Daniel's statement:
"Given a statement S of ZFC and a number n, is there a proof of S that is shorter than n?"
Is trivially in NP.
On the other hand, this is no use for telling you whether main task, solving unsolved mathematics problems, is in NP, since there may be very short mathematical statements that nonetheless have very long proofs. In fact, I believe it has even been proven that the growth rate of maximum proof length exceeds O(n), and as far as I know it may well be exponential.
Many experts suspect that there is no polynomial-time solution to the so-called NP-complete problems, though no-one has yet been able to rigorously prove this and there remains the possibility that a polynomial-time algorithm will one day emerge. However unlikely this is, today I would like to invite LW to play a game I played with with some colleagues called what-would-you-do-with-a-polynomial-time-solution-to-3SAT? 3SAT is, of course, one of the most famous of the NP-complete problems and a solution to 3SAT would also constitute a solution to *all* the problems in NP. This includes lots of fun planning problems (e.g. travelling salesman) as well as the problem of performing exact inference in (general) Bayesian networks. What's the most fun you could have?