Song's Complex System

Algebra Level 5

Let \(a, b, c\) be complex numbers satisfying

\[ a + b + c = abc = 1\]


\[ \frac{ab + bc + ac}{3} = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} \]

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

This problem is posed by Zi Song Y.

Details and assumptions

You may read up on Absolute Value.

\(n\) is allowed to be 1.


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