ELMO SL 2012 N9

Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares?

David Yang.

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.