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\title{Balkan Mathematical Olympiad 2013}
\author{UK leader's report: GCS}
\date{August 2013}
\begin{document}
\maketitle

The Balkan Mathematical Olympiad was held in Cyprus from 28 June until 3 July. 
The UK Team Leader was Dr Geoff Smith of the University of Bath and the Deputy Leader 
was Dr Gerry Leversha, formerly of St Paul's School. The United Kingdom 
participates as a guest nation in this competition, and we have 
a self-imposed rule that we will not send a student to it more than once. 
This creates a lot of churn,
and gives many students the experience of international competition.

In normal circumstances, the Balkan MO is held in May, but this year
that proved impossible. The Cyprus Mathematical Society stepped
in to organize an emergency edition of the competition a couple of
months later than usual. This had the effect that some regular guest
countries, such as France and Saudi Arabia for example, 
were not in a position to participate. Unsurprisingly, the political dispute
concerning the division of the island of Cyprus led to further complications.

There were unusual and complicated bureaucratic problems associated
with taking this team to Cyprus. Thanks are due to Dr Don Collins, the BMOS
treasurer, and Rachel Greenhalgh, Director of UKMT, who worked 
tirelessly and successfully to overcome the difficulties. Further
thanks are due to the families, teachers and guardians of the students 
concerned for working together so effectively at speed.

The team was as follows: 

\medskip

$\begin{array}{ll}
\mbox{Oliver Feng}&	\mbox{Eton College, Berkshire}\\
\mbox{William Gao}&\mbox{Merchiston Castle School, Colinton, Edinburgh}\\
\mbox{Frank Han}&\mbox{Dulwich College, London}\\
\mbox{Maria Holdcroft}&	\mbox{Willink School, Berkshire}\\
\mbox{Freddie Illingworth}&	\mbox{Magdalen College School, Oxford}\\
\mbox{Warren Li}& \mbox{Fulford School, York}
\end{array}$

\medskip

The team labels were subject to a permutation beyond UK control.
The results were as follows 

\begin{center}
http://www.bmo2013.eu/index.php/en/contest/olympiad-results
\end{center}
and the performance of the British team is shown in the following table:
\medskip

$\begin{array}{llrrrrrl}
        &                  & \mbox{P1} &  \mbox{P2}&  \mbox{P3} &  \mbox{P4} & \Sigma & \mbox{Medal}\\
\mbox{UNK1}	&\mbox{Oliver Feng}&	10&	10&	0&	0&	20&	\mbox{Silver}\\
\mbox{UNK2}	&\mbox{William Gao}&	2&	3&	1&	0&	6&\\	
\mbox{UNK3}	&\mbox{Frank Han} &	10&	8&	10&	1&	29& \mbox{Silver}\\
\mbox{UNK4}	&\mbox{Freddie Illingworth}&	10&	10&	10&	4&34&\mbox{Gold}\\
\mbox{UNK5}	&\mbox{Maria Holdcroft} &	10&	10&	10&	0&30&\mbox{Silver}\\
\mbox{UNK6}	&\mbox{Warren Li} &	0&	3&	10&	1&	14&\mbox{Bronze}
\end{array}$

\medskip

Here are the problems of the competition.

\medskip

\noindent \textbf{Problem 1\ }In a triangle $ABC$, the 
excircle $\omega_a$ opposite $A$ touches $AB$ at 
$P$ and $AC$ at $Q$, and the excircle $\omega_b$ opposite $B$ touches 
$BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of 
$C$ onto $MN$, and let $L$ be the projection of $C$ onto $PQ$.  
Show that the quadrilateral $MKLP$ is cyclic.  

\medskip

\noindent \textbf{Problem 2\ }Determine all positive integers $x, y$ and $z$ such that 
$x^5 + 4^y = 2013^z$.

