DE-MO 2016 Problem 4

Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation $m \cdot \binom{2n}{n} = \binom{m^2}{2}.$ (German MO 2016 - Problem 4)

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.