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Complex Number Sum and Products

Source: 2018 AIME II #5

March 23, 2018
complex numbersAIMEAIME I

Problem Statement

Suppose that xx, yy, and zz are complex numbers such that xy=−80−320ixy = -80-320i, yz=60yz = 60, and zx=−96+24izx = -96+24i, where i=−1i = \sqrt{-1}. Then there are real numbers aa and bb such that x+y+z=a+bix+y+z = a+bi. Find a2+b2a^2 + b^2.
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