I think that this is not fair, and can be abused. It's not fair because not all the people started solving problems in the same time, so if someone start solving problems in 2050 even if he does all the problems he will not be at the first place. And it can be abused if someone makes 100 fake accounts and resubmits his solutions.
My suggestion is to change that rule in this manner: instead of "number_of_people_who_have_solved p" it should be "number_of_people_who_have_solved p in faster running time then yours" In that way also you will encourage users to submit more quality solutions 
I sincerely hope that you will consider my suggestion,
My regards
Marko