The Clay Mathematics Institute put together the Millennium Prize in 2000, listing seven important math problems no one had been able to answer. If solved, they could unlock doors to new areas of mathematics and new real-world technologies. Three years ago, an eccentric Russian guy solved one of them but turned down the money. (!)
The problems are hard, but there’s no reason why anyone in the world can’t think about it.
If those look too hard, a different group has smaller prizes for smaller problems.