MathDB

Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before.
A real

2024 \times 2024

matrix

A

is called nice if

(Av, v) = 1

for every vector

v\in \mathbb{R}^{2024}

with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix

Problem Statement