Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. Oleksii Masalitin, Ukraine

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

Recall that for an arbitrary prime $p$, we define a primitive root modulo $p$ as an integer $r$ for which the least positive integer $v$ such that $r^{v}\equiv 1\pmod{p}$ is $p-1$. Prove or disprove the following statement: For every prime $p>2023$, there exists positive integers $1\leqslant a<b<c<p$ such that $a,b$ and $c$ are primitive roots modulo $p$ but $abc$ is not a primitive root modulo $p$.

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$Show that there exist infinitely many prime number that divide at least one number in this sequences

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: every term in the sequence is less than or equal to $2^{2023}$, and there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]

In a test, $201$ students are trying to solve $6$ problems.We know that for each of $5$ first problems, there are at least $140$ students, who can solve it. Moreover, there is exactly $60$ students, who can solve $6^{th}$ problem. Show that there exist $2$ students, such that two of them combined are able to solve all $6$ question. (For example, number $1$ do $1,2,3,4$ and number $2$ do $3,5,6$)

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. Proposed by Tahjib Hossain Khan, Bangladesh

Let triangle \( ABC \) be an acute-angled triangle. Square \( AEFB \) and \( ADGC \) lie outside triangle \( ABC \). \( BD \) intersects \( CE \) at point \( H \), and \( BG \) intersects \( CF \) at point \( I \). The circumcircle of triangle \( BFI \) intersects the circumcircle of triangle \( CGI \) again at point \( K \). Prove that line segment \( HK \) bisects \( BC \).

Let $ABC$ be an acute triangle with altitudes $AD,BE$ and $CF$. Denote $\omega_1,\omega_2$ the circumcircles of $\triangle AEB, \triangle AFC$, respectively. Suppose the line through $A$ parallel to $EF$ intersects $\omega_1$ and $\omega_2$ at $P$ and $Q$, respectively. Show that the circumcenter of $\triangle PQD$ lies on $AD$

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

A polynomial $A(x)$ is said to be simple if $A(x)$ is divisible by $x$ but not divisible by $x^2$. Suppose that a polynomial $P(x)$ has a simple polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists a simple polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{2023}$.