Each square of the chessboard was painted in one of eight colors in such a way that the number of squares colored by all the colors are equal. Is it always possible to put $8$ rooks not threatening each other on multi-colored cells?

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. The segment $AL$ intersects the inscribed circle for the second time at point $S$. Line $KL$ intersects the circumscribed circle of triangle $ASK$ for the second at point $M$. Prove that $PL = PM$.

Grisha wrote $n$ different natural numbers, the sum of which does not exceed $S$. The saboteur added to each of them a number from the half-interval $[0, 1)$. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than $1$. At what minimum $S$ can Grisha ensure that the sabotage will definitely not be succeeded?

Consider three pairwise intersecting circles $\omega_1$, $\omega_2$ and $\omega_3$. Let their three common chords intersect at point $R$. We denote by $O_1$ the center of the circumcircle of a triangle formed by some triple common points of $\omega_1$ & $\omega_2$, $\omega_2$ & $\omega_3$ and $\omega_3$ & $\omega_1$. and we denote by $O_2$ the center of the circumcircle of the triangle formed by the second intersection points of the same pairs of circles. Prove that points $R$, $O_1$ and $O_2$ are collinear.

Given three natural numbers greater than $100$, that are pairwise coprime and such that the square of the difference of any two of them is divisible by the third and any of them is less than the product of the other two. Prove that these numbers are squares of some natural numbers.

We have six positive numbers $a_1, a_2, \ldots , a_6$ such that $a_1a_2\ldots a_6 =1$. Prove that: $$ \frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.$$

You are given a convex polygon with perimeter $24\sqrt{3} + 4\pi$. If there exists a pair of points dividing the perimeter in half such that the distance between them is equal to $24$, Prove that there exists a pair of points dividing the perimeter in half such that the distance between them does not exceed $12$.

Consider the graph $G$ with $100$ vertices, and the minimum odd cycle goes through $13$ vertices. Prove that the vertices of the graph can be colored in $6$ colors in a way that no two adjacent vertices have the same color.

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.

Among $33$ balls, there are $2$ radioactive ones. You can put several balls in the detector and it will show if the both radioactive balls are among the balls. What is the smallest number that we have to use the detector so that one can certainly find at least one of the radioactive balls?

We call natural numbers $n$ and $k$ are similar if they are multiples of square of a number greater than $1$. Let $f(n)$ denote the number of numbers from $1$ to $n$ similar to $n$ (for example, $f(16)=4$, since the number $16$ is similar to $4$, $8$, $12$ and $16$). What integer values can the quotient $\frac{n}{f(n)}$ take?

In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.

For positive numbers $x$, $y$, and $z$, we know that $x + y^2 + z^3 = 1$. Prove that $$\frac{x}{2 + xy} + \frac{y}{2 + yz} + \frac{z}{2 + zx} > \frac{1}{2} .$$