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 17:14, 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
