\documentclass[11pt]{scrartcl}
\usepackage{evan}
\begin{document}
\title{APMO 2009/1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 33}
\maketitle
\section*{Problem}
Consider the following operation on positive real numbers written on a blackboard:
Choose a number $r$ written on the blackboard, erase that number,
and then write a pair of positive real numbers $a$ and $b$
satisfying the condition $2 r^2 = ab$ on the board.
Assume that you start out with just one positive real number $r$ on the blackboard,
and apply this operation $k^2 - 1$ times to end up with $k^2$ positive real numbers,
not necessarily distinct.
Show that there exists a number on the board which does not exceed $kr$.
\section*{Video}
\href{https://www.youtube.com/watch?v=uj93tNL8f7M&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/uj93tNL8f7M}}
\section*{External Link}
\url{https://aops.com/community/p1434404}
\newpage
\section*{Solution}
The problem follows from one observation:
\begin{claim*}
[Main claim]
The sum of the squares of reciprocals is always nondecreasing
under this operation.
\end{claim*}
\begin{proof}
Clear.
\end{proof}
Hence, if $x_1$, \dots, $x_n$ are the numbers on the board,
when $n = k^2$, they satisfy
\[ \frac{1}{r^2} \le \sum \frac{1}{x_n^2} \]
which implies $\min(x_1, \dots, x_n) \le {\sqrt n}r = kr$.
\end{document}