In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.

In a room there are $2023$ vases numbered from $1$ to $2023$. In each vase we want to put a note with a positive integer from $1$, $2$ $...$ , $2023$ on it. The numbers on the notes do [u]not[/u] necessarily have to be distinct. The following should now apply to each vase. Look at the note inside the vase, find the (not necessarily different) vase with the number written on the note, and look at the note inside this vase. Then the average of the numbers on the two notes must be exactly equal to the number of the first selected vase. For example, if we put a note with the number $5$ in vase $13$, then vase $5$ should contain a note with the number $21$ on it: after all, the average of $5$ and $ 21$ is $13$. Determine all possible ways to provide each vase with a note.

Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.

The numbers from $1$ to $2000^2$ were written on a board. Vasya choose $2000$ of them whose sum of them equal to two thousandth of the sum of all numbers. Proof that his friend, Petya, will be able to color each of the remaining numbers by one of other $1999$ colors so that the sum of numbers with each of total $2000$ colors are the same.

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

If in the formula $ C \equal{} \frac {en}{R\plus{}nr}$, $n$ is increased while $ e$, $R$ and $r$ are kept constant, then $C$: $\textbf{(A)}\ \text{Increases} \qquad \textbf{(B)}\ \text{Decreases} \qquad \textbf{(C)}\ \text{Remains constant} \qquad \textbf{(D)}\ \text{Increases and then decreases} \qquad\\ \textbf{(E)}\ \text{Decreases and then increases}$

Evaluate the infinite product \[ \prod_{k = 2}^{\infty} \left( 1 - 4 \sin^2 \frac{\pi}{3\cdot 2^{k}} \right) . \]

Let there be a finite number of straight lines in the plane, none of which are three in one point to cut. Show that the intersections of these straight lines can be colored with $3$ colors so that that no two points of the same color are adjacent on any of the straight lines. (Two points of intersection are called [i]adjacent [/i] if they both lie on one of the finitely many straight lines and there is no other such intersection on their connecting line.)

Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves. Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves. For which values of $n$ do they need the same number of moves?

Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? [i]Proposed by Nathan Xiong[/i]

Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?

Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.