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Let a_1, a_2, \ldots , a_n be distinct positive integers and let M be a set of n - 1 positive integers not containing s = a_1 + a_2 + \ldots + a_n. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a_1, a_2, \ldots , a_n in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.

Proposed by Dmitry Khramtsov, Russia

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