\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{latexsym}
\title{Balkan Mathematical Olympiad 2010}
\date{Chi\c sin\u au, Moldova}
\author{UK leader's report}
\begin{document}
\maketitle

The Balkan Mathematical Olympiad is a mathematics competition
for children at secondary school. It is held annually, at a site which
moves around the region. In recent years, a few non-Balkan countries,
and countries of marginal Balkanity, 
have taken to participating unofficially, either as an end in itself,
or as valuable practice for the International Mathematical Olympiad
held each July.

The 27th competition was held in 
Chi\c sin\u au, Moldova during May~2-May~8 2010. 
Moldova is the poorest country in
Europe, but the warmth of the welcome, and the efficient
organization of the competition made this hard to believe.
I would like to mention all the main organizers by name, but
with characteristic modesty, the website does not
specify their identities.

The UK has a self-imposed rule that no student may participate in the
Balkan Mathematical Olympiad more than once. This ensures that lots 
of students gain international experience, but it also means that
we send teams which are weaker than they could be. Thus UK scores
in this competition are usually quite low. However, the team of 2010
rose to the occasion, and put in a very creditable performance.

Here are the members of the UK team.

\vspace{0.5cm}

\begin{center}
\begin{tabular}{ll} 
Benjamin Elliott & Godalming College\\ 
Richard Freeland & Winchester College \\
Sahl Khan & St Paul's School \\
Jordan Millar & Regent House School, Northern Ireland\\ 
Sergei Patiakin & Dame Alice Owen's School\\ 
Jack Smith & King's School, Grantham 
\end{tabular}
\end{center}

\vspace{0.5cm}

Leader was Dr Geoff Smith of the University of Bath, and 
Deputy Leader was Jacqui Lewis of St Julian's School, Lisbon, Portugal.
The involvement of the UK team in this competition 
is sponsored by Winton Capital Management.

The paper
lasts 4 hours 30 minutes, and each of the four questions is marked out of 10
points. The medal boundaries are determined by the results
of official participants, and this year the cut-offs were 12, 25 and 35.
Problem 4 proved very taxing, and no student produced a complete solution
to it during the exam. Here are the results of the British students.

\vspace{0.5cm}

\begin{tabular}{lcccccl}
Name & P1 & P2 & P3 & P4 & Total & Award\\ \hline 
Benjamin Elliott & 10 & 0 & 2 & 7 & 19 & Bronze\\ 
Richard Freeland & 10 & 10 & 10 & 0 & 30 & Silver\\ 
Sahl Khan & 10 & 1 & 0 & 0 & 11 & Honourable Mention\\ 
Jordan Millar & 10 & 0 & 1 & 0 & 11 & Honourable Mention\\ 
Sergei Patiakin & 10 & 10 & 5 & 2 & 27 & Silver\\ 
Jack Smith & 10 & 0 & 5 & 2 & 17 & Bronze
\end{tabular}

\vspace{0.5cm}

More information concerning the event can be found at
\begin{center}
{\tt http://www.math.md/bmo2010/index.php}
\end{center}
Here is the paper, sat during 4 hours 30 minutes.

\vspace{0.5cm}
 
\begin{enumerate}
\item Let $a, b$ and $c$ be positive real numbers. Prove that

\[ a^2b(b-c)/(a+b) + b^2c(c-a)/(b+c) + c^2a(a-b)/(c+a) \geq 0.\]

\item Let $ABC$ be an acute triangle with orthocentre $H$. 
Let $M$ be the midpoint of $AC$. Let $C_1$ 
on $AB$ be the foot of the perpendicular from $C$, and let 
$H_1$ be the reflection of $H$ in $AB$. Let the points 
$P, Q$ and $R$ be the orthogonal projections of $C_1$ onto the lines 
$AH_1, AC$ and $CB$, respectively. Let $M_1$ be the point such 
that the circumcentre of triangle $PQR$ is the midpoint of the segment 
$MM_1$.

Prove that $M_1$ lies on $BH_1$.

