2000 AIME II Problems/Problem 9
Given that is a complex number such that , find the least integer that is greater than .
Using the quadratic equation on , we have .
There are other ways we can come to this conclusion. Note that if is on the unit circle in the complex plane, then and . We have and . Alternatively, we could let and solve to get .
Using De Moivre's Theorem we have , , so .
We want .
Finally, the least integer greater than is .
Let . Notice that we have
must be (or else if you take the magnitude would not be the same). Therefore, and plugging into the desired expression, we get . Therefore, the least integer greater is
~solution by williamgolly
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