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Characterization of complex numbers that satisfy a³=b³ or a̅=b.

Source: Romanian District Olympiad 2011, Grade X, Problem 3

October 8, 2018
complex numbersalgebra

Problem Statement

Let be two complex numbers a,b. a,b. Show that the following affirmations are equivalent:
(i) \text{(i)} there are four numbers x1,x2,x3,x4∈C x_1,x_2,x_3,x_4\in\mathbb{C} such that ∣x1∣=∣x3∣,∣x2∣=∣x4∣, \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, and x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a, \forall j_1\in\{ 1,2\} , \forall j_2\in\{ 3,4\} .
(ii)a3=b3 \text{(ii)} a^3=b^3 or b=a‾ b=\overline{a} (the conjugate of a).
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