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\title{International Mathematical Olympiad 2010}
\date{Almaty and Astana, Kazakhstan}
\author{UK leader's report}
\begin{document}
\maketitle

This year the annual world championship of secondary school mathematics was held
at various sites in Kazakhstan during July 5 -- July 14th. The six students sit
two exams, each of duration four hours 30 minutes under stictly controlled conditions.
This year the UK side consisted of six boys, and unusually five of them are 
being educated in the maintained sector.

The team was selected by competitive examination, and consisted of the following six people.

\vspace{0.5cm}

\begin{tabular}{p{4.3cm}p{0.5cm}p{9cm}}
UNK1 Luke Betts      & & Hills Road 6th, Cambridge\\
UNK2 Nathan Brown    & & King Edward's Camp Hill, Birmingham\\
UNK3 Andrew Carlotti & & Sir Roger Manwood’s School, Kent\\
UNK4 Richard Freeland& & Winchester College, Hampshire\\
UNK5 Sergei Patiakin & & Dame Alice Owen’s School, Hertfordshire\\
UNK6 Aled Walker     & & King Edward's Camp Hill, Birmingham
\end{tabular}

\vspace{0.5cm}

The reserves were Andrew Hyer of Westminster School, London, and 
Jordan Millar of Regent House School, Northern Ireland. The provenance
of the team and reserves by chance includes all four countries of the United Kingdom.
The team were accompanied by Dr Geoff Smith, University of Bath as leader; Dr James
Cranch, University of Leicester as deputy; Dr Joseph Myers, CodeSourcery as observer with
leader; Dr Ceri Fiddes, Millfield School as observer with students. For the pre-IMO
phase, when Dr Fiddes was unavailable, Miss Jacqui Lewis of St Julian's School, Lisbon
acted as observer with students.

The results of our team members were as follows.

\vspace{0.5cm}

\begin{tabular}{|p{4.3cm}p{0.3cm}p{0.3cm} p{0.3cm}p{0.3cm}p{0.3cm}p{0.3cm}|p{0.5cm}|p{3.8cm}|}
\hline
 & P1 & P2 & P3 & P4 & P5 & P6 & $\Sigma$ & award\\ 
\hline
UNK1 Luke Betts      & 7 & 1 & 1 & 7 & 1 & 0 & 17 & bronze\\
UNK2 Nathan Brown    & 7 & 0 & 1 & 7 & 7 & 0 & 22 & silver\\
UNK3 Andrew Carlotti & 7 & 0 & 0 & 7 & 0 & 6 & 20 & bronze\\
UNK4 Richard Freeland& 7 & 0 & 0 & 7 & 0 & 0 & 14 & honourable mention\\
UNK5 Sergei Patiakin & 7 & 0 & 6 & 7 & 7 & 0 & 27 & gold\\
UNK6 Aled Walker     & 7 & 0 & 0 & 7 & 0 & 0 & 14 & honourable mention\\
\hline
totals          & 42 & 1 & 8 & 42 & 15 & 6 & 114 & \\
\hline
\end{tabular}

\vspace{0.5cm}

The medal cut-offs were 15 for bronze, 21 for silver and 27 for gold.

\vspace{0.5cm}

This result had us ranked 25th in terms of points scored, at the back of the
peloton. A dramatic underperformance on Problem 2, a geometry question, served
to undermine some excellent work on the far more demanding problems 3, 5 and 6.
The scores of the leading countries were as follows:
1 People's Republic of China 197;
2 Russian Federation 169;	
3 United States of America 168;	
4 Republic of Korea	156;
5= Kazakhstan 148;	
5= Thailand 148;	
7 Japan	141;
8 Turkey 139;	
9 Germany	138;
10 Serbia	135;	
11= Vietnam	133;	
11= Italy	133;	
13= Canada	129;	
13= Hungary	129;
15 Australia 128;	
16= Romania	127;
16= Islamic Republic of Iran	127;	
18 Peru 124;	
19 Taiwan 123;
20 Hong Kong 121;	
21 Bulgaria 118;
22= Ukraine	117;	
22= Singapore 117;	
24 Poland	116;	
25 United Kingdom	114;
26 Uzbekistan 112;
27 Belarus	110;
28 Azerbaijan 109;
29 New Zealand 106;
30 France	105.

As usual, the performance of China was excellent, and they had the only student with a perfect score. 
Australia performed very well this year,
and the results of New Zealand are truly exceptional, given their modest population base
and their performances in the past. 
I should also draw your attention to the performance
of Saudi Arabia. The Kingdom is making a great training effort,
and this year secured 55 marks and two bronze medals. This is
more than double the number of marks they obtained in all five
previous participations combined. 

As for the UK, 
if we had performed well on Problem 2, and say
scored 21 points, then that would have given an overall score of 134. As I have
noted before, the rank statistic is extremely sensitive to small perturbations in
performance for those teams which finish in the range 10th--30th. Two of our students
have that rarest of distinctions, a double honourable mention. In most years, 14 points
is enough to get a bronze medal, and they can count themselves unlucky to have lost out.

