MathNet 2009-part-05 — 172 problems (EN–VI) Question 162

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

(The original version of this problem was suggested by Dmitry Khramtsov from Russian Federation.
The current version was strengthened by Christian Reiher from Germany.)