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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