Are there any math contests paid in Bitcoin? Are there any hard math problems or math contests that have rewards paid in Bitcoin?
There is a site where one can submit bounties for proofs of theorems in Coq formal proof management system, and get paid in bitcoin:


proofmarket.org


Additionally there are other sites that have done competitions in the past:


mathgate.info treasure hunt


Another project in development (but at the moment not specifically tied to bitcoin) is ProofPeer

Additional information about these proof markets can be found in this article:  http://blogchain.fr/programmarket-en/