Some time ago we posted about an online math research collaboration project initiated by Timothy Gowers which you, your dog, and everyone else on the Internet was welcome to participate in. It turned out to be remarkably successful. Not only was the original problem solved but various other results were obtained as well.
They are currently working on writing up the results of that project as a research paper for publication. The author of the paper is D. H. J. Polymath. Timothy Gowers is planning to run another polymath project this fall (keep an eye out for it!).
In the meanwhile, Terence Tao is running a mini-polymath project. The problem is the sixth (and hardest) problem from the 2009 International Math Olympiad. The problem is:
Problem 6. Let be distinct positive integers and let be a set of positive integers not containing . A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .
The good news is that the IMO is an international competition for high school students. This means the above problem is meant to be solvable by students of this age and, in particular, is known to have a solution (which doesn’t happen when you’re doing research!). The bad news is that the IMO is for the best of the best of high school math students from around the world and only 6 or so out of 500 were able to solve it.
If you’re feeling good about your math kung-fu, take a crack at the problem. If not so much, then swing on by Terence Tao’s blog and join in with the crowd working on solving it right now.