Let $f(x), g(x)$ be real polynomials of degrees $2$ and $3$, respectively. Could it happen that $f(g(x))$ has $6$ distinct roots, which are powers of $2$?

Given is a triangle $ABC$ with median $BM$. The point $D$ lies on the line $AC$ after $C$, such that $BD=2CD$. The circle $(BMC)$ meets the segment $BD$ at $N$. Show that $AC+BM>2MN$.

The infinite periodic fractions $\frac{a} {b}$ and $\frac{c} {d}$ with $(a, b)=(c, d)=1$ are such that every finite block of digits in the first fraction after the decimal point appears in the second fraction as well (again after the decimal point). Show that $b=d$.

What is the minimal number of operations needed to repaint a entirely white grid $100 \times 100$ to be entirely black, if on one move we can choose $99$ cells from any row or column and change their color?

Let $a>1$ be a positive integer and let $f(n)=n+[a\{n\sqrt{2}\}]$. Show that there exists a positive integer $n$, such that $f(f(n))=f(n)$, but $f(n) \neq n$.

Given is a triangle $ABC$. Let $X$ be the reflection of $B$ in $AC$ and $Y$ is the reflection of $C$ in $AB$. The tangent to $(XAY)$ at $A$ meets $XY$ and $BC$ at $E, F$. Show that $AE=AF$.

Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?

Do there exist $2023$ nonzero reals, not necessarily distinct, such that the fractional part of each number is equal to the sum of the rest $2022$ numbers?

Let $ABC$ be a triangle with $\angle B=120^{o}$. Let $D$ be point on the $B$-angle bisector, such that $\angle ADB=2\angle ACB$. Point $E$ lies on the segment $AB$, so that $AE=AD$. Show that $EC=ED$.

Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.

One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into $n^2$ identical square cells. In each of these $2n^2$ cells is written a number from $1$ to $k$. Find the smallest $k$,for which the following properties hold simultaneously: (i) on the red side, any two numbers in different rows are distinct; (ii) on the blue side, any two numbers in different columns are different; (iii) for each of the $n^2$ squares of the partition, the number on the blue side is not equal to the number on the red side.

Let $x_0, x_1, \ldots, x_{n-1}, x_n=x_0$ be reals and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. The numbers $y_i$ for $i=0,1, \ldots, n-1$ are chosen such that $y_i$ is between $x_i$ and $x_{i+1}$. Prove that $\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i)$ can attain both positive and negative values, by varying the $y_i$.

There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.

Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.

Let $a, b>1$ be reals such that $a+\frac{1}{a^2} \geq 5b-\frac{3}{b^2}$. Show that $a>5b-\frac{4}{b^2}$.

A few (at least $5$) integers are put on a circle, such that each of them is divisible by the sum of its neighbors. If the sum of all numbers is positive, what is its minimal value?

Let $M$ be the midpoint of $AC$ in an acute triangle $ABC$. Let $K$ be a point on the minor arc $AC$, such that $\angle AKM=90^{o}$. Let $BK \cap AM=X$ and the $A$-altitude meets $BM$ at $Y$. Show that $XY \parallel AB$.

For a positive integer $n$ and a nonzero digit $d$, let $f(n, d)$ be the smallest positive integer $k$, such that $kn$ starts with $d$. What is the maximal value of $f(n, d)$, over all positive integers $n$ and nonzero digits $d$?