IMO 1967 LL GDR15

Origin: GDR

Problem

Suppose tan \alpha = p/q, where p and q are integers and q ̸= 0. Prove that the number tan \beta for which tan 2\beta = tan 3\alpha is rational only when p2 + q2 is the square of an integer.

Solution

Given that tan \alpha \inQ, we have that tan \beta is rational if and only if tan \gamma is rational, where \gamma = \beta −\alpha and 2\gamma = \alpha. Putting t = tan \gamma we obtain p q = tan 2\gamma = 2t 1−t2 , which leads to the quadratic equation pt2+2qt−p = 0. This equation has rational solutions if and only if its discriminant 4(p2+q2) is a perfect square, and the result follows.