Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.

Prove that for every real number $x>0$ and each integer $n>0$ we have \[x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)\]

For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational. (a) Prove the claim for $r \ge \frac43$ and $r \le 0$. (b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold.

Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$. (Martin Panák)

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy][asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy][/asy]

Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the arc ${BAC}$. The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$ at points ${K, L}$ ${(|AK| <|AL|)}$. Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$. Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .