Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:

“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-14.

A similar solution was submitted by 하석민 (수리과학과 2017학번, +3). Late solutions are not graded.

Let \(I, J\) be connected open intervals such that \(I \cap J\) is a nonempty proper sub-interval of both \(I\) and\(J\). For instance, \(I = (0, 2)\) and \(J = (1, 3)\) form an example.

Let \(f\) (\(g\), resp.) be an orientation-preserving homeomorphism of the real line \(\mathbb{R}\) such that the set of points of \(\mathbb{R}\) which are not fixed by \(f\) (\(g\), resp.) is precisely \(I\) (\(J\), resp.).

Show that for large enough integer \(n\), the group generated by \(f^n, g^n\) is isomorphic to the group with the following presentation

\[ <a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>. \]

The best solution was submitted by 김동률 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-12.

Let \( A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \} \). Consider the following property, which we call Property R:

“If each of the points in \(A\) is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of \(|A_{a, b}|\) such that \( A_{a, b} \) has Property R but \( A_{a-1, b} \) and \( A_{a, b-1} \) do not.

The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2019-13.

An incorrect solution was received. Late solutions are not graded.