Problem

Source: ELMO Shortlist 2011, A8

Tags: algebra, polynomial, induction, complex numbers, algebra proposed



Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. Evan O'Dorney.