Another method to solve the grasshopper problem (the International Mathematical Olympiad)

TL;DR

This paper presents a novel polynomial-based approach to the grasshopper problem from the IMO, extending previous methods and analyzing specific cases to address the problem's complexities.

Contribution

It introduces an original polynomial method inspired by Kos's work, providing new insights and partial solutions for the grasshopper problem.

Findings

The method successfully solves cases for n=3, 4, 5

Identifies singularities affecting the solution

Offers a new perspective on the problem's structure

Abstract

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon's combinatorial Nullstellensatz. We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.