Question:
Find the number of pairs $(z_1,z_2)$ of complex numbers such that:

$z_1 z_2$ is pure imaginary
$\frac{z_1}{z_2}$ is real
$|z_1| = |z_2| = 1.$

Answer:
Since $|z_1| = |z_2| = 1,$ $|z_1 z_2| = 1.$  Let
\[z_1 z_2 = si,\]where $s \in \{-1, 1\}.$

Similarly, $\left| \frac{z_1}{z_2} \right| = 1.$  Let
\[\frac{z_1}{z_2} = t,\]where $t \in \{-1, 1\}.$

Multiplying these equations, we get $z_1^2 = sti.$  This equation has two solutions.

Thus, there are two choices of $s,$ two choices of $t,$ and two choices of $z_1,$ giving us $\boxed{8}$ possible pairs $(z_1,z_2).$