Difference between revisions of "2020 AIME I Problems/Problem 8"
Promatheus (talk | contribs) m (→Solution 3 (Solution 1 faster)) |
Cooljoseph (talk | contribs) (→Solution 1 (Coordinates)) |
||
Line 4: | Line 4: | ||
== Solution 1 (Coordinates) == | == Solution 1 (Coordinates) == | ||
+ | <asy> | ||
+ | size(8cm); | ||
+ | pair O, A, B, C, D, F, G, P, X; | ||
+ | O = (0, 0); | ||
+ | A = (5, 0); | ||
+ | X = (8, 0); | ||
+ | P = (5, 5 / sqrt(3)); | ||
+ | B = rotate(-120, A) * ((O + A) / 2); | ||
+ | C = rotate(-120, B) * ((A + B) / 2); | ||
+ | D = rotate(-120, C) * ((B + C) / 2); | ||
+ | F = rotate(-120, D) * ((C + D) / 2); | ||
+ | G = rotate(-120, F) * ((D + F) / 2); | ||
+ | draw(O -- A -- B -- C -- D -- F -- G); | ||
+ | draw(A -- X, dashed); | ||
+ | markscalefactor = 0.05; | ||
+ | path angle = anglemark(X, A, B); | ||
+ | draw(angle); | ||
+ | |||
+ | dot(P); | ||
+ | dot(O); | ||
+ | |||
+ | label("$O$", O, W); | ||
+ | label("$P$", P, E); | ||
+ | label("$60^\circ$", angle, ENE*3); | ||
+ | </asy> | ||
+ | |||
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. | 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. | ||
Revision as of 12:43, 13 March 2020
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 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
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
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.