Decimate the decimals

Algebra Level 4

Let \(x,y,z\) be positive numbers such that \(x+y+z=2\) and \(xy+yz+zx=1\). Given that \[\large x^{20.17}+y^{20.17}+z^{20.17}\] achieves its maximum value when \((x,y,z)=(X,Y,Z)\), and that \[XYZ=\frac{a}{b},\] where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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