Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.

A number is called Norwegian if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

Let $n>3$ be an integer. If $x_1<x_2<\ldots<x_{n+2}$ are reals with $x_1=0$, $x_2=1$ and $x_3>2$, what is the maximal value of $$(\frac{x_{n+1}+x_{n+2}-1}{x_{n+1}(x_{n+2}-1)})\cdot (\sum_{i=1}^{n}\frac{(x_{i+2}-x_{i+1})(x_{i+1}-x_i)}{x_{i+2}-x_i})?$$

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.

Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

If $d$ is a positive integer such that $d \mid 5+2022^{2022}$, show that $d=2x^2+2xy+3y^2$ for some $x, y \in \mathbb{Z}$ iff $d \equiv 3,7 \pmod {20}$.

Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. We say that a tree is majestic if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.