Difference between revisions of "2020 AIME I Problems/Problem 8"
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− | + | ==Problem== | |
+ | A bug walks all day and sleeps all night. On the first day, it starts at point <math>O</math>, faces east, and walks a distance of <math>5</math> units due east. Each night the bug rotates <math>60^\circ</math> 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 <math>P</math>. Then <math>OP^2=\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
− | == | + | ==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. | ||
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-molocyxu | -molocyxu | ||
− | == Solution 2 (Complex) == | + | ==Solution 2 (Complex)== |
− | We | + | We place the ant at the origin of the complex plane with its first move being in the positive real direction. Then the ant's journey can be represented as the infinite series |
− | + | <cmath>5\left(1 + \frac{e^{\frac{i\pi}{3}}}{2} + \left(\frac{e^{\frac{i\pi}{3}}}{2}\right)^2 + \cdots\right)</cmath> | |
− | <cmath>|\ | + | Using the formula for an infinite geometric series, this is equal to |
− | and | + | <cmath>\frac{5}{1 - \frac12e^{\frac{i\pi}{3}}} = \frac{5}{1 - \frac{1 + i\sqrt{3}}{4}} = \frac{20}{3 - i\sqrt{3}} = 5 + \frac{5i\sqrt{3}}{3}</cmath> |
+ | We are looking for the square of the modulus of this value: | ||
+ | <cmath>\left|\frac{5 + 5i\sqrt{3}}{3}\right|^2 = 25 + \frac{25}{3} = \frac{100}{3}</cmath> | ||
+ | so the answer is <math>100 + 3 = \boxed{103}</math>. | ||
+ | |||
+ | ==Solution 3 (Solution 1 Faster)== | ||
+ | The ant goes in the opposite direction every <math>3</math> moves, going <math>(1/2)^3=1/8</math> the distance backwards. Using geometric series, he travels <math>1-1/8+1/64-1/512...=(7/8)(1+1/64+1/4096...)=(7/8)(64/63)=8/9</math> the distance of the first three moves over infinity moves. Now, we use coordinates meaning <math>(5+5/4-5/8, 0+5\sqrt3/4+5\sqrt3/8)</math> or <math>(45/8, 15\sqrt3/8)</math>. Multiplying these by <math>8/9</math>, we get <math>(5, 5\sqrt3/3)</math> <math>\implies</math> <math>\boxed{103}</math> . | ||
+ | |||
+ | ~Lcz | ||
+ | |||
+ | ==Solution 4 (Official MAA 1)== | ||
+ | Suppose that the bug starts at the origin <math>(0,0)</math> and travels a distance of <math>a</math> units due east on the first day, and that there is a real number <math>r</math> with <math>0<r < 1</math> such that each day after the first, the bug walks <math>r</math> times as far as the previous day. On day <math>n</math>, the bug travels along the vector <math>\pmb v_{n}</math> that has magnitude <math>ar^{n-1}</math> and direction <math>\langle\cos(n\cdot 60^\circ),\sin(n\cdot 60^\circ)\rangle</math>. Then <math>P</math> is the terminal point of the infinite sum of the vectors <math>\pmb v_{1}+\pmb v_{2}+\pmb v_3+\cdots</math>. The <math>x</math>-coordinate of this sum is | ||
+ | <cmath>a\big(\!\cos0^\circ+r\cos60^\circ + r^{2}\cos120^\circ+r^{3}\cos180^\circ+r^{4}\cos240^\circ</cmath> | ||
+ | <cmath>+r^{5}\cos300^\circ+r^{6}\cos360^\circ+\cdots\big).</cmath> | ||
+ | Because the angles repeat after 6 terms, this sum is equal to | ||
+ | <cmath>aS(1+r^{6}+r^{12}+r^{18}+\cdots)=\frac{aS}{1-r^{6}},</cmath> | ||
+ | where | ||
+ | <cmath>S=\cos0^\circ+r\cos60^\circ + r^{2}\cos120^\circ+r^{3}\cos180^\circ+r^{4}\cos240^\circ+ r^{5}\cos300^\circ.</cmath> | ||
+ | Similarly, the <math>y</math>-coordinate of <math>P</math> will be <math>{\frac{aT}{1-r^{6}}}</math>, where | ||
+ | <cmath>T=\sin0^\circ+r\sin60^\circ + r^{2}\sin120^\circ+r^{3}\sin180^\circ+r^{4}\sin240^\circ+ r^{5}\sin300^\circ.</cmath>In this case <math>r=\frac12</math> and <math>a = 5</math>, so | ||
+ | <cmath>S=1+\frac14-\frac18-\frac18-\frac1{32}+\frac1{64}=\frac{63}{64},</cmath> | ||
