The XYZ Equations

Algebra Level 5

x2y2+y2z2=5xyzy2z2+z2x2=17xyzz2x2+x2y2=20xyz \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) (x, y, z) such that x,y,x,y, and zz are between -10 and 10 inclusive are solutions to the system of equations above?

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