A combinatorial solution to LIMO 2010 question 10

TL;DR

This paper presents a purely combinatorial proof for the nilpotency of a matrix family, offering an alternative to the traditional Cayley-Hamilton theorem-based solutions in a mathematical olympiad problem.

Contribution

It introduces a novel combinatorial approach to proving nilpotency, diverging from standard algebraic methods used in prior solutions.

Findings

Provides a new combinatorial proof for matrix nilpotency

Demonstrates an alternative method to Cayley-Hamilton theorem

Enhances understanding of matrix properties through combinatorics

Abstract

Problem 10 of the Landelijke Interuniversitaire Mathematische Olympiade 2010 asks for a proof that all matrices in a certain family are nilpotent. Both model solutions prove this using the Cayley-Hamilton theorem. I give a purely combinatorial proof.