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\author{@GeoffBath, G.C.Smith@bath.ac.uk}
\title{UK Leader's report, Balkan MO, 2012}
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The UK team in the Balkan Mathematical Olympiad 2012
in Antalya (http://bmo2012.tubitak.gov.tr/) 
consisted of six boys. They were:

\vskip 0.5cm

\begin{tabular}{rl}Robin Elliott&	Reading School\\
Gabriel Gendler	& Queen Elizabeth's School, Barnet\\
Daniel Hu	& City of London School\\
Matthew Jasper	& St Crispin's School, Wokingham\\
Matei Mandache	& Loughborough Grammar School\\
Harry Metrebian	& Winchester College
\end{tabular}

\vskip 0.5cm 

The team was led by Dr Geoff Smith of the University of Bath, 
with deputy Dr Gerry Leversha, formerly of St Paul's School, Barnes.
The event began on April 26 and ended on May 2.

The problems were as follows. The language and notation has been slightly
modified.

\vskip 0.5cm

\noindent \textbf{Problem 1\ } Let $A, B$ and $C$ be points 
lying on a circle $\Gamma$ with centre $O$. Assume that
$\angle ABC > 90^\circ$. 
Let $D$ be the point of intersection of the line 
$AB$ with the line perpendicular to
$AC$ at $C$. Let $l$ be the line through $D$ 
which is perpendicular to $AO$. Let $E$ be the point of
intersection of $l$ with the line $AC$, and let $F$ 
be the point of intersection of $\Gamma$ with $l$ that lies
between $D$ and $E$.
Prove that the circumcircles of triangles $BFE$ and $CFD$ 
are tangent at $F$.

\vskip 0.5cm

\noindent \textbf{Problem 2\ } Prove that
\[ \sum_{\rm \scriptstyle cyc}
(x + y) \sqrt{(z+x)(z+y)} \geq 4(xy + yz + zx),\]
for all positive real numbers $x, y$ and $z$.
\emph{The notation above means that the left-hand side is
\[(x + y)
\sqrt{
(z + x)(z + y)} + (y + z)
\sqrt{
(x + y)(x + z)} + (z + x)
\sqrt{
(y + z)(y + x)}.\]}

\vskip 0.2cm

\noindent \textbf{Problem 3\ } Let $n$ be a positive integer. 
Let $P_n = \{2^n, 2^{n-1}\cdot 3,  2^{n-2}\cdot 3^2, \ldots,  3^n\}$.
For each
subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of 
$X$, with the convention that $S_{\emptyset} = 0$
where $\emptyset$ is the empty set. 
Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1} - 2^{n+1}$.
	
Prove that there is a subset $Y$ of $P_n$ such that 
$0  \leq y - S_Y < 2^n$.

\vskip 0.5cm

\noindent \textbf{Problem 4\ } Let $\mathbb N$ be 
the set of positive integers. Find all functions 
$f : \mathbb N \longrightarrow \mathbb N$ such that
the following conditions both hold:

\begin{enumerate}
\item[(i)] $f(n!) = f(n)!$ for every positive integer $n$,
\item[(ii)] $m - n$ divides $f(m) - f(n)$ whenever $m$ and $n$ 
are positive integers.
\end{enumerate}

\vskip 0.5cm 

Each problem was worth 10 marks, and the exam was sat over 4 hours 30
minutes. The results of the British students were
as follows.

\vskip 0.5cm

\begin{tabular}{|r|rrrr|l|r|}\hline
Name &	P1 & P2 & P3 & P4 & Total & Medal \\
\hline Robin Elliott& 	8 & 10&	8 & 1 & 27 &Bronze Medal\\
Gabriel Gendler&	1&	9& 10 & 0 & 20 & Bronze Medal\\
Daniel Hu &	10 & 0&	10 & 10	& 30& Silver Medal\\
Matthew Jasper&4 &0 & 10 & 1 &15& Honourable Mention\\
Matei Mandache& 	10 &10& 9 & 4&	33 &Silver Medal\\
Harry Metrebian&	10& 4&	0 & 0 & 14 & Honourable Mention\\
\hline
\end{tabular}

\vskip 0.5cm 

This was a very creditable performance, especially since the UK
has a self-imposed rule that no student can attend the Balkan MO
more than once. The current strongest players at maths competitions
were therefore left at home (including two IMO gold medallists).
This Balkan side was very much a team of the future. Although Robin
will be going to the University of Cambridge this year, the other
team members all have at least one more year of eligibility for
international mathematics competitions, and it seems very likely
that several of them will attend IMO 2013 in Colombia. 

