Prove that $z_1z_2z_3 = 1$ for complex numbers $z_1$, $z_2$, $z_3$. Let $z_1$, $z_2$, $z_3$ be distinct complex numbers such that $|z_1| = |z_2| = |z_3| > 0$. If $z_1 + z_2z_3$, $z_2 + z_3z_1$, and $z_3 + z_1z_2$ are real numbers, prove that $z_1z_2z_3 = 1$.
I believe that this problem comes from 1979 Romanian Math Olympiad (State Competition, 10th grade).
I tried several approaches: The first was to substitute $a + ib$, $c + id$, and $f + ig$ for $z_1$, $z_2$, and $z_3$, respectively, and bash. I used $(1+z_1)(1+z_2)(1+z_3)$, which didn't really get me anywhere.
After a couple more attempts at substitutions and such, I hit a wall and didn't really know what to do.

Gosh... well, I just looked at the three-way symmetry, noted all points must lie on a circle around the origin, and deduced $z_1 = 1$, $z_2 = e^{2 i \pi/3}$ and $z_3 = e^{-2 i \pi/3}$, and confirmed these worked.

I suppose one could work through the algebra more rigorously, but on an exam... I think this is fine!
