Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. Proposed by Ilir Snopce

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x,y>0$ we have $$f(xy+f(x))=yf(x)+x.$$ Proposed by Nikola Velov

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$. Proposed by Nikola Velov

Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$. Proposed by Nikola Velov

There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a friendly triplet if any two students in the group are friends. Determine the maximal possible number of friendly triplets. Proposed by Nikola Velov

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: • For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$. • $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$. Proposed by Nikola Velov

Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$. Determine the largest $k$ (depending on $n$) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other. Proposed by Nikola Velov

Does the equation $$z(y-x)(x+y)=x^3$$have finitely many solutions in the set of positive integers? Proposed by Nikola Velov