Someone with a lot of bitcoins could do a public contest.  Brute forcing a 256 bit ECDSA key (has 128 bit security) is infeasible so nobody would attempt it (well not anyone with any real power) however one could make the problem simpler.

For example move 1,000 BTC to an address and provide publicly half of the private key.  This would mean someone could brute force the key on average in 2^64 attempts.  The blockchain will be proof of if someone accomplished it or not.  Even that would be an incredibly difficult problem, but it is feasible (back to the whole infeasible vs improbable).

If/when someone can steal those coins just keep in mind that brute forcing a full key would be 2^64 = 18,446,744,073,709,600,000 x harder/longer.