BG-MO 2020 Problem 4

Are there positive integers $m>4$ and $n$ , such that a) ${m \choose 3}=n^2;$ b) ${m \choose 4}=n^2+9?$

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.