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2021 WSMO Team Round/Problem 14

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Problem

Suppose that $x$ is a complex number such that $x+\frac{1}{x}=\frac{\sqrt{6}+\sqrt{2}}{2}$ and the imaginary part of $x$ is nonnegative. Find the sum of the five smallest nonnegative integers $n$ such that $x^{n}+\frac{1}{x^n}$ is an integer.

Solution

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