On the Grasshopper Problem with Signed Jumps

Abstract

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, was the following. Let a1, a2, …, an be distinct positive integers and letℳ be a set of n − 1 positive integers not containing s = a1 + a2 + ⋯ + an. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, …, an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point inℳ. In this paper we consider a variant of the IMO problem when the numbers a1, a2, …, an can be negative as well. We find the sharp minimum of the cardinality of the set ℳ which blocks the grasshopper, in terms of n. In contrast with the Olympiad problem where the known solutions are purely combinatorial, for the solution of the modified problem we use the polynomial method.