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 .

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- Jul 23rd 2009, 12:17 PM #1

- Jul 24th 2009, 04:05 AM #2

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- Jul 24th 2009, 06:23 AM #3
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 in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .

- Jul 24th 2009, 10:05 AM #4

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- Jul 24th 2009, 12:12 PM #5

- Jul 24th 2009, 02:28 PM #6
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.)

- Jul 24th 2009, 03:01 PM #7