\medskip

\noindent \textbf{Problem 3\ }Let $S$ be the set of positive real numbers. Find all functions 
$f : S^3 \longrightarrow S$ such that, for all positive real numbers 
$x, y, z$ and $k$, the following three conditions are satisfied: 

\begin{enumerate}
\item[(a)]  $xf(x,y,z) = zf(z,y,x)$ 
\item[(b)] $f(x,yk,k^2z) = kf(x,y,z)$, 
\item[(c)] $f(1,k,k + 1) = k + 1$.  
\end{enumerate}

\medskip

\noindent \textbf{Problem 4\ }In a mathematical competition some competitors are friends; 
friendship is always mutual, that is to say that when $A$ is a friend of $B$, 
then also $B$ is a friend of $A$. We say that $n \geq 3$ 
different competitors $A_1 , A_2 ,\ldots , A_n$ 
form a \emph{weakly-friendly cycle} if $A_i$ is not a friend of $A_{i +1}$, for 
$1 \leq i \leq n$ $(A_{n +1} = A_1)$, and there are no other pairs 
of non-friends among the components of this cycle.  
The following property is satisfied: 

\emph{for every competitor $C$, and every weakly-friendly cycle $\mathcal S$ 
of competitors not including $C$, the set of competitors $D$ in $\mathcal S$ 
which are not friends of $C$ has at most one element.}  

Prove that all competitors of this mathematical competition can be 
arranged into three rooms, such that every two competitors that 
are in the same room are friends.  

\section*{Diary}

\noindent \textbf{Friday June 28th\ }I arrive first at 
Heathrow Terminal 1, and just as I am settling in, receive a telephone call from
a legal person associated with Morgan Matthews's
fictional reworking of \emph{Beautiful Young Minds},
a romantic comedy in which, in a novel twist, 
love triumphs. The film-makers
want the freedom to defame (or portray accurately, I don't know
which would be worse) a fictional UK IMO leader, and 
in consequence I must sign a piece of paper to licence
calumny, accuracy and similar dangers. In fact I have already
signed a piece of paper to this effect, but it turns out
that was the wrong piece of paper, and I must sign the right
one. This is hardly a convenient moment, but I agree 
to sign and send a scan of the new piece of paper 
to the legal people from my forthcoming Cypriot HQ in Nicosia.

The team arrive, William's flight from Scotland
being a little late. There is no time to take lunch,
but people with sufficiently empty lives that they
read the terms and conditions of their plane tickets
(and we have some) assure the rest of us that there
will be a meal on the plane.

We take off, and a eventually mini-meal arrives in the form 
of a small tray of pasta, with a little sauce, a sliced 
mushroom and a chicken fragment. Fortunately I do
not have a large appetite, but I worry about the others.
The modest nature of this meal will cause problems later.

We arrive about 9pm local time, and by the time we 
clear the formalities, it is dark. The Macedonians
are waiting for us, to share our bus. The plan is
to drive north to Nicosia, drop the leaders (Vesna of
FYROM and me), and then take both sets of students to
the mountains. Vesna and I scuttle into the Hilton
at 10:20pm, check-in, dump our bags and just manage to
get to the restaurant before it closes.

The rest of group had a more difficult time. They drove west
up mountain roads until, very late, they arrived at the
Rodon Hotel resort in Agros. No food was waiting for them.
Fortunately I have not written \emph{cannibal} on the dietary
requirements section of the relevant form, so Gerry is well placed
to go into serious howling mode. Somehow he managed to get through 
to Greg Makrides, the head of most things (including
the Balkan Mathematical Olympiad) in Cyprus, and very
quickly a cold collation appeared. Well done Gerry, Greg and the Rodon Hotel.

Meanwhile, back at the Nicosia Hilton, I 
have received the shortlist of problems. All five UK
submissions are on the shortlist, and two of them are my 
own personal efforts. In a characteristic display
of good taste, my questions will not be selected by the jury.

\noindent \textbf{Saturday June 29th\ }
I have breakfast outside by the pool, and run into
my old friend Massimo, the Italian leader. The leaders then 
gather for a brief jury meeting to eliminate a problem or two
for being known, and then depart for Agros to go to
the 30th Balkan MO opening ceremony. Having seen the
shortlist, I must not be intimate with my deputy or students,
but we wave and smile across the room. There is a gratifying
amount of folkloric dancing, and a play about Pythagoras
is put on by local schoolchildren. We are formally welcomed,
and then the leaders are sent back to Nicosia.

