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\title{Balkan Mathematical Olympiad XXVIII}
\date{May 4--8 2011, Ia\textcommabelow{s}i , Romania}
\author{Leader's Report by Geoff Smith }
\begin{document}
\maketitle\thispagestyle{empty}
This year the twenty eighth edition of the
Balkan Mathematical Olympiad was held in Ia\textcommabelow{s}i, Romania.
The name of this town is pronounced ``Yash''. Ia\textcommabelow{s}i
is a major city in Romanian terms, and is situated
close to the border with Moldova. However, the old region
of Moldavia spreads across the frontier, and Ia\textcommabelow{s}i is in it.
The UK selects a team of six students to attend each BaMO,
but this year Sherry Jiang was unable to attend because
of paperwork issues. The Romanian authorities were a model
of helpfulness, and they were not the source of the problem.
The team which travelled was as follows.
\begin{verbatim}
GBR1 James Aaronson St. Paul's School, London
GBR2 Sam Cappleman-Lynes Shebbear College, Devon
GBR3 Martin Chan Westminster School, London
GBR5 John (Hyunjik) Kim Hampton School, London
GBR6 Joshua (Yeuk) Lam The Leys School, Cambridge
\end{verbatim}
Note that we were designated GBR rather than UNK by the organizers.
Future organizers please note that we have a marginal preference for UNK,
but this is not a major issue. We adhered to our longstanding policy
of not allowing any student to participate in the Balkan competition
more than once. This ensures that lots of students get experience
in an international mathematics competition. GBR1 and GBR6
are in the current group of eight students contesting
the six places on the IMO team in Amsterdam. Thus it is not
a complete surprise that the marks in Ia\textcommabelow{s}i fell as they did.
This was our best ever team performance at the Balkan MO.
Moreover, GBR1 James
Aaronson was placed first in the competition, and that is
clearly our best individual performance in this event.
\begin{verbatim}
Name P1 P2 P3 P4 Total Medal
James Aaronson 10 4 10 10 34 Gold
Sam Cappleman-Lynes 10 0 4 0 14 Bronze
Martin Chan 10 0 5 0 15 Bronze
John Hyunjik Kim 10 0 4 0 14 Bronze
Joshua Lam 10 1 10 1 22 Silver
\end{verbatim}
I am Geoff Smith, University of Bath, and acted as Leader in Ia\textcommabelow{s}i. The
Deputy Leader, which for us is a pastoral post, was Rosaleen ``Rosie''
Wiltshire of Wootton Bassett School. It was Rosie's first trip to an
overseas maths competition. She did a great job, and seemed to enjoy
the experience very much. Indeed, the students were also a happy band,
and we had a marvellous time. There seemed to be a significant overlap
between the people who organized the Romanian Masters of Mathematics
in February, and the
people who ran the Balkan Competition in May. They did first-rate work.
Certain key figures are internationally
known including Sever Moldoveanu and Radu Gologan, but there is a small
army of Romanian volunteers who make these competitions possible, and
we all owe them a great debt of gratitude.
There is another country, the United Kingdom, which is also blessed
with a tremendous volunteer base, and a few dedicated professionals,
who work to make our participation in international competitions
possible. They also deserve virtual applause.
\section*{The Problems}
\noindent \textbf{Problem 1\ }Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid
and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and
$G$ respectively, and $l$ is the line through $G$ parallel to $AB$. The feet of the
perpendiculars from $E$ onto the lines $l$ and $CD$ are $H$ and $K$, respectively.
Prove that the lines $EF$ and $HK$ are perpendicular.
\vskip 1cm
\noindent \textbf{Problem 2\ }Given real numbers $x, y, z$ such that $x + y + z = 0$, show
that
\[ \frac{x(x + 2)}{2x^2 + 1}
+
\frac{y(y + 2)}{2y^2 + 1}
+
\frac{z(z + 2)}{2z^2 + 1}
\geq 0.\]
When does equality hold?
\vskip 1cm
\noindent \textbf{Problem 3\ }Let $S$ be a finite set of positive integers which has the
following property: if $x$ is a member of $S$, then so are all positive divisors
of $x$. A non-empty subset $T$ of $S$ is good if whenever $x, y \in T$ and $x < y$,
the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is
bad if whenever $x, y \in T$ and $x < y$, the ratio $y/x$ is not a power of a prime
number. We agree that a singleton subset of $S$ is both good and bad. Let
$k$ be the largest possible size of a good subset of S. Prove that $k$ is also the
smallest number of pairwise-disjoint bad subsets whose union is $S$.