\item A {\em strip} of width $w$ is the set of points in the plane 
which are on, or between, two parallel lines distance $w$ apart. 
Let $S$ be a finite set of $n$ ($n \geq 3$) points in the plane, 
such that any three different points from $S$ can be covered 
by a strip of width 1.

Prove that $S$ can be covered by a strip of width $2$.

\item For each positive integer $n$ ($n \geq 2$), let $f(n)$ denote the 
sum of all positive integers which are at most $n$ and are not relatively prime to $n$. Show that $f(n+p) \neq f(n)$ for each such $n$ and for every prime $p$.
\end{enumerate}
\section*{Acknowledgements}
I thank the small army of coaches and examiners who help to prepare
our teams, the students who compete in our competitions, and the families
whose plans are disrupted by mathematics competitions. This includes
my own. The support of the Leeds Office of UKMT has been 
invaluable, and of course we are all very grateful for the continuing
financial support of {\em Winton Capital Management} whose backing
for UK participation in the Balkan Mathematical Olympiad and 
the Romanian Master of Mathematics competitions has been unwavering, despite
recent vicissitudes in the finance industry.
\section*{Leader's Diary}
The Leader's diary is inspired by reality, but events are 
subject to arbitrary exaggeration and distortion.

\vspace{0.5cm}

\noindent \textbf{May 2\ }It is Sunday, and our plane is scheduled to
leave Heathrow Terminal~4 at lunchtime. I plan to catch a train from Bath to
Reading, and transfer to the airport by bus. All goes well for the 
first 3 minutes, until I notice that the train has turned off the main line,
and is heading for Portsmouth. A jovial conductor informs me that
this is a scheduled diversion, and that we should reach Reading on
time. He reckons without a sequence of what are termed 
{\em engineering overruns.} We reach Reading via Newbury 45 minutes
late. There is no choice, and I catch a cab to Terminal 4, the fare being
two limbs.

I am the last of the party to arrive. We take an excellent and uneventful
TAROM S.A. flight to Bucharest where we meet up with the French team 
in the transit lounge. The Italians have taken a direct flight to Moldova.

Our luggage is checked through
to the Moldovan capital Chi\c sin\u au. We use a 50 seat propeller driven
plane to reach our destination. 

The entrance formalities are light, and our bags arrive quickly. 
I read the customs regulations with interest, not wishing  
to get thrown into gaol. It turns out that you
can bring in arbitrarily large finite amounts of foreign 
currency. The euros in my wallet
are therefore legal.

We arrive to be warmly greeted by several representatives of the competition,
including our guide Florentin. He has never visited an English speaking
country, but has a wonderful facility for languages. He came second in
a Moldovan national English competition. He is young, and very enthusiastic.

We are first driven to the Leogrand Hotel in the centre of town, so the
team can check in. This is a posh place, with several uniformed doormen.
It has an opulent foyer, and a sweeping staircase up to the mezzanine floor.
I try to disguise my disappointment at the absence of an indoor waterfall,
but we must somehow make do.
We share the hotel with the teams from France, Italy, Kazakhstan
and Saudi Arabia.
I bid the
team and Jacqui farewell, and accompanied by the French leader Claude Deschamps,
I am driven to the secret jury site, east of Chi\c sin\u au and just outside the
city limits. We are among the last leaders to arrive, but we are just in time for a late
supper. I look around, and see many friendly faces from previous IMOs.
We receive copies of
the problems shortlist, and repair to our rooms to study the options. 

There is a conventional hotel building,
but at the back there is a garden, complete with a swan pond and 
picturesque and romantic farming machinery. Of course it is dark, so I 
am not aware of all this yet, and am temporarily a sort of
Moldovan Bishop Berkeley. Following a path made of decking, I am eventually led
to my faux rustic cabin. This is excellent in almost all respects, but there are issues
concerning illumination. There one large room, and a bathroom. The bathroom
has a light in it, which is a blessing, and a heated wall. The sleeping end of the big room has no
light at all. The other end has a light, but no visible light-switch (remember it is
dark). To skip ahead, I will discover a switch behind a sofa next day, but
for the moment I am stuffed. Well, I am very sleepy, so it doesn't matter.

