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\title{IMO 2013 Student Report}
\date{15 -- 29 July 2013}
\author{Warren Li}
\maketitle
\section{Introduction}
This year, the International Mathematical Olympiad was held in Santa Marta, Colombia from 17th July to 29th July. The International Mathematical Olympiad (or IMO for short) is an annual competition where aspiring young mathematicians from across the globe can meet each other and sit two difficult 4.5 hour exams with 3 questions each.
The purpose of this report is to relate the events of the IMO and the pre-IMO camp from the student perspective, and, together with Geoff Smith's report as leader, readers can hopefully get a good idea of what IMO 2013 was like. This year our deputy leader, Dominic Yeo, who incidentally began the tradition of writing an unofficial student report 6 years ago, is also producing a diary on his blog \footnote{This is split into four parts, the first of which is \url{http://eventuallyalmosteverywhere.wordpress.com/2013/07/17/imo-2013-part-one-travel-and-training/}}.
Firstly I will show the reader the 6 problems we encountered during this year's IMO, and of course our results.
\section{Problems}
{\it Day 1}
\begin{enumerate}
\item Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1,m_2,\ldots,m_k$ (not necessarily different) such that
$$1+\frac{2^k-1}{n} = \left ( 1 + \frac{1}{m_1} \right ) \left ( 1 + \frac{1}{m_2} \right ) \cdots \left ( 1 + \frac{1}{m_k} \right ).$$
\item A configuration of $4027$ points in the plane is called {\it Colombian} if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is {\it good} for a Colombian configuration if the following two conditions are satisfied:
\begin{itemize}
\item no line passes through any point of the configuration;
\item no region contains points of both colours
\end{itemize}
Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.
\item Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that the triangle $ABC$ is right-angled.
\end{enumerate}
\noindent
{\it Day 2}
\begin{enumerate} \addtocounter{enumi}{3}
\item Let $ABC$ be an acute-angled triangle with orthocentre $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of $CWN$, and let $Y$ be the point on $\omega_2$ such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.
\item Let $\mathbb{Q}_{>0}$ be the set of positive rational numbers. Let $f : \mathbb{Q}_{>0} \mapsto \mathbb R$ be a function satisfying the following three conditions:
\begin{itemize}
\item[(i)] for all $x,y \in \mathbb{Q}_{>0}$, we have $f(x)f(y) \geq f(xy)$;
\item[(ii)] for all $x,y \in \mathbb{Q}_{>0}$, we have $f(x+y) \geq f(x) + f(y)$;
\item[(iii)] there exists a rational number $a>1$ such that $f(a)=a$.
\end{itemize}
Prove that $f(x)=x$ for all $x \in \mathbb{Q}_{>0}$.
\item Let $n \geq 3$ be an integer, and consider a circle with $n+1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0,1,\ldots,n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called {\it beautiful} if, for any four labels $a