On the grasshopper problem with signed jumps

TL;DR

This paper explores a variation of the IMO grasshopper problem, demonstrating that removing the positivity constraint on jump lengths enables the use of polynomial methods for a solution.

Contribution

It introduces a modified version of the problem without the positivity restriction, allowing polynomial techniques to be applied for the first time.

Findings

Polynomial method can solve the problem without positivity constraints

Elementary solutions are known only with positive integers

Removing positivity enables algebraic approaches

Abstract

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, was the following. Let $a_1,a_2,...,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+...+a_n$. A grasshopper is to jump along the real axis, starting at the point 0 and making $n$ jumps to the right with lengths $a_1,a_2,...,a_n$ 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 problem was discussed in many on-line forums, as well by communities of students as by senior mathematicians. Though there have been attempts to solve the problem using Noga Alon's famous Combinatorial Nullstellensatz, up to now all known solutions to the IMO problem are elementary and inductive. In this paper we show that if the condition that the numbers $a_1,...a_n$ are positive is omitted, it allows us to apply the polynomial method to solve the modified problem.