Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$for all $x>0$. Show that $f(x)=x$ for all $x>0$.

A rich knight has a chest and a lot of coins, so every day he puts into the chest some quantity of coins - among the numbers $1, 2, \ldots, 100$. If there exist two days on which he added equal quantities of coins (say, $k$ coins) and he has added in total at most $100k$ coins on the days between these two days, he stops putting coins into the chest. Prove that this will necessarily happen eventually.

a) (version for grades 10-11) Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle. b) (version for grades 8-9) Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?

Let $n$ be a positive integer greater than $1$ and let us call an arbitrary set of cells in a $n\times n$ square $\textit{good}$ if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint $\textit{good}$ sets required to cover the entire table without the main diagonal?

A quadrilateral $ABCD$ has an incircle $\Gamma$. The points $X, Y$ are chosen so that $AX-CX=AB-BC$, $BX-DX=BC-CD$, $CY-AY=AD-DC$ and $DY-BY=AB-AD$. Given that the center of $\Gamma$ lies on $XY$, show that $AC, BD, XY$ are concurrent.

Let $X$ denotes the set of integers from $1$ to $239$. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of $10$ elements on the board from the set $X$. The magician’s assistant looks at them and adds $k$ more elements from $X$ to the existing sequence. After that the spectator replaces three of these $k+10$ numbers by random elements of $X$ (it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of $k+10$ numbers and without error names the original $10$ numbers written by the spectator. Find the minimal possible $k$ for which the trick is possible.

Let $n>3$ be a positive integer satisfying $2^n+1=3p$, where $p$ is a prime. Let $s_0=\frac{2^{n-2}+1}{3}$ and $s_i=s_{i-1}^2-2$ for $i>0$. Show that $p \mid 2s_{n-2}-3$.

There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?

There are $169$ non-zero digits written around a circle. Prove that they can be split into $14$ non-empty blocks of consecutive digits so that among the $14$ natural numbers formed by the digits in those blocks, at least $13$ of them are divisible by $13$ (the digits in each block are read in clockwise direction).

Let $I$ be the incenter of a triangle $ABC$. The points $X, Y$ lie on the prolongations of the lines $IB, IC$ after $I$ so that $\angle IAX=\angle IBA$ and $\angle IAY=\angle ICA$. Show that the line through the midpoints of $IA$ and $XY$ passes through the circumcenter of $ABC$.

Let $a, b, c$ be reals such that $$a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.$$Show that $$3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.$$

Prove that there exists a positive integer $k>100$, such that for any set $A$ of $k$ positive reals, there exists a subset $B$ of $100$ numbers, so that none of the sums of at least two numbers in $B$ is in the set $A$.

Let $x_1, x_2, \ldots$ be a sequence of $0,1$, such that it satisfies the following three conditions: 1) $x_2=x_{100}=1$, $x_i=0$ for $1 \leq i \leq 100$ and $i \neq 2,100$; 2) $x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}$ for $51 \leq n \leq 100$; 3) $x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}$ for $n>100$. Show that the sequence is periodic.