\vskip 1cm
\noindent \textbf{Problem 4\ }Let $ABCDEF$ be a convex hexagon of area 1, whose opposite
sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine
the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs
to determine the vertices of another triangle. Show that the area of at least
one of these two triangles is at least $3/2$.
\vskip 1cm
The time allowed was 4 hours 30 minutes, and each Problem was worth 10 marks. Problems~1 and 3
are good questions. Problem~2 proved very hard under exam conditions. Problem~4
is even more attractive than it appears at first sight, and admits many lovely
solutions. Problem~1 is by David Monk of Edinburgh. Problem~2 is from Greece, and the other two
problems have Bulgarian provenance.
\section*{Leader's Diary}
\subsection*{May 3\ }We gather in Terminal 5 at Heathrow Airport. We have only five students
because, unfortunately, there are technical difficulties concerning
Sherry Jiang's travel documents.
Rosie Wiltshire, who sensibly lives in the eponymous county, is in
charge of discipline and good order. I arrive late, but laden with
fetching royal blue UKMT Balkan MO polo shirts. We quickly verify
that the students are not carrying geometry sets in their hand luggage.
Taking the passports, I attempt to use a machine to generate
boarding cards. However, the device quickly senses my limited
grasp of its interface, gives up with good grace, and helpfully
suggests that I go to a check-in desk. This works well, and we
soon find ourselves in the departure lounge.
We are scheduled to arrive late at night, and an airline meal will
not necessarily take the edge off the appetite of the students, so
we decide to eat. We quickly agree to use an oriental restaurant.
A couple of us go for bowls of excellent ramen. GBR1 James Aaronson
has serious food issues, exemplified by his order of chicken
curry with no curry. As most of us tuck in to our meals, GBR3
Martin Chan quietly points out that he has been to the restaurant before.
We ask him if he enjoyed it, and he replied that he and his family all
had food poisoning. I took the bait, and asked Martin why he did
not point this out when we were choosing a restaurant. He replied
that he expected to be outvoted, so did not offer any advice.
In fact the food was excellent and wholesome (this time).
At length we boarded the flight, and travelled across the Netherlands, Germany,
the Czech Republic, Slovakia, Hungary and finally Romania. We are
met at the airport by a hotel bus, and transfer to the RIN~4. There are
two RIN hotels near Henry Coanda (Otopeni) Airport, including
the juxtaposed RIN~3 and RIN~4. The hotels have correspondingly many stars,
so the arrangement serves to humiliate users of RIN~3.
At the check-in, late at night and tired, there is confusion over
whether or not the rooms have been paid for in advance. The chappie
gets quite excited until I produce the UKMT credit card, and then he soon
calms down. Rosie ushers the students to their rooms, and I sneak off to mine.
\subsection*{Thursday May 4\ }This is contest arrival day, and we have a mid-morning
flight from the same airport. At the gate we meet the Italian delegation
and a small party from Indonesia led by Anton Wardaja. It turns out that the Italians
did not diligently book early, as we had, and several of them
have been forced to travel business class on this internal flight
from Bucharest to Ia\textcommabelow{s}i. Alas this will not involve
reclining seats, champagne, quail, caviar and truffles, but rather
there is a curtain which gets pulled across, and you pay two and half times
more. I am pained by the humiliation, and resolve that never again will
we be upstaged by grandstanding Italians, and that UKMT treasure
must be squandered to this end. The airplane has propellers.
Ia\textcommabelow{s}i International Airport has a provincial feel. There is
class of primary schoolchildren at the edge of the tarmac, and for
a moment I fear that the folkloric dancing is about to begin.
Happily it turns out that they are there to see the flying machines.
We walk into the airport terminal, and wait for the luggage to
be handed in by a side door. After a few moments we gather our bags,
and step outside. There are buses and personal cars to meet us. I wave
farewell, and am driven to my hotel. This is the \emph{Ciric}, an airport
hotel in fact, so the journey takes about 5 minutes. The hotel
is beautifully located next to a lake. There is virtually no noise
from the airport
since the planes are infrequent, small, and are not jets.
