HMMT November, 2015

Algebra Difficulty 4.9 AIME Prove it United States

Problem:
Find the smallest positive integer $n$ such that there exists a complex number $z$ , with positive real and imaginary part, satisfying $z^{n} = (\bar{z})^{n}$ .

Solution

Solution:
Since $|z| = |\bar{z}|$ we may divide by $|z|$ and assume that $|z| = 1$ . Then $\bar{z} = \frac{1}{z}$ , so we are looking for the smallest positive integer $n$ such that there is a $2n^{\text{th}}$ root of unity in the first quadrant. Clearly there is a sixth root of unity in the first quadrant but no fourth or second roots of unity, so $n = 3$ is the smallest.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.