Difference between revisions of "1980 AHSME Problems/Problem 13"
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== Solution == | == Solution == | ||
− | <math>\fbox{}</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> |
+ | |||
+ | == Solution == | ||
+ | We can represent the bug's position on the coordinate plane using complex numbers. The first move the bug makes is <math>1</math>, the second <math>i/2</math>, the third <math>-1/4</math>, and so on. It becomes clear that the distance the bug travels is an infinite geometric series with initial term 1, and common ratio <math>i/2</math>. | ||
+ | |||
+ | Thus, applying the infinite geometric series formula: <math>\frac{1}{1-\frac{i}{2}} = \frac{1}{\frac{2-i}{2}} = \frac{2}{2-i} = \frac{2(2+i)}{(2-i)(2+i)} = \frac{4 + 2i}{5}</math> | ||
+ | |||
+ | This is equivalent to the <math>x</math> coordinate being <math>4/5</math> and the <math>y</math> coordinate being <math>2/5</math>, so the answer is <math>\fbox{B}</math> | ||
+ | |||
+ | ~ jaspersun | ||
== See also == | == See also == |
Latest revision as of 00:03, 4 September 2024
Contents
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
Solution
We can represent the bug's position on the coordinate plane using complex numbers. The first move the bug makes is , the second
, the third
, and so on. It becomes clear that the distance the bug travels is an infinite geometric series with initial term 1, and common ratio
.
Thus, applying the infinite geometric series formula:
This is equivalent to the coordinate being
and the
coordinate being
, so the answer is
~ jaspersun
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.