1980 AHSME Problems/Problem 17
Contents
Problem
Given that , for how many integers
is
an integer?
Solution 1
Expanding yields
. This quantity is an integer if and only if
, that is, if
. Therefore, there are
such values of
.
-aopspandy, edited by j314andrews
Solution
Since we have an imaginary term, we can think about rotations. We are in the first and second quadrant, so we only need to think about angles from 0 to exclusive. Specifically,
, where
is an integer. Therefore, the only angles which can work are
and
.
Now we just need to see if these angles can be represented by .
and
work, since they form a 45-45-90 triangle, and
works, since it doesn't have a real component.
So, the answer is .
~ jaspersun
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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