Newton's Sums with pi?

Algebra Level 5

\[x+y+z=1\\ { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=\pi\\ { x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }={ \pi }^{ 2 }+3{ z }^{ 2 }-3z+1\\ \]

This set of equations is true for three complex numbers \(x, y, z\).

If \(xyz-xy=\frac { A }{ B } {\pi }^{ C }\) for positive integers \(A, B, C\) and \(\gcd(A,B)=1\), then find \(A+B+C\).

First do this.

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