TR-MO 2015 Problem 1

$m$ and $n$ are positive integers. If the number $k=\dfrac{(m+n)^2}{4m(m-n)^2+4}$ is an integer, prove that $k$ is a perfect square.

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.