My room is on the first floor (that is the second floor in some
cultures). I have a balcony from which you can view the lake, albeit over
a large roof. Having unpacked, I pop downstairs in search of leaders,
lunch and the shortlist. Leaders and organizers are easy to find.
The first leader I meet gives me the traditional greeting,
``Are you J\'ozsef Pelik\'an?''
There is a long narrow room which contains computers, coffee and
bananas, and I settle in and chat with colleagues. At length someone suggests
lunch, and I realise that the bananas are merely a prelude.
During lunch, many more people arrive, including Fawzi al Thukair from
Saudi Arabia, and the Saudi coach Titu Andreescu. There is much
shaking of hands and many warm smiles.
There has been considerable confusion over the Italian Observer
Maria Colombo. Maria was a contestant in IMOs 2005--7 (BGS). There were
a few language issues at the airport, and she had been taken, incorrectly,
to the students' site. Now she arrived. She is assigned the use of
the second bed in the Italian leader's room. The Italian leader
Massimo Gobbino is a respectable man of a certain age, and such
an arrangement violates taboos (even Italian ones apparently).
The hotel quickly sorts out a more appropriate arrangement, but
Massimo and Maria have to endure serious teasing.
On return to the UK, forensic investigation of our records will reveal
that Maria has form. She was originally assigned to be
involved in domestic arrangements
suitable for a \emph{m\'enage \`a trois} with British students
Alison Zhu and Dominic
Yeo in the Balkan Mathematical Olympiad of 2007.
In the late afternoon we get conference bags and shortlists, and
I retire to study the proposed problems. Many of the problems seem harder than last
year, but then I get more stupid monotonically.
In the evening there is a banquet in an outbuilding at
the lakeside. A flight of swallows is also enjoying its menu,
swooping low over the surface to take insects. I am seated
opposite Azimbay Kamila and Anton Wardaja, respective
leaders of Kazakhstan and Indonesia. We introduce ourselves, and
of course I say that I am ``Geoff from Britain''. I am asked if
I know ``Smith''. I am tempted to answer, correctly, ``no, but I do
know Pelik\'an'', but there are limits to how much confusion
a conversation can bear.
The handful of leaders who
drink alcohol are served the local spirit from recycled plastic
water flagons. After a while, I feel very ecological, and
go to my room to view the shortlist from a more philosophical
point of view.
\subsection*{Friday May 5\ }I get up early to work on the shortlist some more. The story is
that we have only a couple of number theory problems, one of which appeals
to quadratic reciprocity in a way which seems both unavoidable and unacceptable.
There are some very nice combinatorics problems, but again, not that many.
If you screw up your eyes you can pretend that one of them
is a number theory problem. Another one looks suspiciously like geometry,
but the Selection Committee has obviously been forced to make some
slightly \emph{outr\'e} classifications in attempting to foster the
illusion that we have a full spread of problems.
We have some very attractive geometry problems of a wide range
of difficulty. There are also lots of inequalities. I have neither talent for,
nor interest in, inequalities. This lot all look impossible.
First thing after breakfast we get in a bus to go to the opening
ceremony. As we wait, one of our hosts casually mentions that there
will be politicians present. I reach into my inside pocket to retrieve
the tie I keep for such situations. We are sat at the front of a hall,
with luminaries on stage, and the students and deputies behind.
In fact the deputies get a little mixed up with the leaders. At the IMO,
sirens would sound, and special forces would come crashing through the
walls to make sure that such a thing could never happen. However, the Balkan
Mathematical Olympiad is a more relaxed event. The compere is
the chief inspector of schools for the region. She power dresses, has
power hair, and hands out chocolate bars recklessly. I can see that
she must have a substantial fan base. She gave especially large
amounts of chocolate to Indonesia, since that country had made the
longest journey.
Leaders are each asked to give
a short speech, under a minute in length. Massimo Gobbino
of Italy is hilarious. He starts by promising to speak for no
more than 15 minutes. After the opening ceremony we are even allowed to
meet the students, but I make sure that this happens in public
surrounded by other leaders, so as to avoid giving the impression that
any information about the shortlist is being transferred. This makes
me uncomfortable, and I am pleased to slip away.
Back at the jury hotel, the jury meets, and the familiar ``beauty contest''
form is handed out. I counsel the jury that it would be wise to delay filling
in this form as long as possible. This is because we want
well-informed leaders to contribute to this survey.
