IMC 1996 · Problem 2

Let $\theta$ be a positive real number and let $\cosh t = \dfrac{e^t + e^{-t}}{2}$ denote the hyperbolic cosine. Show that if $k \in \mathbb{N}$ and both $\cosh k\theta$ and $\cosh (k+1)\theta$ are rational, then so is $\cosh \theta$.