Thanks to his strong showing in this Balkan MO,  Matei Mandache 
has been added to 
the small group of students in contention for the six places
in the UK team at the 2012 IMO 
in Mar del Plata in July.

The competition was brilliantly organised by 
the Scientific and Technological Research Council of Turkey (T\"{U}B\.{I}TAK).
I thank our Turkish hosts for many kindnesses, and the UK deputy Gerry 
Leversha, who shows little sign of slowing down in retirement. Of
course one must acknowledge the help and support of the staff and 
volunteers of UKMT, and the families of the young people
involved. Finally, I would like to thank the team itself. They behaved
in an exemplary manner throughout, and were a credit to themselves, their
families and their schools.

On a personal note, it was an exceptional pleasure that my former
Ph.D student, Prof Dr Huseyin Aydin, Dean of the Faculty of
Science of Erzurum Technical University
(http://www.erzurum.edu.tr/etu\underline{\ }/huseyin-aydin.aspx) 
was able to drop
in to the competition to renew our friendship. We will be trying
to set up academic links between Bath and Erzurum, a magnificent city
on the ``silk road'' in Eastern Turkey.

\section*{Diary}

As usual, in a maths competition where almost everything goes well, 
it is natural to draw attention to occasional imperfections for comic effect.
This does not detract from the greater truth, that the competition was a
tremendous 
success, and was enjoyed by all participants.

The diary is just one person's perspective, and the students will have
experienced a completely different event.

\subsection*{Thursday April 26}

I begin by letting my dongle get me into trouble. While
clearing
my email on the train, I omit to notice that we are pulling
into Reading station until the train actually stops.
Since I must change trains here, this is the time for a serious
panic. I unplug everything and thrust it into my bag, and 
scamper for the door making pathetic whimpering noises, scattering 
pens, change and spare spectacles from various pockets. A
train official takes pity on me, and holds the door and therefore the
train, while passengers kindly pass me my possessions. The conductor
even waits while I check that I still have my wallet and passport before 
allowing the train to move off. 

The second train journey is a sequence of happy surprises. I assume that
I must have lost various items in my dramatic departure from the
express train. However, as time goes by, I keep finding that more and more
items are in my possession in odd pockets and the inner recesses of the bag.

I arrive at Gatwick airport in good time, at about the same time as Gerry, 
and we go downstairs from the departure lounge for an all-day breakfast.
We then return to meet the students and their families,
and I hand out union flags, team shirts and lapel pins.

The journey to Turkey was more arduous than expected. The easyJet flight from 
Gatwick to Antalya turned out to be monopolised by a golf club.
This was not a golf club for the middle-classes at play. Rather, it
would be the golf club of choice for the characters from the TV soap-opera
\emph{Eastenders}. When they arrived on board, there was a little
crooning and some mildly boisterous behaviour. In fact they must have 
pre-fuelled before take-off, and were merely pretending to be
sober. As soon as the plane took-off, the booze orders started flying, and 
two hours into the flight, they had drunk the plane dry. By this time, 
the volume of their senseless and ceaseless gibbering had become 
horribly oppressive. Even placing your fingers in your ears gave little
respite.

As we landed, one of the other humans on the flight tried 
to point out to a golfer how unreasonable it was to hold a 
party on a scheduled flight. He was admonished for his elitist
attitudes, and told to fly first-class in future.

All credit to the easyJet crew. They continued to act in a completely
professional manner, despite the unfolding scene. It has been over a 
week now, and I still have a buzzing in my ears. The males shouted
and the females shrieked, in a squalid parody of conversation.
As you can imagine, Gerry took it all in his stride, exuding the
relaxed bonhomie with which his friends are so familiar. 

I rather
hoped that a golf suppression unit of the Turkish police would be waiting 
for them in Antalya,
equipped with dogs, nightsticks and knuckle-dusters. 
Unfortunately this was not the case. It turns out that
Belek, the holiday resort suburb of Antalya, has 21 golf courses.
Actually building golf
courses to attract them seems foolhardy. 

