MathDB

IMO LongList 1967, The Democratic Republic Of Germany 3

Source: IMO LongList 1967, The Democratic Republic Of Germany 3

December 16, 2004

trigonometrynumber theoryTrigonometric EquationsDiophantine equationIMO ShortlistIMO Longlist

Problem Statement

Suppose

\tan \alpha = \dfrac{p}{q}

, where

p

and

q

are integers and

q \neq 0

. Prove that the number

\tan \beta

for which

\tan {2 \beta} = \tan {3 \alpha}

is rational only when

p^2 + q^2

is the square of an integer.