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C(omplex) numbers

Source: Romanian District Olympiad, grade 10, p3

March 10, 2024
algebracomplex numbers

Problem Statement

Let a,b,c∈C∖{0}a,b,c\in\mathbb{C}\setminus\left\{0\right\} such that ∣a∣=∣b∣=∣c∣|a|=|b|=|c| and A=a+b+cA=a+b+c respectively B=abcB=abc are both real numbers. Prove that Cn=an+bn+cn C_n=a^n+b^n+c^n is also a real number,, (∀)n∈N.(\forall)n\in\mathbb{N}.
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