We all have to pay the 10 pound entry fee for Turkey, gathering
little visa stickers in our passports. Finally we make it
through customs and immigration at about 10:30pm Turkish time. 
Disturbingly, there is no-one to meet us. I feign calm.
When arriving at IMOs as a solitary leader, there have been 
occasions when I was not met, and some initiative was required
to make contact with the organizers. However, when turning up with
students, you do expect to be met.

After a while, I phoned the UKMT executive director in Leeds, largely
to mitigate the loneliness. I began to formulate a plan: wait for an hour,
and if no-one shows up, then check in to a hotel. At length, a 
young man called Sal turns up. He is the UK guide. There is 
much apologizing, but I am pretty grumpy by now, the 
barbarians on the plane having exhausted my goodwill toward the
human race (and other primates). It turns out that while the 
UK easyJet plane was on time, lots of
other countries are arriving late, and this has disrupted the host country's
meet and greet schedule.

I am put on a bus with other leaders, and we plunge into the night. 
We do some more picking up at another terminal, and at last head off down the 
motorway. We pass a bus in the same livery as our own, so I look inside,
and sure enough Gerry and the students are all slumped in their seats,
looking the worse for wear. They will not know that they were under 
observation until they read this report.

When I arrive at the \emph{Bel Conti Resort Hotel} it is close to midnight.
I scan the lobby for Neanderthals, but it seems clear.
We are told that food will be available until 1:30am. Having a 
distrusting nature, I dump my bags, and head straight to the restaurant.
It is closing up, but I manage to grab a plate of food just before it shuts. 
Replete, I saunter
upstairs to the bar, to discover less cynical but more hungry leaders who
are now casting around for nourishment, frustrated that 
the restaurant is shut. There is a promise that the hotel
will open for breakfast at 1am, but in fact it won't, and by then I am 
tucked up in bed.

Our hosts have provided a lavish collection of presents for
participants. This is not only the year of the London Olympics,
it is also the year of the rucksack. I was given one in the Gulf
Mathematical Olympiad, one more at the European Girls' Mathematical
Olympiad, and now a third at the Balkan Mathematical Olympiad. 
These are excellent and useful gifts, many thanks. However,
future organizers please note: go easy on the rucksacks for 
a few years. 
The olive oil and scented soap are wonderful.

\subsection*{Friday April 27} 

I find the first night in a new bed rather difficult, so I don't get
to sleep until about 5am. I sleep through my alarm, and wake up at 
about 8:40am
Turkish time. I rush downstairs, have a quick breakfast, then
dash back to my room for a shower. I can dress down for the 
opening ceremony, since I am not required on stage. A short
bus journey past countless hotels takes us to the venue.

The speeches at the opening ceremony are pleasantly short.
There is a tentative announcement that Turkey might be able
to host EGMO, the European Girls' Mathematical Olympiad, in 2014.
This is excellent news. 
We have a magnificent display of folkloric dancing, and 
connoisseurs of the form settle in for a real treat. Unusually,
there is no parade of the nations.

The organizers and leaders return to their own hotel, to
engage in problem selection. I look around and am delighted
to see so many old friends leading and 
observing various teams. A Balkan MO
jury is a very male environment.
There is some manly hugging. 
We establish a schedule, and
agree to meet again after lunch. This gives us a few more precious
hours to study the problems.

In the afternoon we set the paper easily enough. For
problem 1 we select G1 from Romania. Problems 2 and 4
are Saudi submissions, and Problem 3 was from the UK.
The UK submission was made in Bath.
It was a joint effort, and was constructed from an
issue which arose during Carolyn Ashurst's Ph.D research.
I am her supervisor, and as we resolved the
difficulty, it became clear that the mathematics involved
would make a suitable problem for an international competition.
I therefore submitted it to the UK International Problems Committee,
and forgot all about it. I was pleasantly surprised
to see it again when opening the
Balkan MO shortlist for 2012. The combinatorics shortlist was
unusually short, and consisted of just this one question! There were
lots of submissions for Algebra and Geometry, and a few for Number Theory.

