\documentclass[11pt]{scrartcl}
\usepackage{evan}

\begin{document}
\title{Russia 2011/9.8}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 43}
\maketitle

\section*{Problem}
There are some counters in some cells of a $100 \times 100$ board.
A cell $C$ is \emph{nice} if there are an even number of counters
in cells which share a side with $C$. Can exactly one cell be nice?

\section*{Video}
\href{https://www.youtube.com/watch?v=FozF63KHw6E&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/FozF63KHw6E}}

\section*{External Link}
\url{https://aops.com/community/p2262408}

\newpage

\section*{Solution}
A \emph{good set} is a set $S$ of cells such that
any cell is adjacent to an even number of cells in $S$.

\begin{claim*}
  For any cell $C$, there exists a good set $S$ containing $C$
  with an even number of elements.
\end{claim*}
\begin{proof}
  All sets of the following shape are good
  (red squares mark the points of $S$):
  \[
    \begin{bmatrix}
      . & {\color{red}\blacksquare} & . & {\color{red}\blacksquare} & . & . & . & . & . & . \\
      {\color{red}\blacksquare} & . & . & . & {\color{red}\blacksquare} & . & . & . & . & . \\
      . & . & . & . & . & {\color{red}\blacksquare} & . & . & . & . \\
      {\color{red}\blacksquare} & . & . & . & . & . & {\color{red}\blacksquare} & . & . & . \\
      . & {\color{red}\blacksquare} & . & . & . & . & . & {\color{red}\blacksquare} & . & . \\
      . & . & {\color{red}\blacksquare} & . & . & . & . & . & {\color{red}\blacksquare} & . \\
      . & . & . & {\color{red}\blacksquare} & . & . & . & . & . & {\color{red}\blacksquare} \\
      . & . & . & . & {\color{red}\blacksquare} & . & . & . & . & . \\
      . & . & . & . & . & {\color{red}\blacksquare} & . & . & . & {\color{red}\blacksquare} \\
      . & . & . & . & . & . & {\color{red}\blacksquare} & . & {\color{red}\blacksquare} & . \\
    \end{bmatrix}
  \]
  Descriptively, the cells look like a ``tilted rectangle''.
  By varying the dimensions and orientation of the rectangle,
  we can cover any cell.
  (There is an edge case including just the major diagonal.)
\end{proof}

If there is exactly one nice cell $C$,
then consider the set $S$ and consider
\[ \sum_{\text{counter } x}
  \# \left( \text{cells in $S$ adjacent to $x$} \right). \]
On the one hand, it is even by the definition of good set.
On the other hand, counting by the cells, the total should be odd,
contradiction.

\begin{remark*}
  By pairing this result with the tiling from IMO 1999/3 or EGMO 2020/2,
  it follows that there also cannot be exactly one non-nice cell.
\end{remark*}

\end{document}