+ | <cmath>T=0+\frac{\sqrt3}4+\frac{\sqrt3}8+0-\frac{\sqrt3}{32}-\frac{\sqrt3}{64}=\frac{21\sqrt3}{64},</cmath>and the coordinates of <math>P</math> are | ||
+ | <cmath>\left(\frac{5S}{1-\frac1{64}}, \frac{5T}{1-\frac1{64}}\right)=\left(5,\frac{5\sqrt3}{3}\right).</cmath>Thus the square of the distance from the origin to <math>P</math> is <math>25+\frac{25}3=\frac{100}3</math>. The requested sum is <math>100+3=103</math>. | ||
+ | |||
+ | ==Solution 5 (Official MAA 2)== | ||
+ | Let point <math>O</math> be the origin in the complex plane. Point <math>P</math> is the complex sum <math>5(1+z+z^2+\cdots) = \frac{5}{1-z}</math>, where <math>z=\frac{1+i\sqrt3}4</math>. The distance squared is<cmath> {OP}^2=\left|\frac5{1-\frac{1+i\sqrt3}4}\right|^{2}= | ||
+ | \frac{(4\cdot5)^2}{\left|4-(1+i\sqrt3)\right|^2}=\frac{400}{9+3}=\frac{100}3.</cmath> | ||
+ | |||
+ | ==Solution 6 (no coordinates or complex numbers)== | ||
+ | The bug goes forward and backward in three directions: straight east, northeast, and northwest. It travels <math>5-\frac{5}{8}+\frac{5}{64}-\cdots=\frac{40}{9}</math> units east. Thus, it goes northeast <math>\frac{\frac{40}{9}}{2}=\frac{20}{9}</math> units northeast and <math>\frac{10}{9}</math> units northwest. Now, the bug travels a total of <math>\frac{40}{9}+\frac{10}{9}-\frac{5}{9}=\frac{45}{9}=5</math> units east and a total of <math>\frac{10\sqrt{3}}{9}+\frac{5\sqrt{3}}{9}=\frac{15\sqrt{3}}{9}=\frac{5\sqrt{3}}{3}</math> units north because of the 30-60-90 right triangles formed. Now, <math>OP^2=5^2+\frac{5^2}{3}=\frac{100}{3}</math> by the Pythagorean Theorem, and the answer is <math>\boxed{103}</math>. | ||
+ | |||
+ | -integralarefun | ||
+ | |||
+ | ==Video Solution with Motion in Python== | ||
+ | https://youtu.be/RrMsMw_ZrBU | ||
+ | |||
+ | Moving Bug in AIME solution with python turtle scripts | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://www.youtube.com/watch?v=BtMBSoZ3cMQ | ||
+ | |||
+ | -avn | ||
==See Also== | ==See Also== |
Latest revision as of 20:44, 29 November 2021
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 place the ant at the origin of the complex plane with its first move being in the positive real direction. Then the ant's journey can be represented as the infinite series Using the formula for an infinite geometric series, this is equal to We are looking for the square of the modulus of this value: so the answer is .
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
Solution 4 (Official MAA 1)
Suppose that the bug starts at the origin and travels a distance of units due east on the first day, and that there is a real number with such that each day after the first, the bug walks times as far as the previous day. On day , the bug travels along the vector that has magnitude and direction . Then is the terminal point of the infinite sum of the vectors . The -coordinate of this sum is Because the angles repeat after 6 terms, this sum is equal to where Similarly, the -coordinate of will be , where In this case and , so and the coordinates of are Thus the square of the distance from the origin to is . The requested sum is .
Solution 5 (Official MAA 2)
Let point be the origin in the complex plane. Point is the complex sum , where . The distance squared is
Solution 6 (no coordinates or complex numbers)
The bug goes forward and backward in three directions: straight east, northeast, and northwest. It travels units east. Thus, it goes northeast units northeast and units northwest. Now, the bug travels a total of units east and a total of units north because of the 30-60-90 right triangles formed. Now, by the Pythagorean Theorem, and the answer is .
-integralarefun
Video Solution with Motion in Python
Moving Bug in AIME solution with python turtle scripts
Video Solution
https://www.youtube.com/watch?v=BtMBSoZ3cMQ
-avn
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.