As the only native speaker of English on the jury, I convene
an English Language Committee based on people's willingness
to sit next to me and drink coffee on the verandah.
We construct the official version
of the paper.
This is subsequently modified and approved by the jury,
and an immaculate typeset version is rapidly produced
by the Italian leader, Massimo Gobbino. 

I am now ahead of the game, and can head to the bar for a 
complimentary gin and tonic, but other leaders now have to
work hard on producing different language versions. Once
those are approved, we hear mark scheme proposals from the four
problem captains.

\subsection*{Saturday April 28} 
Leaders have to pack early in the morning, and we transfer to the
students' site, arriving just as the exam begins.
It turns out that we have moved to the 
\emph{Letoonia Hotel}. Is this the Turkish word for Lithuania?
This is one of a city of hotels that have been constructed to separate
Ivan from his petroroubles, hence the name (or not, 
depending on what the name means).

We have a jury room very close to the exam hall, and I am
delighted to find Gerry again. The organizers have
helpfully provided two seats to a table, so he can 
join me in the jury room. We have some coffee, and await
the first student query.

A UK student breaks the ice to ask if 0 is a natural number.
After a minor grammatical correction from Gerry, I propose the
reply
\emph{``Z\'ero n'est pas positif en Angleterre''}, but wiser heads
prevail, and instead we send back the answer ``no''.
There are three pillars of British identity: we drive on the left,
the Queen is on the banknotes, and zero is not positive. 
There are mercifully few questions, so there is time
to go back to my room and unpack.

Comparison of my room and that of deputy Gerry Leversha does not
improve my humour. I have a fine bedroom, and the bed satisfies
my minimum dimensional requirement that it should be wider than it is
long. However, Gerry's bedroom is altogether more
splendid. It has a rural annexe, a sort of conservatory which 
spreads out into his grounds, and is filled with soft furnishings.
This will not do at all.

As the exam finishes, the story of an apparent scandal breaks. 
The French leader, Claude Deschamps, is spreading the news.
Our Balkan Problem 4 had been used three days previously in the
American national competition, the USAMO. 

Given the time difference, it seems that the problem had become available 
on \emph{Mathlinks} late in the evening of April 25th in Europe. On 
April 26th everyone was on the move, so there was only a limited opportunity
for students to view the solution. Investigation revealed that, as far as we
could tell, only a very small
number of students had read the problem on the internet in advance of the
Balkan Mathematical Olympiad, and of those, an even smaller number had
read the solution.

The jury chair, Azer Kerimov, has to make a quick decision. The choices are
to proceed with the competition, or to hold an emergency jury meeting
and to discuss possible extraordinary action (the discounting of all
solutions to Problem 4). I counsel him to proceed without a meeting,
as if nothing has happened.
I am sure that many other leaders also gave advice. If we had convened a
jury meeting at that stage, then there would be almost limitless scope for 
accusation and vituperation. It would take time to find out how
this double submission had happened, and I took the view that 
people had important work to do (marking and co-ordinating the scripts),
and that there would be an ample opportunity for an
inquest at the final jury meeting. 

The students' scripts arrived in the afternoon, and Gerry and I settled
in for close textual analysis. Gerry is a geometry specialist, and 
it made sense to put him in charge of Problem 1. I would lead the
co-ordination on the other three problems. It quickly became clear that
we had a rather good set of scripts, given that we had an inexperienced
team. 
  
\subsection*{Sunday April 29} This was a day for serious tourism. 
In the morning we visited 
Perga (or Perge), where the geometer Apollonius used to hang his hat
in about 250 BC. Perga has excellent rubble, and I find the
connection with the foundations of conic sections enchanting.
An ice-cream seems in order for obvious reasons.

We proceed to a park which contains in impressive waterfall,
and finally, in a sun-toasted state, we are dropped in central 
Antalya for shopping. Gentlemen of a certain age repair to 
a shaded coffee house and start to work their way through the menu.

\subsection*{Monday April 30} 
It is co-ordination day, the day when the marking police will establish
that we have marked our scripts correctly. 
Since the UK has submitted problem 3,
we must be involved in the co-ordination of that problem
for both Turkey and the guest team Turkey B, since a country
cannot co-ordinate itself unsupervised. This makes the day very
busy indeed.

