Let $A$ be a real $n \times n$ matrix such that $A^3 = 0$.

(a) Prove that there is a unique real $n \times n$ matrix $X$ that satisfies the equation $$X + AX + XA^2 = A.$$ (b) Express $X$ in terms of $A$.

(proposed by Bekhzod Kurbonboev, Institute of Mathematics, Tashkent)

Let $n$ and $k$ be fixed positive integers, and let $a$ be an arbitrary non-negative integer. Choose a random $k$-element subset $X$ of $\{1, 2, \dots, k+a\}$ uniformly (i.e., all $k$-element subsets are chosen with the same probability) and, independently of $X$, choose a random $n$-element subset $Y$ of $\{1, \dots, k+n+a\}$ uniformly.

Prove that the probability $$\mathsf{P} \bigl( \min(Y) > \max(X) \bigr)$$ does not depend on $a$.

(proposed by Fedor Petrov, St. Petersburg State University)

We say that a positive real number $d$ is good if there exists an infinite sequence $a_1, a_2, a_3, \dots \in (0, d)$ such that for each $n$, the points $a_1, \dots, a_n$ partition the interval $[0, d]$ into segments of length at most $1/n$ each. Find $$\sup \{ d \mid d \text{ is good} \}.$$ (proposed by Josef Tkadlec)

Let $f : \mathbb{R} \to \mathbb{R}$ be a function. Suppose that for every $\varepsilon > 0$, there exists a function $g : \mathbb{R} \to (0, \infty)$ such that for every pair $(x, y)$ of real numbers, $$\text{if } |x - y| < \min\{ g(x), g(y) \}, \text{ then } |f(x) - f(y)| < \varepsilon.$$ Prove that $f$ is the pointwise limit of a sequence of continuous $\mathbb{R} \to \mathbb{R}$ functions, i.e., there is a sequence $h_1, h_2, \dots$ of continuous $\mathbb{R} \to \mathbb{R}$ functions such that $\lim\limits_{n \to \infty} h_n(x) = f(x)$ for every $x \in \mathbb{R}$.

(proposed by Camille Mau, Nanyang Technological University, Singapore)

Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that $$2021 B = A^m + B^2.$$ Prove that $|\det A| \le 1$.

(proposed by Rafael Filipe dos Santos, Instituto Militar de Engenharia, Rio de Janeiro)

For a prime number $p$, let $\operatorname{GL}_2(\mathbb{Z}/ p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\varphi : \operatorname{GL}_2(\mathbb{Z}/p\mathbb{Z}) \to S_p$.

(proposed by Thiago Landim, Sorbonne University, Paris)

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{ z : |z| \le 1 \}$. Let $f : D \to \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$\bigl| f(0) \bigr| \le \max_{|z|=1} \bigl| f(z) p(z) \bigr|.$$ (proposed by Lars Hörmander)

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

(proposed by Alexander Polyanskii, Moscow Institute of Physics and Technology; based on results of Paul Erdős and Moshe Rosenfeld)