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basic complex algebra

Source: Romanian District Olympiad, Grade X, Problem 2

September 24, 2018
complex numbersalgebra

Problem Statement

Let z1,z2,z3∈C z_1,z_2,z_3\in\mathbb{C} such that \text{(i)}  \left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1 \text{(ii)}  z_1+z_2+z_3\neq 0 \text{(iii)}  z_1^2 +z_2^2+z_3^2 =0.
Show that ∣z13+z23+z33∣=1. \left| z_1^3+z_2^3+z_3^3\right| = 1.
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