Problem

Source: Romania National Olympiad 2000, Grade X, Problem 2

Tags: algebra, complex numbers



Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$then $ z_1^6=z_2^6 $