\sqrt{3}}{8}</math> up. | ||
+ | Fourth move: <math>\frac{5}{8}</math> left. | ||
+ | Fifth move: <math>\frac{5}{32}</math> left, <math>\frac{5\sqrt{3}}{32}</math> down. | ||
+ | Sixth move: <math>\frac{5}{64}</math> right, <math>\frac{5\sqrt{3}}{64}</math> down. | ||
+ | |||
+ | Total of x-coordinate: <math>5 + \frac{5}{4} - \frac{5}{8} - \frac{5}{8} - \frac{5}{32} + \frac{5}{64} = \frac{315}{64}</math>. | ||
+ | Total of y-coordinate: <math>0 + \frac{5\sqrt{3}}{4} + \frac{5\sqrt{3}}{8} + 0 - \frac{5\sqrt{3}}{32} - \frac{5\sqrt{3}}{64} = \frac{105\sqrt{3}}{64}</math>. | ||
+ | |||
+ | After this cycle of six moves, all moves repeat with a factor of <math>(\frac{1}{2})^6 = \frac{1}{64}</math>. Using the formula for a geometric series, multiplying each sequence by <math>\frac{1}{1-\frac{1}{64}} = \frac{64}{63}</math> will give us the point <math>P</math>. | ||
+ | |||
+ | <math>\frac{315}{64} \cdot \frac{64}{63} = 5</math>, <math>\frac{105\sqrt{3}}{64} \cdot \frac{64}{63} = \frac{5\sqrt{3}}{3}</math>. | ||
+ | Therefore, the coordinates of point <math>P</math> are <math>(5,\frac{5\sqrt{3}}{3})</math>, so using the Pythagorean Theorem, <math>OP^2 = \frac{100}{3}</math>, for an answer of <math>\boxed{103}</math>. | ||
==See Also== | ==See Also== |
Revision as of 16:14, 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 .
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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