Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3 + 3n)^2}{n^6 - 64}.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan

A four-digit number $YEAR$ is called very good if the system $$\begin{align*} Yx + Ey + Az + Rw &= Y \\ Rx + Yy + Ez + Aw &= E \\ Ax + Ry + Yz + Ew &= A \\ Ex + Ay + Rz + Yw &= R \end{align*}$$ of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century.

(The 21st century starts in 2001 and ends in 2100.)

Proposed by Tomáš Bárta, Charles University, Prague

Let $f : (-1, 1) \to \mathbb{R}$ be a twice differentiable function such that $$2 f'(x) + x f''(x) \ge 1 \quad \text{for } x \in (-1, 1).$$ Prove that $$\int_{-1}^{1} x f(x)\,dx \ge \frac13.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan

Define the sequence $a_0, a_1, \dots$ of numbers by the following recurrence: $$a_0 = 1, \quad a_1 = 2, \quad (n + 3) a_{n+2} = (6n + 9) a_{n+1} - n a_n \quad \text{for } n \ge 0.$$ Prove that all terms of this sequence are integers.

Proposed by Khakimboy Egamberganov, ICTP, Italy

Determine whether there exist an odd positive integer $n$ and $n \times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:

(1) $\det(B) = 1$;

(2) $AB = BA$;

(3) $A^4 + 4 A^2 B^2 + 16 B^4 = 2019 I$.

(Here $I$ denotes the $n \times n$ identity matrix.)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

Let $f, g : \mathbb{R} \longrightarrow \mathbb{R}$ be continuous functions such that $g$ is differentiable. Assume that $\bigl( f(0) - g'(0) \bigr) \bigl( g'(1) - f(1) \bigr) > 0$. Show that there exists a point $c \in (0, 1)$ such that $f(c) = g'(c)$.

Proposed by Fereshteh Malek, K. N. Toosi University of Technology

Let $C = \{4, 6, 8, 9, 10, \dots\}$ be the set of composite positive integers. For each $n \in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges: $$\tag{1} \sum_{n \in C} \left( \frac{a_n}{n} \right)^n.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

Let $x_1, \dots, x_n$ be real numbers. For any set $I \subset \{1, 2, \dots, n\}$ let $s(I) = \sum\limits_{i \in I} x_i$. Assume that the function $I \mapsto s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1, 2, \dots, n\}$. Prove that the number of sets $I \subset \{1, 2, \dots, n\}$ for which $s(I) = 2019$ does not exceed $1.7^n$.

Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University

Determine all positive integers $n$ for which there exist $n \times n$ real invertible matrices $A$ and $B$ that satisfy $AB - BA = B^2 A$.

Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

2019 points are chosen at random, independently, and distributed uniformly in the unit disc $\{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 \le 1 \}$. Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices?

Proposed by Fedor Petrov, St. Petersburg State University