In the afternoon the leaders see
the solutions to the problems
on the shortlist, and then we choose the problems for the
exam paper. A pretty geometry problem
from Bulgaria is selected in position 1, and a Serbian 
elementary number theory problem in position 2. Jack Smith's 
UK submission, a tricky and unusual functional
equation problem was selected in position 3. Finally 
a Serbian question was chosen as Problem 4. 

Massimo and I form the English Language Committee, and 
polish the language and notation of the problems. We
then present our work to the rest of the jury, who suggest
further improvements.

I then scurry off for a few hours rest while the rest of the jury
go to work to set the paper in eleven other languages.
At the end of the process we are invited to pack our own 
students' envelopes with exam papers in appropriate languages,
and associated instruction sheets. I am not sure that this is a very
good idea because it presents a clear opportunity to breach security.
A dishonest leader (with dishonest students) could slip in
a note containing summary solutions. Perhaps this practice 
should not be encouraged?

I am beginning to be a little mystified by the Nicosia 
Hilton. It is a classy hotel, and is very comfortable. However,
apart from the Balkan Maths Olympiad people, there are very few guests.
It is an expensive place, and of course the Balkan MO is not picking 
up bar bills. I notice that a fairly normal bottle of wine is on sale
in the bar for \euro{44}. This all looks very strange, with the bar
so expensive that no-one is using it, the high-rollers having
disappeared since the economic crisis. Why do they not drop their prices
to more modest levels and make some money? 

\noindent \textbf{Sunday June 30th\ }
We have breakfast early, check-out from the Hilton, and pile into
a bus to transfer to the mountains. We close the curtains and doze.
On arrival at the Rodon Hotel, I pass deputy Gerry Leversha en route
to the jury room, and shake his hand. 

The jury is summoned to the exam room to witness the
exam papers being distributed. I try to imagine why this is 
a good idea, but cannot work it out. There appears to be only
one clock in the ell-shaped jury room, and it is situated in a place
that makes it maximally difficult to see. The organizers improvise
brilliantly, and an itinerant invigilator wears a clock on his chest
for the duration of the exam.

The jury retires to its rather small room.
and at 10am the exam begins. The jury does not have to wait 
long for the first question. 
It is from a British student, who wants to know the definition of
a \emph{projection} in the context of the geometry question. I propose
a formal abstract definition, and it is approved by the jury and sent
back.
There are only a few questions sent in during the 30 minute 
questions window, so it seems that the wording on the papers
is quite clear. 

There is a break for coffee, and I go to find Gerry. 
I find him in the \emph{al fresco} coffee lounge, on a large balcony
with a fantastic view south over Cyprus. In the far distance
there is a patch of water just visible, and consulting a map
we realise that we are looking at the eastern coast of the 
Akrotiri peninsula (Cheltenham-super-Mare). 

We have a meeting with the co-ordinators who collect
alternative solutions and propose marking schemes. I 
hand in my solution to the geometry problem which,
somewhat predictably, 
uses areal co-ordinates and reduces the problem to 
linear algebra. It turns out that areal methods are
not often used in this part of the world, and I 
later give a brief impromptu tutorial on areal techniques
to those who are interested.

At the end of the exam, the British students pile out, and they
have had very different experiences. William Gao has had an off-day,
but has some part marks, whereas Maria Holdcroft, Freddie Illingworth
and Frank Han claim to have done about three questions with Freddie
having perhaps a little more, Oliver
Feng has done a couple, and Warren Li about one and a half.
Rumour has it that no-one has solved Problem 4 
completely. Therefore the Gold Medal boundary looks certain 
to be in the low 30s, so Freddie might get a gold, and
both Maria and Frank can be sure of getting at least silver medals.
Warren and Oliver look likely to get bronze medals. I don't tell
William, but the bronze medal cut-off at the Balkan MO is usually
quite low, so there is an outside chance that he will get a bronze
medal if his fragments can earn enough marks. Unfortunately, they don't.