\noindent \textbf{May 3\ }After an excellent breakfast, we work on problems 
for a while, and then select the paper. There are some very good problems
on the shortlist, but I really wish we had more time to think about the
problems. Only the official Balkan Mathematical Olympiad countries can 
vote on the jury, but everyone is welcome to air their views. The jury
chair is Valeriu Gu\c tu, and he is very warm, businesslike and fair-minded.

We select a paper. See \begin{center}
{\tt http://www.bmoc.maths.org/}
\end{center} 

Problem 1 is a classic cyclically symmetric 3-variable inequality. The
official solution is short and neat, but it will turn out to have 
very many solutions. This is a fairly easy problem by the standards of international
mathematics competitions.
Problem 2 is a geometry problem. The official solution makes the problem seem harder
than it is. Also the way the problem is posed actually disguises what is really going on, and
of course that is deliberate. 
Problem 3 is a combinatorial geometry problem.
Later I will read a wonderful solution on 
Mathlinks (Art of Problem Solving) by {\em Stifler} from St Petersburg which
involves a slick move which leads to many fewer technical details than the
usual solution.
Problem 4 is a hard number theory problem, and seems to require a solver to have 
several good ideas. 

We jump into a bus and go to a theatre where the students have gathered for the 
opening ceremony. We are penned in the green room to prevent conversation with
the students. The Secretary of State for Education drops in, as they do.
Then we go onto the stage to get clapped by the students, a welcome inversion
of the usual procedure. Then we step down into front row seats to enjoy some children 
engaging in folkloric dancing. After a couple of sets, both the minister and the jury
depart, leaving the students to enjoy the entertainment.

Back at the jury site, I am buoyed by the discovery of a light switch deep behind the sofa,
but am starting to fret about the fate of Crystal Palace FC. By now they have played
their last match of the season against their rivals Sheffield Wednesday, but I have no
way to discover the result. My phone
doesn't function in Moldova, and the jury site does not have internet access.
I scan the staff looking for fans of weak Championship sides, but none is evident. 
For both teams, defeat would mean relegation, and in the case of Palace, 
likely bankruptcy and oblivion. However, here in Moldova, the result of 
this match has the status of Schr\"odinger's cat.

We work on the language of the papers. We sort the English version out first, and
I am drawn in as the local expert. As always, I argue for clarity rather than 
elegance of expression, and the jury is very helpful. In the evening there is
a very late banquet, and the spirits flow, in both senses. 
Fortunately I have practised
drinking vodka with my father-in-law, and so can employ the Russian method, 
the digital form of drinking where glasses are either full or
empty at all times. This has the advantage that it is 
always clear whether or not your glass
needs to be refilled. 
I drink a large amount of bottled water
before going to bed, for which I am truly grateful next morning.


\noindent \textbf{May 4\ }We return to Chi\c sin\u au immediately after breakfast
in order to be able to answer questions of clarification during the first 
30 minutes of the exam. Then we discuss the marking schemes, and it all goes very smoothly.
I meet Jacqui and the students at the end of their exam, we have lunch and I 
check-in to the Leogrand hotel. 
I raise the matter of the Palace-Wednesday game, and as wave functions
clatter to the floor all over the room, it turns out that the Glaziers have 
survived.

I skip the afternoon excursion, and have a long shower
and rest. The students give me individual reports on what they think they have done on the paper.

In the early evening, the scripts become available. 
I walk with Fawzi Al-Thukair and other Saudi leaders to pick up our scripts. The celebrated Titu Andreescu
is now part of the Saudi coaching set-up, and it is very heartening to see him
again. I return to the Leogrand Hotel
and start to mark scripts.

\noindent \textbf{May 5\ }I have breakfast with Jacqui and the students. A
welcome feature of the hotel is that a young lady harpist gives a recital 
throughout the meal. I find this very soothing, and will introduce
daily recitals at home. What is the point of having
children if they cannot play the harp? I spend the rest of the morning doing more marking,
and in the afternoon co-ordination begins.

