An algebra problem by Priyanshu Mishra

Algebra Level 4

If \((x,y,z)\) are non-negative reals and \(xy + yz + zx = 1\),

The maximum value of the expression below is of the form \(\large \frac { a }{ b\sqrt { c } } \).

\(\large\ x\left( 1-{ y }^{ 2 } \right) \left( 1-{ z }^{ 2 } \right) + y\left( 1-{ z }^{ 2 } \right) \left( 1-{ x }^{ 2 } \right) + z\left( 1-{ x }^{ 2 } \right) \left( 1-{ y }^{ 2 } \right) \)

\(a,b,c\) are all positive integers with \(a,b\) coprime and \(c\) square-free.

Find \(a + b + c\).

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