Difference between revisions of "2020 AIME I Problems/Problem 8"
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<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>. | <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>. | 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>. | ||
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+ | -molocyxu | ||
==See Also== | ==See Also== |
Revision as of 17:18, 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
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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