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Without loss of generality, lets assume; x>=y>=z. adding x on all 3 sides: 2x>=x+y>=x+z adding y on all 3 sides: x+y>=2y>=y+z. Adding z on all three sides: x+z>=y+z>=2z.
Cubing on all three sides, and substituting for their respective values from the problem statement we get; 8x^3>=z>=y z>=8y^3>=x y>=x>=8z^3
we get z>=y>=x and since we assumed x>=y>=z; the only plausible soln is x=y=z. Substitiuting; we get three solutions..