An official marking scheme document is issued when we receive the scripts 
in the late afternoon. Gerry is very confident at Euclidean Geometry, so
he takes Problem 1, and I look at the other scripts. 
Now is the time for the students to relax, and for the leaders to work
really hard, and engage in fine textual analysis of the scripts.

\noindent \textbf{Monday July 1st \ }Today is 
co-ordination day. We have a happy time, since the 
co-ordination is expert and fair. Our students, save poor William,
have done interesting work, and Freddie Illingworth had excelled himself.

There is a curious incident during co-ordination. While the UK
is settling in to discuss a (non-geometry) problem, an Italian
geometry script is thrust into my hands by a co-ordinator. 
The student has solved the problem by areal co-ordinates, and I
am asked to verify that the method and calculation are correct.
It looks and feels right, but I do not have the time to give it 
the full going over that I would wish. I give it my provisional
blessing, and when I check later, it is in fact correct.

The jury meets at the end of the day and approves the medal cut-offs
based on the results of the ``official'' nations. 
The cut-offs were 31 for gold, 20 for silver and 8 for bronze.
Unfortunately 
William has not quite managed to get a bronze medal. Life is 
sometimes very unfair, for William is a very talented young mathematician.
Our other team members have done better, and Freddie has secured 
a very clear gold medal. This underscores
the strength of the UK IMO team of 2013, because Freddie is not
in it! Oliver, Frank and Maria all have silver medals (Oliver having no 
margin of safety), and Warren has a solid bronze.

\noindent \textbf{Tuesday July 2nd\ }Today we are going out to play. 
In the morning we all take buses to Nicosia to visit the Archaeological Museum.
This contains an astonishing collection of artifacts spanning 
a huge period. These objects are not properly separated and classified
by era and civilization, so you have to keep your wits about you to 
follow what is going on. However, this is one of the great museums of the 
world, and every visitor to Cyprus with any interest in civilizations
ought to pay a visit.

For lunch we visit the University of Cyprus where we are given a warm welcome, 
and attractive scholarships are offered. 
We listen to a passionate speech about human tragedies associated with
the division of the island delivered by the Rector.

As we had driven through Nicosia, we had, for a while, been
travelling parallel to the \emph{Green Line}, the UN controlled neutral zone
which separates the widely recognised Republic of Cyprus, 
and the territory to the north which is recognized as a 
legitimate political entity only by Turkey. There is a giant
flag of the ``Turkish Republic of Northern Cyprus'' somehow 
painted on, or carved into, the mountains to the north. 

At the moment there are no active hostilities, and I read that
it is possible
to cross the border without too many formalities. At least one 
enterprising team leader actually did this, but in case this 
causes offence, my memory has become hazy as to who this was.
 
We return to the Rodon Hotel for the closing ceremony. The Minister
for Education is there to make a speech and give out medals. We then
step outside and have a charming farewell dinner in the gardens.

\noindent \textbf{Wednesday July 3rd\ }
The return journey to London was relaxing and uneventful, except for
one curious event. After we took off, the
announcement from the cockpit deviated from the usual patter
concerning cruising height, the likely time of arrival and 
impending turbulence.
An officer of Cyprus 
Airways made a speech which made it clear that he had trenchant
views concerning
the division of Cyprus,
and which nation was responsible. 

After we arrive back at Heathrow, I go to the central bus station to 
pick up a rail-air bus to Reading. This proves difficult. The central
bus station is so over-stretched that they cannot assign buses to stops
in advance. My bus is running late, and the number of its stop
is announced one minute before its scheduled departure time. Those
of us passengers who scamper from the waiting room towards the stop
are rewarded with the sight of the bus driving off. Sometimes I
am glad that the UK has robust gun controls.

I go back to the waiting room to express my views somewhat 
forcefully. The central bus station is a disgrace, and I advise 
readers not to use it.
It turns out that there are dedicated rail-air stops outside most
of the terminals, and it is much safer to use those. 

\noindent \textbf{Summary\ }
The Cyprus Mathematical Society did a wonderful job of putting on this 
emergency competition at very short notice, and I thank all concerned
for this fantastic effort, and for the warmth of the welcome that we received
in both Nicosia and Agros.
\end{document}

