The XYZ Equations

Algebra Level 5

\[ \begin{array}{l l} x^2 y^2 + y^2 z^2 = 5 xyz\\ y^2 z^2 + z^2 x^2 = 17 xyz \\ z^2 x^2 + x^2 y^2 = 20 xyz \\ \end{array} \]

How many ordered sets of integers \( (x, y, z)\) such that \(x,y,\) and \(z\) are between -10 and 10 inclusive are solutions to the system of equations above?

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