Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?

Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

Let $ABCD$ be a cyclic quadrilateral such that the circles with diameters $AB$ and $CD$ touch at $S$. If $M, N$ are the midpoints of $AB, CD$, prove that the perpendicular through $M$ to $MN$ meets $CS$ on the circumcircle of $ABCD$.

Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?

In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.

Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.

A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.

Given are $50$ distinct sets of positive integers, each of size $30$, such that every $30$ of them have a common element. Prove that all of them have a common element.

In a triangle $ABC$, let $BD$ be its altitude and let $H$ be its orthocenter. The perpendicular bisector of of $HD$ meets $(BCD)$ at $P, Q$. Prove that $\angle APB+\angle AQB=180^{o}$

The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Prove that for all positive reals $x, y, z$, the inequality $(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0$ is satisfied.

We write pairs of integers on a blackboard. Initially, the pair $(1,2)$ is written. On a move, if $(a, b)$ is on the blackboard, we can add $(-a, -b)$ or $(-b, a+b)$. In addition, if $(a, b)$ and $(c, d)$ are written on the blackboard, we can add $(a+c, b+d)$. Can we reach $(2022, 2023)$?

Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.

Given is a triangle $ABC$ with circumcenter $O$. Points $D, E$ are chosen on the angle bisector of $\angle ABC$ such that $EA=EB, DB=DC$. If $P, Q$ are the circumcenters of $(AOE), (COD)$, prove that either the line $PQ$ coincides with $AC$ or $PQCA$ is cyclic.

If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.

Given is a simple connected graph with $2n$ vertices. Prove that its vertices can be colored with two colors so that if there are $k$ edges connecting vertices with different colors and $m$ edges connecting vertices with the same color, then $k-m \geq n$.