Our student Gabriel Gendler has managed to solve Problem 2, except
that he has used an invalid symmetry argument. All statements which
he claims to be true are true, though the co-ordinators 
are suspicious of one result at first. I supply a valid proof of that
statement, and we agree to meet later. In the interim, I manage to come
up with a valid argument which is actually a simplified version of
Gendler's false reasoning, obtained by cunningly leaving out the bits that 
are wrong. This reduces the penalty for the error, and he gets 9/10 on 
that problem. 

On problem 3, the same UK student, Gabriel Gendler is in trouble again.
Is it just chance that the same student is involved? Well, no, not 
really. This time he has used a controversial form of induction,
and to the innocent eye there seems to be a small gap in the induction step. 
In fact there is no gap at all, but the co-ordinators do not realise this.
It turns out that the co-ordinators are reading the script as
if it were written in English. In fact, it is written in Gendlerese,
a modified version of the language in which quantifiers and parenthetic
remarks interact to introduce multiple levels of ambiguity. If you read on,
Gabriel does make his original intentions perfectly clear, but he
does not do that until about page 3. When I show this to the
co-ordinators, they immediately understand that the student's solution
is completely correct and that he deserves full marks (and that the same
cannot be said for his use of English).

The co-ordinations of the other scripts were relatively uneventful.
Matei lost a mark for overlooking a triviality on Problem 2, but we were
expecting that.

The final jury meeting was convened after dinner.
It opened with an unreserved apology from Fawzi Al-Thukair,
leader of Saudi Arabia, for the double submission 
of Problem 4 on number theory. The four problems of the contest had
been submitted by, in sequence, Romania, Saudi Arabia, the United 
Kingdom and Saudi Arabia again. 

I had been in contact with the US IMO leader Zuming Feng by email. 
From what he told me, and from Fawzi's statement, I 
have constructed the following simplified but, I hope, accurate
summary of events.

There were a number of internationally known 
trainers who had worked on the maths enrichment programme for
the Kingdom of Saudi Arabia. The problem in question had been constructed 
by a member or members
of this team some time ago, and in fact the problem had been submitted to
IMO 2010 in Kazakhstan, but had not been put on the shortlist. The problem
was therefore returned to the submitting nation.
Some, but not all, of this team of trainers ceased to 
work with the KSA in the (northern) autumn 
of 2011, and the two groups were no longer in regular 
communication with one another.
There was at least one person in each group who felt that they had 
``ownership'' of the problem, and the problem was submitted 
independently to both the
Balkan MO (February 2012) and the USAMO (December 2011), with neither
party being aware of what the other had done.

Without being privy to the (doubtless complex) history of the
invention and development of the problem, it is hard to know
which of the two (or more) rival claimants to ownership of its
intellectual property rights has the better case. 
Collectively, the people involved caused a foul-up, but clearly there are 
a multiplicity of views as to who carries the most responsibility.

It is not the case that the problem was somehow
recycled from the USAMO to the Balkan MO. That is completely ridiculous.
The accident happened, and it was a 50-50 shot as to whether the 
Balkan MO or the USAMO would happen first. Had the competitions been separated
by one fewer day in time, then it would been most unlikely that 
any of the students would have viewed the solution on the internet, since
they would have been having an early night before the Balkan competition. 
On the other hand, had the competitions been separated by one more day, then 
large numbers of leaders at the Balkan MO would have seen the problem
on Mathlinks, and the jury would not have selected it for the Balkan
contest. 

Thus we were very unlucky to experience this accident. However,
that is how accidents happen.

\subsection*{Tuesday May 1}
Bitter experience has it that if students are left in charge
of their passports, only bad things can happen. Therefore 
we confiscate travel documents whenever possible, and lock them in hotel 
safes. This morning Gerry decides to test his safe, just to make sure that he 
will be able to extract the passports in good time for the return
journey tomorrow. After battling with the safe for some time and getting
nowhere, he eventually resorts to calling hotel reception. There
is a little spare time this morning, so this is the ideal moment to
explore the resources of the hotel's engineering section.

A sequence of experts arrive, all thoroughly familiar with the
problem, but to little effect until the big man arrives. He tries all 
the usual 
technical fixes, before putting the safe on the floor and hitting it.
This works. 

Gerry is to be commended both for testing his defunct safe with 24 hours
to spare, and for finding someone with the appropriate sophisticated skills. 

