IMO 1967 - Longlist Problems Question 15

(GDR 3) Suppose $\tan\alpha=p/q$, where $p$ and $q$ are integers and $q\ne0$. Prove that the number $\tan\beta$ for which $\tan2\beta=\tan3\alpha$ is rational only when $p^2+q^2$ is the square of an integer.