For this to happen, people need time both to study the
problems, and to discuss the merits of the various proposals.
We agree not to hand in the survey until the afternoon. We break up
into small groups, but some of us prefer to work alone.
After lunch, I feel that my opinions have at least positive weight, and
the survey now makes sense. There is a delay while the data
is processed, and finally we get to see, in statistical form, the mind of
the jury. There is agreement that there are very few easy problems.
It will eventually turn out that the geometry problems labelled $G1$ and $G2$ are
both submissions from the UK. Problem~$G1$ is chosen.
Its author is the prolific IMO problem composer David Monk of
Edinburgh. We select a Greek Problem~2, an inequality with an
official solution which looks like a sequence of magical
moves, with the inequality neatly hidden at the end of the rainbow.
I wonder how many students will be able to follow the Yellow Brick Road.
Some people take the view that the students are smarter than we are,
which is certainly true. However, I fear that this will
end badly.
For Problem~3 we select a combinatorics problem which has
a whiff of number theory. The first half is not that hard,
but the second half requires
a more cunning idea. This problem will clearly generate lots of part marks.
The combinatorial geometry
question which attracts interest for Problem~4 is provoking an
arms race. The official solution is very clever, but it does not
seem that beautiful. On the other hand, the problem itself is very attractive.
Everyone in the room can see that we must be missing a sweet solution.
Massimo gets in first by suggesting the use of an affine transformation
to generate a couple of right-angles. He then finishes it off with some professional
algebra.
We choose the problem, but then I start peddling my solution,
which involves stealing Massimo's ideas but using them differently. I propose
using an affine transformation to generate a pair of oppositely oriented
equilateral triangles, possibly of different sizes. You then move their centroids together
in a way which preserves the integrity of the problem, and which I leave the reader to
rediscover. Finally you deploy some \emph{origami}. I am pretty pleased with this, but
my initial solution had dodgy parts. A dialectic exchange over dinner with
the Serbian leader Milos Stojakovic rendered the argument robust. Still, GBR1 James Aaronson
will put me to shame, and come up with the ``proof from the book''.
There is a translation phase. A few of us form the English Language Committee,
including the French leader Claude Deschamps. We are under pressure to produce
the wording at speed. We decide to adopt an anthropomorphic attitude to hexagons,
so that we can say ``whose diagonals'' rather than ``with diagonals''. The Saudi
leader Fawzi quietly questions this usage, and I agree with him
that we are stretching things a bit. However, a maths exam paper is not a
piece of fine prose. It should be as economical with language as is consistent
with complete clarity.
\subsection*{Saturday May 6\ }It is the morning of the exam. We have a very hurried breakfast,
and scamper into town. We were warned that traffic would be heavy. Such advice
could only come from someone more familiar with congestion in Ia\textcommabelow{s}i, rather than, say,
London. We arrived very quickly. The Hotel Astoria is opposite a sex-shop, which
sells a wide range of implausible items. The Astoria contains a
``London Pub'', with the traditional Shamrock logo and the word ``Irish''
written on top. The corridor which leads to the pub is called the
``London Boulevard'' which lends a Gallic air to this traditional
Irish London Paris pub. It is clear that the word ``London'' bestows mystery, just as the word
``continental'' is used in the UK, to denote something other-worldly, exotic
and intriguing (such as having a croissant for breakfast).
The Astoria is internally linked to a second hotel, the Traian, where we will
have meals. Opposite the Traian is ``Cowboys and Angels: English Gentleman's Club''.
It is pleasing that my country's reputation has spread so far, but perhaps
we had better not explore these issues further.
We quickly move to the exam site, a school along the road. The jury assembles
at 9:30 just as the exam starts. The deputies are allowed to join us, and this
gives me the chance to catch up with Rosie. She reports that the students
are well cared for, and there have been no unpleasant incidents. Apparently
we have three excellent guides and
they get on very well with the team.
For the first half-hour, students are allowed
to ask written questions of clarification. Each leader who is asked a question
proposes an answer to the jury. They discuss it, and approve a specific wording
for the answer. Quite often a student is simply confused, and we answer ``read the
question again''. A student sends a question which asks if he or she has
understood the question correctly, rephrasing it for our benefit. The student
has a perfect understanding of the problem. This is
a tricky one, since we don't want to give uneven assistance.
