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\title{International Mathematical Olympiad 2013}
\author{UK leader's report: GCS}
\date{August 2013}
\begin{document}
\maketitle
The 54th International Mathematical Olympiad was held on the Caribbean coast of Colombia
in July 2013. The students stayed in Santa Marta, and the leaders in Barranquilla, at the mouth
of the Rio Magdalena.
The UK Team Leader was Dr Geoff Smith of the University of Bath and the Deputy Leader
was Dominic Yeo of Worcester College, Oxford. The person in charge of pastoral
matters is our Observer C, Bev Detoeuf from the Leeds Office of UKMT.
The team was as follows:
\medskip
$\begin{array}{ll}
\mbox{Andrew Carlotti}& \mbox{Sir Roger Manwood's School, Kent}\\
\mbox{Gabriel Gendler}&\mbox{Queen Elizabeth's School, London}\\
\mbox{Daniel Hu}&\mbox{City of London School}\\
\mbox{Sahl Khan}&\mbox{St Paul's School, London}\\
\mbox{Warren Li}& \mbox{Fulford School, York}\\
\mbox{Matei Mandache}&\mbox{Loughborough Grammar School}
\end{array}$
\medskip
The reserves were
\medskip
$\begin{array}{ll}
\mbox{Frank Han}& \mbox{Dulwich College, London}\\
\mbox{Maria Holdcroft}&\mbox{Willink School, Berkshire}\\
\mbox{Freddie Illingworth}&\mbox{Magdalen College School, Oxford}\\
\end{array}$
\medskip
Andrew Carlotti was competing for the fourth time, and by securing a gold medal
at IMO 2013, he now has the best IMO medal
record of any British student.
He has one bronze medal and three gold medals, won during 2010--13.
This takes him
above Simon Norton (3 gold medals, 2 special prizes) and the late
John Rickard (3 gold medals,
3 special prizes).
The performance of the British team of 2013 is shown in the following table:
\medskip
$\begin{array}{llrrrrrrrl}
& & P1 & P2 & P3 & P4 & P5 & P6 & \Sigma & \mbox{Medal}\\
\mbox{UNK1} &\mbox{Andrew Carlotti}&7&7&0&7&7&6&34&\mbox{Gold} \\
\mbox{UNK2} &\mbox{Gabriel Gendler}&7&5&0&7&6&0&25&\mbox{Silver}\\
\mbox{UNK3} &\mbox{Daniel Hu} &7&7&0&7&7&2&30&\mbox{Silver}\\
\mbox{UNK4} &\mbox{Sahl Khan} &7&0&0&7&7&0&21&\mbox{Bronze}\\
\mbox{UNK5} &\mbox{Warren Li} &7&7&0&7&7&0&28&\mbox{Silver}\\
\mbox{UNK6} &\mbox{Matei Mandache} &7&7&0&7&7&5&33&\mbox{Gold}
\end{array}$
\medskip
The cut-offs were 15 for bronze, 24 for silver and 31 for gold.
There were 97 participating nations.
The unofficial ranking of countries by total scores has
the UK in 9th position overall, 2nd among European nations (behind
Russia), and in 1st position among the nations of the European
Union (by some
margin).
This represents the best team performance by a British side at an IMO since
1996 when we finished 5th.
Here are the top 30 places at IMO 2013. The full table can be found at
\begin{center}
{\tt http://www.imo-official.org/year\underline\ country\underline\ r.aspx?year=2013}
\end{center}
\medskip
1. China (208), 2. South Korea (204), 3. USA (190), 4. Russia (187),
5. North Korea (184),
6. Singapore (182), 7. Vietnam (180), 8. Taiwan (176)
9. United Kingdom (171),
10. Iran (168),
11. Canada (163),
11. Japan (163),
13. Israel (161),
13. Thailand (161),
15. Australia (148),
16. Ukraine (146),
17. Mexico (139),
17. Turkey (139),
19. Indonesia (138),
20. Italy (137),
21. France (136),
22. Belarus (134),
22. Hungary (134),
22. Romania (134),
25. Netherlands (133),
26. Peru (132),
27. Germany (127),
28. Brazil (124),
29. India (122),
30. Croatia (119).
Of the remaining nations, Anglophone and Commonwealth scores
include 31. Hong
Kong (117), 31. Malaysia (117), 48. New Zealand (77),
56. Sri Lanka (65), 58. South Africa (64), 61. Bangladesh (60),
64. Cyprus (52), 76. Ireland (33), 79. Pakistan (25),
84. Nigeria (18), 86. Trinidad and Tobago (16), 95. Uganda (1).
Italy are to be congratulated for finishing top of the nations
using the \emph{Euro}. France finished above Germany, a singular event which
had not happened since German re-unification.
There seems to be a general trend that some of the countries of
central and eastern Europe are getting lower IMO rankings, whereas the
nations of the Far East are doing very well recently. Indonesia
obtained an excellent result, a sharp improvement on their
previous performances, and Israel secured their best ranking position since 2000.
As often happens, China sent a very strong team, and are to be congratulated
for winning the event.
\section*{Problems of Day 1}
\begin{enumerate}
\item Prove that for any pair of positive integers $k$ and $n$, there exist $k$
positive integers $m_1, m_2, \dots, m_k$ (not necessarily different) such that
\[
1+\frac{2^k-1}n \;=\;
\left(1+\frac1{m_1}\right)\left(1+\frac1{m_2}\right) \cdots \left(1+\frac1{m_k}\right).
\]
\item A configuration of 4027 points in the plane is called \emph{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 \emph{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
triangle $ABC$ is right-angled.
\vspace{\baselineskip}
\emph{The \emph{excircle} of triangle $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$. The excircles opposite $B$ and $C$ are similarly defined.}
\end{enumerate}
\section*{Problems of 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~$CWM$,
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 $f\colon \mathbb{Q}_{>0}\to \mathbb R$ be a function satisfying the following three conditions:
\begin{enumerate}\renewcommand{\labelenumi}{(\roman{enumi})}
\item
for all $x, y\in \mathbb{Q}_{>0}$, we have $f(x)f(y) \geq f(xy)$;
\item
for all $x, y\in \mathbb{Q}_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
\item
there exists a rational number $a>1$ such that $f(a)=a$.
\end{enumerate}
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 \emph{beautiful} if, for any four labels $a