Day 1
Given is an equilateral triangle $ABC$ with circumcenter $O$. Let $D$ be a point on to minor arc $BC$ of its circumcircle such that $DB>DC$. The perpendicular bisector of $OD$ meets the circumcircle at $E, F$, with $E$ lying on the minor arc $BC$. The lines $BE$ and $CF$ meet at $P$. Prove that $PD \perp BC$.
Let $S=\{13, 133, \cdots\}$ be the set of the positive integers of the form $133 \cdots 3$. Consider a horizontal row of $2022$ cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a $2022$-digit number $N$. Borja wins if $N$ is divisible by a number in $S$, otherwise Ana wins. Find which player has a winning strategy and describe it.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(yf(x))+f(x-1)=f(x)f(y)$ and $|f(x)|<2022$ for all $0<x<1$.
Day 2
Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following: If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right; If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left; In any other case, he stays on the same square and does not move any further. For each $n$, determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $P$ and $Q$ be points in the half plane defined by $BC$ containing $A$, such that $BP$ and $CQ$ are tangents to $\Gamma$ and $PB = BC = CQ$. Let $K$ and $L$ be points on the external bisector of the angle $\angle CAB$ , such that $BK = BA, CL = CA$. Let $M$ be the intersection point of the lines $PK$ and $QL$. Prove that $MK=ML$.
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.