A grasshopper starts at is to jump right in any order where each and are integers. Let be a set of numbers not containing . Prove that the order can be chosen in such a way so that the grasshopper never lands on any point in .
Before spending too much time pondering this puzzle, you might like to know that it was the "hard" problem in this year's math olympiad, and has been the object of a massive cooperative solving venture on Terence Tao's blog. (I haven't been following this project carefully, but I believe that it hasn't yet reached a solution, though Prof. Tao claims to have found a solution after seven hours' thought, and it seems that three of the olympiad competitors scored full marks on the question.)