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

\begin{document}
\title{Shortlist 2009 A1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 8}
\maketitle

\section*{Problem}
Find the largest possible integer $k$,
such that the following statement is true:

Let $2009$ arbitrary non-degenerated triangles be given.
In every triangle the three sides are coloured,
such that one is blue, one is red and one is white.
Now, for every colour separately,
let us sort the lengths of the sides to obtain
\begin{itemize}
  \ii[] $b_1 \le b_2 \le \dots \le b_{2009}$, the lengths of the blue sides,
  \ii[] $r_1 \le r_2 \le \dots \le r_{2009}$, the lengths of the red sides,
  \ii[and] $w_1 \le w_2 \le \dots \le w_{2009}$, the lengths of the white sides.
\end{itemize}
Then there are at least $k$ indices $j$ with the property
that $b_j$, $r_j$, $w_j$ form the sides of a nondegenerate triangle.



\section*{External Link}
\url{https://aops.com/community/p1932911}

\newpage

\section*{Solution}
The answer is $k=1$.

To see at least one triangle must be formed,
simply note that the longest sides $b_{2009}$, $r_{2009}$, $w_{2009}$
of each color necessarily form a triangle;
for if $w_{2009} > b_{2009} + r_{2009}$ (say)
then no triangle can have side $w_{2009}$.

Conversely, we give a construction where only $j=2009$ works:
\[
  \begin{array}{rrrr r rr}
    \triangle\#1 \hspace*{2ex}
      & \triangle\#2 \hspace*{2ex}
      & \triangle\#3 \hspace*{2ex}
      & \triangle\#4 \hspace*{2ex}
      & \dots
      & \triangle\#2008 \hspace*{2ex}
      & \triangle\#2009 \hspace*{2ex} \\ \hline
    b_1 = 10000 & b_2 = 10001 & b_3 = 10002 & b_4 = 10003
      & \dots & b_{2008} = 12007 & b_{2009} = 12008  \\
    w_{2009} = 14200 & w_1 = 10000 & w_2 = 10001 & w_3 = 10002
      & \dots & w_{2007} = 12006 & w_{2008} = 12007 \\
    r_{2009} = 24199 & r_1 = 20000 & r_2 = 20002 & r_3 = 20004
      & \dots & r_{2007} = 24012 & r_{2008} = 24014
  \end{array}
\]
This counterexample shows $k=1$ is the largest integer
for which the statement is true.

\end{document}
