Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.

Let $a,b,c$ be distinct positive integers and let $r,s,t$ be positive integers such that: $ab+1=r^2,ac+1=s^2,bc+1=t^2$ Prove that it is not possible that all three fractions$ \frac{rt}{s}, \frac{rs}{t}, \frac{st}{r}$ are integers.

Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$ a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist? b) For every positive integer $m$ find how many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ exist such that $a_n=m.$

Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$$$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$

Find all pairs of positive integers $(n,k)$ for which the number $m=1^{2k+1}+2^{2k+1}+\cdots+n^{2k+1}$ is divisible by $n+2.$

Show that if $2023$ real numbers $x_1,x_2,\dots,x_{2023}$ satisfy $x_1\geq x_2\geq\dots\geq x_{2023}\geq0,$ then $$x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.$$When does the equality take place?

Find all integers $ n $ $(n\geq2)$ with the property: for every $ n $ distinct disks in a plane with at least a common point one of the disks contains the center of another disk.

Let $ABC$ be an acute triangle with orthocenter $ H $ and $AB<AC.$ Let $\Omega_1$ be a circle with diameter $AC$ and $\Omega_2$ a circle with diameter $ AB.$ Line $BH$ intersects $\Omega_1$ in points $ D $ and $E$ such that $E$ is not on segment $BH.$ Line $ CH $ intersects $\Omega_2$ in points $ F $ and $G$ such that $G$ is not on segment $CH.$ Prove that the lines $EG, DF$ and $BC$ are concurrent.

Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?

Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$

Find all sets $ A$ of nonnegative integers with the property: if for the nonnegative intergers $m$ and $ n $ we have $m+n\in A$ then $m\cdot n\in A.$

The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and $$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$