\documentclass[11pt]{scrartcl}
\usepackage{evan}
\begin{document}
\title{Shortlist 2007 C1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 2}
\maketitle
\section*{Problem}
Let $n \ge 1$ be an integer. Find all sequences
$a_1$, $a_2$, \dots, $a_{n^2 + n}$ consisting of $0$ and $1$ such that
\[ a_{i+1} + a_{i+2} + \dots + a_{i+n}
< a_{i+n+1} + a_{i+n+2} + \dots + a_{i+2n} \]
for all $0 \le i \le n^2-n$.
\section*{External Link}
\url{https://aops.com/community/p1187174}
\newpage
\section*{Solution}
We give an example for $n=5$ which generalizes readily:
\[ 00000 \mid 00001 \mid 00011 \mid 00111 \mid 01111 \mid 11111. \]
It's obvious this works.
One can actually prove this is the only one.
Now:
\begin{itemize}
\ii First, split the $n^2+n$ numbers into $n+1$ blocks of size $n$
(as in the example above).
Then evidently, they must have $0$, $1$, \dots, $n$ ones in that order.
\ii TODO finish this up.
\end{itemize}
\end{document}