OIM 2022

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 $0<x<1$ .