Roots and coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?
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Problem



If z1z2z3=1 and z1+z2+z3=1z1+1z2+1z3 then show that at least one of these numbers must be 1.

Now for the complexity! When are the other numbers real and when are they complex?