# 3 Mathematicians = Triple Trouble

Algebra Level 5

$$x,y,z$$ are complex numbers such that $x+y+z=-1$ $x^3+y^3+z^3=-13+12i$ $x^4+y^4+z^4=-13$

Given that $$a^2+b^2+c^2=QR(\text{sum of}\textbf{ all possible values}\text{ of}\ x^5+y^5+z^5)$$, find the minimum value of: $4\left(\dfrac{a}{1-a^2}\right)+4\left(\dfrac{b}{1-b^2}\right)+4\left(\dfrac{c}{1-c^2}\right)$

$$\text{for } a, b, c\ge 0$$

Details and Assumptions:

• Let the sum of all possible values of $$x^5+y^5+z^5$$ be $$p + qi$$ where $$p + qi$$ is a complex number . Then $$QR(\text{sum of all possible values of } x^5+y^5+z^5) = |p + q|$$.

• $$|p+q|$$ is the absolute value of $$p+q$$.

• $$i$$$$\text{ is an imaginary number which is equal to } \sqrt{-1}$$.

• $$QR(\text{sum of all possible values of } x^5+y^5+z^5)$$ has been created purely for this question.

This is part of Ordered Disorder.

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