I'll admit that I've made no progress to solve this one. It is way too hard. I guess I must do some stuffs with elliptic curve to solve it but I got nowhere
So, here is the problem:
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?
Proposed by David Yang
Original thread: https://mathoverflow.net/questions/95120/number-of-x-such-that-m-x2-and-n-x2-are-both-squares
AoPS thread: http://www.artofproblemsolving.com/community/c6h486950p2731268