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2009 IMO Problems/Problem 6

Revision as of 05:33, 23 July 2009 by Bugi (talk | contribs) (Created page with '== Problem == Let <math>a_1,a_2,\ldots,a_n</math> be distinct positive integers and let <math>M</math> be a set of <math>n-1</math> positive integers not containing <math>s=a_1+…')
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Problem

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 $M$.

Author: Dmitry Khramtsov, Russia

--Bugi 10:33, 23 July 2009 (UTC)Bugi

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