If \[z_1 z_2 z_3 = 1\] and \[z_1 + z_2 + z_3 = \frac{1}{z_1} + \frac{1}{z_2} +\frac{1}{z_3}\] 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?