Later in the morning, the competition set out for the \c Celebi Marina, 
a harbour you can see easily enough on Google Maps, a little to the
West of Antalya. We boarded a very large boat from the rear.

There were parrots to photograph, and loud music, but it was possible to
find quieter parts of the boat in which to hide. There was serious drama
as we brought the boat close to Antalya city centre. Almost everyone seemed
oblivious to the situation, but there was a swimmer in the water next to
a buoy, waving frantically in a ``come hither'' fashion. The boat picked him up.
He was wearing a wet-suit and flippers, and he climbed aboard the back of
the boat. He was given a mobile phone, and then things happened in short
order. I spoke with a crew member, who explained that he was phoning the 
police because another swimmer was missing. The boat then took him close to 
the shore, where he was met by a police launch. Our pleasure palace then 
moved out to sea, and I have no idea if this story had a happy ending.

Further along the coast, we came close to the shore to view
a waterfall plunging from a cliff top, straight into the sea.

We returned to the hotel to learn that the medal
ceremony would begin in half an hour. I ran to my
room, took a shower, and put on a suit and tie.

I sit in the front row because I am supposed to make a speech.
When my name is called, the UK students make a deal of noise, for
I had not told them that this would happen. I manage to get
off stage before the rotten fruit starts to fly.

Then there was much congratulating, medal and certificate collecting,
and at last the competition was over.

In the evening there is a farewell banquet, with waiters and a band.
Initially the band is a touch loud, and jury chair Azer finds
me outside, muttering about ``the threshold of pain'', ``white noise'',
``the Geneva convention'' 
and so on. He sensibly decides to make the band turn the volume down.
After they do that, the musicians even get applause between songs,
rather than the resentful stunned silences which had greeted their earlier 
intermissions.

Actually, the UK and the Saudis had the best experience, because
they put the two Kingdoms at the far end of the hall from the band.
It was all those Republics who formed the buffer that had
to soak up the worst of it.

The deputy leader of France, Igor Kortchemski, 
engaged in some spirited dancing, and
was no doubt preparing to do excellent work to improve Franco-Turkish 
relations. 

 
\subsection*{Wednesday May 2}
We meet in the lobby in time to catch the bus which will transfer us to
Terminal 2 of the airport. Gerry reports seeing a guest
and a technician hard at work hitting a safe as he made his
way to reception. 

The transport supremo informs us that we are being left in the hands of the
driver, and we are told that when he comes to the correct Terminal 2, 
he will stick up two fingers, thereby signalling that it is time for us
to depart.

There is one airport, but three terminals in Antalya, in an oblique
theological reference (St Paul and St Barnabas visited Perga together). 
The terminals are 
widely separated. Therefore it is crucial to get off at the correct one.
We roll along the Turkish roads, through the relentless landscape of
\emph{soi-disant} 5 star resort hotels, 
thrown up in the past 20 years to serve 
the needs of the Russian tourist industry. 

At length we draw up in front of a terminal, and the driver raises the
traditional farewell salute. Trusting folk start to exit the bus. At this
point I shout from the back that the terminal building is
not numbered, but has the chilling word `domestic' written across its front. 
Some people leave the bus, to interrogate the terminal. After a few minutes,
they return with no definite news, but we decide, on balance, that it is
not Terminal 2. We set off again, and after 10 minutes we arrive at a 
building which is labelled T2 on the traffic signs, and so we alight.

Luggage in tow, we make for the Air Berlin check-in. Back in Leeds, Bev
had spent the previous afternoon trying, in vain, to get the Air Berlin
computer to spit out boarding passes. It turns out that everyone else
has the same problem, so there are queues. We cunningly select the queue run by
a new and diffident member of staff, so it all takes an age. 

Our journey home involves changing planes in Nuremberg. We celebrate
our visit to Bavaria with ice-creams. The second leg of the journey
involves an underpopulated flight to Gatwick airport. As we settle into
our seats, our names were read out on the tannoy. We had to climb half-way 
down the steps, and identify our luggage so that it could be loaded aboard.
Evidently they have sufficiently few passengers travelling from
Turkey to London that their procedures are not finely honed.

The final flight is uneventful, and we enjoy several happy family reunions 
at Gatwick.
\end{document}


