1980 AHSME Problems/Problem 13
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 1
Each step the bug takes is half as long as its previous step. So each horizontal step is as long as and in the opposite direction of the previous horizontal step. Similarly, each vertical step is
as long as and in the opposite direction the previous vertical step.
Therefore, the -coordinate is the sum of an infinite geometric series with first term
and common ratio
, so it must be
. Similarly, the
-coordinate is the sum of an infinite geometric series with first term
and common ratio
, so it must be
. Therefore, the bug's path approaches
Solution 2 (Complex Plane)
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 | ||
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