I suggest ``do not read the question again'', but the jury settles for a
more sensible response.
A British student asked if a prime number was a prime power. The jury accepted
most of my proposed answer, but struck out the word ``idiot''. Another British
student, Martin Chan, had a question. Remember him? He was the one who tried
to poison us all in Heathrow. Problem~1 concerned a cyclic quadrilateral
which was not a trapezoid. His question concerned the difference between
American and British usage of the word ``trapezoid'', and the related word
``trapezium''. The jury seems reluctant to engage in an etymological dispute,
and we send him the relevant definition of a trapezoid.
When the questions finish, we have a discussion with the
chief co-ordinator about the marking scheme. Calin is a Romanian
topologist with an accent like John Gielgud's. He picked it up
listening to BBC Radio. His beautifully modulated vowels put
mine to shame, and I adopt an appropriate forelock-tugging
frame of mind. The marking scheme is rather good, but with
the hint of decadence that often surfaces at the Balkan MO.
For example, there is considerable laxity concerning
diagram dependency in Problem~1.
We now jump on a bus and go on the leader's outing. This goes
to an orthodox monastery on one of the hills
which surround Ia\textcommabelow{s}i.
We get a monk (Brian Blessed with laryngitis) who shows us icons,
artifacts and giant barrels of dry white wine.
We return to our complex of hotels. Time is getting on, so I go
for lunch before checking-in. After an excellent meal, I am asked
for my room number. I explain that I have not yet checked in,
and this causes consternation. I offer to leave a deposit and return
with my room number, and honour is satisfied.
After lunch I have a chance to meet the UK students for a short
while, and they tell me what they think that they have achieved.
It sounds good, but at this stage I am circumspect.
Dan Schwarz, the Romanian-born problems specialist, has invited me for
a beer in the London Pub. He is also Canadian, so we need to catch
up on the Royal Wedding and similar things.
A rumour enters the pub and speeds its way to our table. The scripts
have arrived. I go looking for them, and discover them in an envelope
behind the reception desk. I dash upstairs and take them in sequence.
Everyone seems to have a perfect solutions to Problem~1, although
John Kim has resorted to a page of trigonometry to demonstrate that two
triangles are similar, ignoring the one line geometric justification
which lesser candidates used. For this he earns the \emph{stylo d'or},
the award for the British student who produced the most impressive
convoluted but correct justification of a statement which admits
a straightforward proof.
Problem~2 is stony ground, and marks will be scarce there. Both
James Aaronson and Josh Lam have full solutions to Problem~3, and
everyone has done the first part. Martin tentatively introduces
the necessary ideas for solving
the second part, but undermines his credibility with asides such as
``I can't prove it''. Only James Aaronson has a full solution to
Problem 4, and his argument is lovely.
Now is the time to indulge my passion for social media, and
I tweet vague good news about the UK scripts
from \textbf{@GeoffBath} using the hashtag
\textbf{\#BalkanMO}. I am looking forward to tweeting from
the Amsterdam IMO in July, giving a running commentary on
IMOAB meetings, especially during confidential and delicate
discussions.
\subsection*{Sunday May 7\ }We will try to complete the competition today, because
five nations need to leave early next morning. Co-ordination is
moved, and starts at 8:30am. This begins with Problem~4. I had not sat down
before the co-ordinators were passing on congratulations for James
Aaronson's wonderful solution.
Next came Problem~2. GBR6 Josh Lam gets a mark for dealing with a special
case. Then the fun begins. I show them some of Sam Cappleman-Lynes
rough work which comes within a whisker of doing what Josh did. It
is true that Sam has not completely nailed it, but I have high hopes.
The co-ordinators stroke their chins, and then one of them says that
the work falls short of getting a mark. ``If only there was a special
reason that we should give the mark. For example, is the student a girl?''
I start to flare up in outrage, when I suddenly realise that the co-ordinator is having fun
by quoting from my IMO 2004 report, when a co-ordinator actually said this
and meant it! We all laugh, and I say to them that they must have
established policy on this matter, and they should simply implement it.
They do, and unfortunately Sam collects a 0.
Then the co-ordinators pull a surprise. I knew that James Aaronson's
script was worth a mark for a good early move. However, he then plunged into a lengthy
attempt to slog out the solution by brute force. It seemed unlikely to me
that this would be worth anything, but they revealed that in fact the
particular path he had chosen could be made to work, he had hacked a
significant way through the jungle, and they were offering a total of 4/10
for his effort. They then teased me by suggesting that it would have been
helpful if he knew the AM-GM inequality, since then he could have
pressed further forward.
