1980 AHSME Problems/Problem 17
Contents
[hide]Problem
Given that , for how many integers is an integer?
Solution
, and this has to be an integer, so the sum of the imaginary parts must be . Since , there are solutions for : and .
-aopspandy
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) | ||
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Followed by Problem 18 | |
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