Difference between revisions of "1980 AHSME Problems/Problem 13"
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== Solution == | == Solution == | ||
| − | <math>\ | + | Writing out the change in <math>x</math> coordinates and then in <math>y</math> coordinates gives the infinite sum <math>1-\frac{1}{4}+\frac{1}{16}-\dots</math> and <math>\frac{1}{2}-\frac{1}{8}+\dots</math> respectively. Using the infinite geometric sum formula, we have <math>\frac{1}{1+\frac{1}{4}}=\frac{4}{5}</math> and <math>\frac{\frac{1}{2}}{1+\frac{1}{4}}=\frac{2}{5}</math>, thus the answer is <math>\left( \frac{4}{5}, \frac{2}{5} \right)</math> |
== See also == | == See also == | ||
Revision as of 17:15, 2 January 2017
Problem
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to 
. Then it makes a 
counterclockwise and travels 
a unit to 
. If it continues in this fashion, each time making a degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
Solution
Writing out the change in 
coordinates and then in 
coordinates gives the infinite sum 
and 
respectively. Using the infinite geometric sum formula, we have 
and 
, thus the answer is 
See also
| 1980 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