Our first two co-ordinations are for Problem 2 and 4. The team are safe with
our guide at a bowling alley, so Jacqui pops in to observe. We have
two correct geometry solutions, and a bonus mark for Sahl Khan for making
a useful observation. In the number theory
Problem 4, Ben Elliott has made significantly more progress than 
the other British students, but I expect him to get about 4/10. The 
co-ordinators spring a pleasant surprise, because they have a quick
way to complete his argument to a full solution, and so they are more impressed
by it than I am. He gets 7 points, and I try to control my involuntary eyebrow movement.

When we have finished co-ordinating the
scripts of the the first three students, the
co-ordinator looks at my notes, and thinks that I have written
2,2,2 next to the names of our last three candidates. In fact
I have written smudge, 2, 2. She concedes 2,2,2 and we start to
wrap up. As she is filling in the forms, I notice that my smudge is not a 2, and
I look at the relevant envelope (belonging to Jordan). He has handed in
no pages. Now, getting 2/10 for an empty script seems a big reward, even
for someone with my negotiating skills. I interrupt her and
point out the difficulty,
and explain that the UK is claiming 0 marks for this student. She produces
a script, and sure enough it deserves 2/10.
I explain that I have not seen this script before.
Perhaps there was a mistake in the photocopying room, but
there is still the mystery as to why the envelope was marked as containing
no pages.

I tax Jordan over dinner. He is pretty sure that he handed nothing in, and is
very impressed with a score of 2/10. I ask him if he did some work in rough, and
perhaps that was somehow inserted in the envelope? He thinks not,
but now he is in a confused state. After dinner, all is explained, and
Jordan loses his marks. It turns out that a student from another
country had not labelled his script, had elected to write in English, and
had handwriting very like that of Jordan Millar. His page had become
incorrectly associated with Jordan. Well, these things happen, and fortunately
the student concerned got his correct marks in the end.

\noindent \textbf{May 6\ }Today Jacqui is 
staying with the students, and perhaps 
this is a good thing because co-ordination is considerably more eventful. 
We begin with Problem 1, for which I am asking full marks for everyone. The authorities
accept that this is the correct mark for five of the six students, but they
are concerned that Jordan Millar's solution might be wrong. This comes
as a surprise to me, so I ask them (with trepidation) to point out the error.
He has a cyclically symmetric expression in three variables, which 
he first simplifies, and then he elects to bust the symmetry 
in a manner which is `without loss of generality'. There is no
doubt that this move is vulgar, but the co-ordinators suspect that
it is illegal.
After some conversation, the co-ordinators accept that this
sordid trick is allowed. At this point I expect them to fold, and hand over
10 marks, but to my horror they then proceed to show concern about another
matter. Later on, Jordan has indulged his appetite
for mathematical thuggery even further, and has deliberately broken the 
homogeneity of the expression,
splitting the problem into two cases. The co-ordinators think
that what he has done is not logically correct. They are so adamant
that I begin to think I am mad, so I ask for a time-out and go upstairs.
As soon as I reflect, I realise that Jordan is right, and I write out a 
formal justification of his move. I rush back, but the leader of
Kazakhstan has begun his co-ordination. I hand over the note containing
my argument. Later in the morning we have another meeting, and 
the co-ordinators now agree that Jordan's move is legitimate, and he gets his
10/10. I am delighted about this because he only has 1 mark on the
other problems.

Problem 3 is relatively straightforward.  Richard Freeland has a solution almost
identical to the official solution and gets 10/10. Sergei Patiakin has 
handed in half a solution. Unfortunately there is a non-obvious fact
which he has simply stated as obvious, and the absence of any
justification costs him 5 marks and so he gets 5/10. 
Jack Smith, on the other hand,
has supplied precisely that part of the argument which
Sergei's script lacked, and nothing else.
It makes perfect sense 
that Jack gets 5/10 marks.

