International Mathematics Competition
for University Students
2026

IMC2026: Day 2, Problem 8

Problem 8. Let \(\displaystyle n\geq 5\), and suppose that \(\displaystyle A = (a_{ij})\) is a real symmetric \(\displaystyle n\times n\) matrix such that

\(\displaystyle a_{ii} = 0 \quad\text{and}\quad a_{ij}\in\{-1,1\} \text{ for }i\neq j . \)

Assume that the scalar products of any two distinct rows of \(\displaystyle A\) have the same value. Let \(\displaystyle \lambda_1,\ldots,\lambda_n\) be the eigenvalues of \(\displaystyle A\). Prove that

\(\displaystyle \sum_{i=1}^n\lvert\lambda_i\rvert \geq 2n-2 \)

and determine all matrices for which equality holds.

Slobodan Filipovski, University of Primorska, Koper

  Solution