Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was. Proposed by Viktor Simjanoski and Nikola Velov

Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials. Proposed by Nikola Velov

We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$. $\textit {Proposed by Ilija Jovcheski}$

Given is an acute triangle $ABC$ with $AB<AC$ with altitudes $BD$ and $CE$. Let the tangents to the circumcircle at $B$ and $C$ meet at $Y$. Let $\omega_1$ be the circle through $A$ tangent to $DE$ at $E$; define $\omega_2$ similarly, and let their intersection point be $X$. Prove that $A, X, Y$ are colinear. $\textit{Proposed by Nikola Velov}$

Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

The numbers 1, 2 and 3 are written on a board. Two friends are playing the following game. A player writes a number that doesn't exceed 2022 and isn't already on the board and is a sum or a product of two numbers that are written on the board. They take turns writing numbers and the winner is the player who writes 2022 on the board. Which player has a winning strategy and why? Proposed by Ilija Jovcheski