In the evening there is the final jury meeting. I mentioned the jury chair
Valeriu Gu\c tu before. There are other people who played important roles
in organizing this event, including Valeriu Baltag. The two Valerius 
have an interesting discussion about the medal boundaries which goes on
for a considerable time. Eventually the jury decides to follow its own rules,
and the cut-offs are selected. These are determined by the performances
of students from countries which are official participants. The bronze
cut-off at 12 means that neither Jordan nor Sahl get a medal, which seems very
harsh, but those are the rules. They do both get Honourable Mentions
for solving a whole problem. 

Ben Elliott and Jack Smith both have well deserved bronze medals. 
If ever Ben varies his strategy of going {\em mis\`ere} on geometry problems,
he will be a force in mathematics competitions. Jack famously
won BMO2 this year, and we bid him a fond farewell as he goes
to university next year. 
Richard Freeland and Sergei Patiakin have safe silver medals, so
this is easily the best performance by a UK team in this competition.

\noindent \textbf{May 7\ }I put my alarm on for 04:00, both
as practice for the early start next day when we will come
home, and also so that I can watch the election results at
the expected crucial time (02:00 UK). As you probably know, at that 
moment nothing was clear, so I went back to sleep. 
As I got up for breakfast, the BBC called the result as a hung 
parliament. 

We visit a monastery in the morning, and
have the closing ceremony in the late afternoon. There is high drama
as Jack becomes queezy on stage. There is the possibility of 
projectile activity (q.v.\ {\em The Exorcist}) but he makes it off stage
in the middle of the presentation, and Jacqui takes him back to the hotel.

This happens just as I am called up on stage to present some bronze
medals. By chance one of the students to whom I present a medal
is Matthew Fitch, who plays for France despite his name. 
After the ceremony Sergei Patiakin gets a text from Jacqui to
the effect that Jack is in bed and resting, and that she will stay
at the hotel with him.

We go to the closing banquet. We leave quite promptly
because of the early start next day. Our plane will leave
at 06:40. The travel organizers want to have a coach pick up 
the UK and France at 04:10 and take us to the airport. Since
the airport is a 20 minute drive from the hotel, and 
the airport small with very limited traffic, Claude and I do
not agree. We reschedule the bus for 04:45. 

\noindent \textbf{May 8\ }At 04:10 I get a call from Valeriu Baltag who is downstairs
with a bus and is wondering where to find the teams of 
France and the UK.
Unfortunately the message about the rescheduled time has
not got through. Valeriu is accompanied by the quiet man.
He speaks no English, but shakes hands at all possible opportunities,
and is in charge of conference cash. These poor chaps have kindly come
down to see us off in the middle of the night, and most of us are 
still in bed. The guides of the UK and France are also 
there, and they will even come to the airport to see us off.

The UK guide, Florentin, deserves a report all to himself. He has
been fantastically helpful at all times, and has made a tremendous
effort to befriend our students. There are comic aspects to the situation,
but everyone likes Florentin. He does follow organizational instructions
with excessive zeal. He must have moved us between the hotel
and the exam site about a dozen times. This is a 15 minute walk through
a small safe town with low traffic density. To Florentin, however,
Chi\c sin\u au was more like Baghdad in 2005. He would 
constantly scan the horizon for threats,
and direct the traffic at junctions.

The hotel serves coffee and croissants from 04:30, so we are able to 
charge up before beginning the journey. We inspect Jack, and decide that he is just 
fit to travel, being a paler shade of green than the previous evening.
It seems that after I went to bed, Kazakh students had organized a 
rather good party
of some sort, which bodes well for IMO 2010.
At the airport we say farewell to Florentin, and are surprised to find that
we will travel to Bucharest in a jet plane.

The journey home is uneventful, and we are met by many happy family members.
Jack now has a pink tinge.
Certificates are handed out in Terminal 4, and there is much clapping. 
I transfer to the main bus station with Jordan, which is next
to his terminal for Belfast, and the game is over for another year.
\end{document}
