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\begin{document}
\title{USAMO 1999/1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 41}
\maketitle
\section*{Problem}
Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions:
\begin{enumerate}
\ii[(a)] every square that does not contain a checker shares a side with one that does;
\ii[(b)] given any pair of squares that contain checkers,
there is a sequence of squares containing checkers,
starting and ending with the given squares,
such that every two consecutive squares of the sequence share a side.
\end{enumerate}
Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.
\section*{Video}
\href{https://www.youtube.com/watch?v=Uv4AtkRVPwo&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/Uv4AtkRVPwo}}
\newpage
\section*{Solution}
Take a spanning tree on the set $V$ of checkers
where the $|V|-1$ edges of the tree are given by orthogonal adjacency.
By condition (a) we have
\[ \sum_{v \in V} (4-\deg v) \ge n^2 - |V| \]
and since $\sum_{v \in V} \deg v = 2(|V|-1)$
we get
\[ 4|V| - \left( 2|V|-2 \right) \ge n^2 - |V| \]
which implies $|V| \ge \frac{n^2-2}{3}$.
\end{document}