Tri-Variable Inequality!

Algebra Level 4

Let \(x,y\) and \(z\) be positive real numbers such that \(xyz=1\). What is the minimum value of:

\[\frac{x^{3}}{(1+y)(1+z)}+\frac{y^{3}}{(1+z)(1+x)}+\frac{z^{3}}{(1+x)(1+y)}\]

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