\documentclass[11pt]{scrartcl}
\usepackage{evan}
\begin{document}
\title{Iberoamerican 2022/2}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 124}
\maketitle
\section*{Problem}
Let $S=\{13, 133, 1333, \dots\}$.
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.
\section*{Video}
\href{https://www.youtube.com/watch?v=doMHBeuDnxQ&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/doMHBeuDnxQ}}
\section*{External Link}
\url{https://aops.com/community/p26230907}
\newpage
\section*{Solution}
Ana wins.
All that's needed is:
\begin{claim*}
On Ana's $k$th move for $k=1,2,\dots,1011$,
Ana can pick the digit to ensure the final number
(no matter what happens after)
is neither a multiple of
\[ X = \underbrace{133\dots33}_{2024-2k} \]
nor
\[ Y = \underbrace{1333\dots33}_{2025-2k}. \]
\end{claim*}
\begin{proof}
$X$ eliminates at most $8$ choices of digits
while $Y$ eliminates at most $1$.
\end{proof}
\end{document}