Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.
Source: Federal Mathematical Competition of Serbia and Montenegro 2004
Tags: number theory
Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.