Given a scalene acute triangle $ABC$, let $M$ be the midpoint of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D, E$ its intersections with the bisector of angle $BAC$. Points $D', E'$ lie on $\omega$ such that $DED'E'$ is a rectangle. Prove that $D', E'$, $M, N$ lie on a single circle.
Let $ABC$ be an isosceles triangle with base $AB$ and $P$ a point on its $C$-altitude. Ray $AP$ meets the circumcircle of the triangle $ABC$ again at $Q \neq A$. The line through $P$ parallel to $AB$ meets the side $BC$ at $R$. Prove that $QR$ bisects the angle $AQB$. (Jaroslav Švrček)
Consider an infinite sequence $a_0, a_1, a_2, \dots$ of integers that satisfies
$$
a_0 \ge 1 \quad \text{and} \quad a_{n+1} \in \{2022a_n - 1, 2022a_n + 1\}
$$
for all indices $n \ge 0$. Prove that any such infinite sequence contains infinitely many composite numbers.
(Martin Melicher)
Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$ and $P$ its intersection of diagonals. Denote by $O_1, O_2$ the circumcenters of triangles $ABP, CDP$, respectively. Prove that $O_1BCO_2$ is a parallelogram.
Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is *connected* if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples.
Let $k \le 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \le k \le 2022$, determine which player (if any) has a winning strategy. (David Hruška)
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
$$
f\left(f(x) + \frac{y+1}{f(y)}\right) = \frac{1}{f(y)} + x + 1
$$
for all $x, y \gt 0$.
(Dominik Burek)
Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $\ell$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$ is tangent to a fixed circle, independent of the choice of $\ell$. (Josef Tkadlec)
Given a positive integer $n$, denote by $\tau(n)$ the number of positive divisors of $n$, and by $\sigma(n)$ the sum of all positive divisors of $n$. Find all positive integers $n$ satisfying
$$
\sigma(n) = \tau(n) \cdot \lceil\sqrt{n}\rceil.
$$
(Here, $\lceil x \rceil$ denotes the smallest integer not less than $x$.)
(Michael Reitmeir)
Let $ABC$ be a triangle with $AB \lt AC$ and circumcenter $O$. The angle bisector of $\angle BAC$ meets the side $BC$ at $D$. The line through $D$ perpendicular to $BC$ meets the segment $AO$ at $X$. Furthermore, let $Y$ be the midpoint of segment $AD$. Prove that points $B, C, X, Y$ lie on a single circle. (Karl Czakler)
Consider 26 letters $A, \dots, Z$. A *string* is a finite sequence consisting of those letters. We say that a string $s$ is *nice* if it contains each of the 26 letters at least once, and each permutation of letters $A, \dots, Z$ occurs in $s$ as a subsequence the same number of times. Prove that:
a. There exists a nice string.
b. Any nice string contains at least 2022 letters.
(Here, a permutation $\pi$ of the 26 letters is a *subsequence* of a string $s = s_1s_2s_3\dots$ if there exist 26 indices $i_1 \lt i_2 \lt \dots \lt i_{26}$ such that $\pi = s_{i_1}s_{i_2}\dots s_{i_{26}}$.) (Václav Rozhoň)
Does there exist a positive integer $n$ such that across all representations $n = ab$, the digits $0, 1, \dots, 9$ are all present as the final digit of $a^b$ at least once?
Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center $B$ intersects $\omega$ at $C$ and $AB$ at $D$. The line $CD$ intersects $\omega$ at the point $E$ ($E \neq C$). The intersection of lines $OE$ and $BC$ is $F$.
a. Prove that the triangle $OBF$ is isosceles.
b. Find the ratio $\frac{FB}{BD}$, given that $D$ is the midpoint of $OB$.
A field is a $2020 \times 2021$ grid with a positive integer written into each cell, such that no number repeats in any row or column. An onion consists of $4$ consecutive cells in a row or column, whose numbers add up to exactly $4 \cdot 2021$. Find the largest possible number of onions on the field.
Let $ABC$ be a triangle where $AB = AC$. Let $E$ be the foot of its altitude from the vertex $B$. Given that either $BEA$ or $BEC$ is an isosceles triangle, find all possibilities of the size of the angle at the vertex $A$ of the triangle $ABC$.
Mari chooses five distinct positive integers not greater than $2021$. From these five numbers, it must be possible to choose two numbers with sum $1919$ in two different ways. Likewise, from these five numbers, it must be possible to choose two numbers with sum $2929$ in two different ways. Find all possibilities of which five numbers Mari may choose.