MathNet 2024-part-04 — 168 problems (EN–VI) Question 104

Let $a, b, c$ be nonzero complex numbers of the same modulus for which the numbers $A = a+b+c$ and $B = abc$ are real. Prove that, for any nonnegative integer number $n$, the number $C_n = a^n + b^n + c^n$ is real.