Song's Complex System

Algebra Level 5

Let a,b,ca, b, c be complex numbers satisfying

a+b+c=abc=1 a + b + c = abc = 1

and

ab+bc+ac3=1a2+1b2+1c2 \frac{ab + bc + ac}{3} = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}

The sum of absolute values of all possible ab+bc+acab + bc + ac can be written as nm \frac{\sqrt{n}}{m}, where nn and mm are positive coprime integers. What is n+mn+m?

This problem is posed by Zi Song Y.

Details and assumptions

You may read up on Absolute Value.

nn is allowed to be 1.

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