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 with lengths

in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in

Proposed by Dmitry Khramtsov, Russia

%V0
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