This is all excellent news. I scan the partial reports of other
countries' scores on display in the jury room.
Each of our students is sure to get a medal, Josh
will surely get a silver medal, and James should be a safe gold.
There are only four or five students who can possibly beat him.
We have arranged that the UK team and guides will go out for lunch.
We meet amongst the pigeons, and I give the students my forecasts
of their medals, and in fact my assessments will prove completely accurate.
We board a tram, which is quite exciting for most Brits. After 15 minutes we
arrive at a shopping mall and are taken to an Italian restaurant.
I return to the school fairly quickly, because the final jury meeting
is scheduled for 2:30. Well, I know this is not going to happen,
since there is always a brutal co-ordination which holds things up,
but I can't help my British programming, and return on time.
It turns out that Bulgaria are involved in a difficult negotiation
over Problem~2. We wait and wait, until the room starts to steam up.
At four o'clock we get permission from the jury chair Radu Gologan
to adjourn for an hour. This enables us to change the air in the room,
and to take showers.
When we return, the Bulgarian co-ordination is finished, and
Rosie is already in
the jury room.
She is rather excited, because the rank list of performances
has been distributed, and GBR1 James Aaronson is in first place.
It is time to begin. We stand for a moment to remember
Jacqui Lewis, the UK observer with students and deputy leader at
very many international competitions, including last year's Balkan
Mathematical Olympiad. She was popular both with students and staff,
and is badly missed.
Then we get down to business.
The long wait has not improved the general mood, and various
heartfelt grievances are aired. These involve issues
of national self-respect, and so it is no wonder that
some people are very engaged in the discussion.
The mischievous leader of
Turkmenistan, Erol Aslan, is sitting on my right, and urges
me to get involved but even I am not that stupid.
I decline the chalice and Erol gives a knowing smile.
Jury chairman Radu Gologan steers us through
the politics and back to our core concern, the administration
of a successful competition.
At length the jury addresses the medal boundaries. The cut-offs
(the thresholds) are
determined by the performances of official Balkan participants.
This gives rise to all sorts of complications, because this procedure
only allows rather loose control of the numbers of medals being won.
Truncation issues arise. The mood of the meeting
switches to a more co-operative and generous spirit, and the chair
negotiates a difficult path which results in decisions which are
a fair approximation to justice.
We let our team know about the preciousness of their gongs, but we don't reveal
James's first place. Later that evening, everything becomes public at
the medal ceremony. Romanian politicians get themselves
photographed next to James Aaronson enjoying what is probably his first Warholic
time-slot.
The students and guides depart for a farewell disco, and the adults make their
way to a restaurant for a final banquet. There is an array of impressive
cutlery, but after one course, cups of coffee are brought round.
This is quite disconcerting, and we wonder if the budget has run out.
However, after the coffee the banquet resumes its expected course.
Rosie and I leave at midnight, and I go back to send some Tweets
and post a message on Ask nRich.
At last the competition is over, so the obligations of complete jury
secrecy are now relaxed.
\subsection*{Monday May 8\ }We catch a lunchtime flight from Ia\textcommabelow{s}i to Bucharest, and
then an afternoon BA flight back to Heathrow.
GBR3 Martin Chan is playing Luddite tetris. He has a random number generator
which he hits every 5 seconds to select the shape of the next tetromino,
and then he quickly draws it in using a pencil. Apparently it is best played
as a competition between two players, presumably using the same sequence
of tetrominoes, to see who can create the larger number of full lines.
GBR2 Sam Cappleman-Lynes
has a plan to stay alone at Heathrow until past 10pm, and then catch a coach
to Exeter, arriving in the middle of the night. We overrule this, and get him
home at a sensible time by using a train.
\section*{Next Year}
At the closing ceremony, the leader of Turkey,
Sahin Emrah,
invited us all to the next Balkan Mathematical Olympiad
to be held in Antalya, Turkey. This is a very big place, population
2m, a holiday town on the
Mediterranean coast with many buildings from
the Ottoman era, and even a few from Roman times. Happily there are daily
flights between Stansted and Antalya using Pegasus Airlines.
\end{document}