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Complex Numbers and Two Constraints

Source: Romanian District Olympiad 2018 - Grade X - Problem 4

March 10, 2018
complex numbers

Problem Statement

Let n≥2n\ge 2 be a natural number. Find all complex numbers zz which simultaneously satisfy the relations
a) zn+zn−1+…+z2+∣z∣=n;\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;
b) ∣z∣n−1+∣z∣n−2+…+∣z∣2+z=nzn.\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.
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