Congratulations are due to all of our students, and one must mention the special 
performances of Nathan and Sergei in obtaining silver and gold medals respectively.

Detailed results and statistics can be found at the official IMO site:

\begin{center}
http://www.imo-official.org
\end{center}

It is with sorrow that I must report that the jury disqualified the team of North Korea following
allegations that some of their students had supplied solutions which were exceptionally close
to those in the official solutions booklet.

Here are the six problems of the 51st Mathematical Olympiad. The problems should be taken
in two groups of three, and for each three questions the time limit is 4 hours 30 minutes.

\begin{enumerate}
\item
Determine all functions $f \colon \mathbb R \to \mathbb R$ such that the equality
$$
  f\bigl(\lfloor x\rfloor y\bigr) = f(x)\bigl\lfloor f(y)\bigr\rfloor
$$
holds for all $x,y \in \mathbb R$.
(Here $\lfloor z\rfloor$ denotes the
greatest integer less than or equal to~$z$.)

\item Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$.  Let $E$ be a point on the arc~$BDC$ and $F$ a point on the side~$BC$ such that $$ \angle BAF = \angle CAE < \frac{1}{2} \angle BAC.  $$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the
lines $DG$ and $EI$ intersect on~$\Gamma$.

\item Let $\mathbb N$ be the set of positive integers. Determine all functions $g\colon
\mathbb N \to \mathbb N$ such that
$${\bigl(g(m)+n\bigr)\bigl(m+g(n)\bigr)}$$
is a perfect square for all $m,n \in \mathbb N$.


\item 
Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$
intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points~$K$,
$L$ and $M$ respectively. The tangent to $\Gamma$ at~$C$ intersects the line
$AB$ at~$S$.
% Prove that $SC=SP$ if and only if $MK=ML$.
Suppose that $SC=SP$. Prove that $MK=ML$.


\item %TRANSLATE THE EXERCISE INTO YOUR OWN LANGUAGE
In each of six boxes $B_1,B_2,B_3,B_4,B_5,B_6$
there is initially one coin. There are two types of operation allowed:
\begin{quote}
  \begin{itemize}
  \item[\textit{Type~}1:] Choose a nonempty box $B_j$ with $1 \le j \le 5$.
    Remove one coin from $B_j$ and add two coins to $B_{j+1}$.
  \item[\textit{Type~}2:] Choose a nonempty box $B_k$ with $1 \le k \le 4$.
    Remove one coin from $B_k$ and exchange the contents of (possibly empty)
    boxes $B_{k+1}$ and $B_{k+2}$.
  \end{itemize}
\end{quote}
Determine whether there is a finite sequence of such operations that results
in boxes $B_1,B_2,B_3,B_4,B_5$ being empty and box $B_6$ containing exactly
$2010^{2010^{2010}}$ coins.  (Note that $a^{b^{c}}=a^{(b^{c})}$.)


\item  %TRANSLATE THE EXERCISE INTO YOUR OWN LANGUAGE
Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers. Suppose that
for some positive integer $s$, we have
$$
  a_n=\max\{a_k+a_{n-k} \,\,|\,\, 1\leq k\leq n-1\}
$$
for all $n>s$.
Prove that there exist positive integers $\ell$ and $N$, with $\ell\leq s$ and
such that $a_n=a_\ell+a_{n-\ell}$ for all $n\geq N$.
\end{enumerate} 
These problems were submitted by
France, 
Hong Kong, USA, Poland, Netherlands and Iran respectively.

\section*{Leader's Diary}
Readers are warned that sometimes this diary is more accurate than 
seems likely. It is intended to entertain and inform, and I trust
that no-one will take offence.

This year the IMO produced its own official diary, and various
editions can be downloaded 
from http://www.imo2010org.kz using the top right tab labelled
``Diary of Olympiad''. These are large files, for the diary is in glorious
colour, and has many pictures. I urge the reader to examine these documents
as an accompaniment to my more subversive commentary. The first
edition has some impressive pictures of the architecture of Astana, and 
much else besides.

\noindent \textbf{June 30\ }I make a very early start to 
get to Heathrow Airport very early.
I am carrying almost all the passports, and the consequences of
being late are too horrible to contemplate. My bags are very heavy 
because they are stuffed with IMO detritus for the team. I reach Paddington on
time and reverse direction using the Heathrow Express, arriving 
at Terminal 1 before 09:00.

I am first to arrive because pre-IMO pastoral supremo Jacqui Lewis
is having trouble finding a Hotel Hoppa bus. Joseph `Combinatorics' Myers
is first to show up, wearing a UK Panama hat 
lovingly preserved from IMO Vietnam.
Then Jacqui arrives, and the team start to trickle in. There is much swapping
of goods to be done, so we open our bags. The sadly absent Vesna Kadelburg has sorted out
some natty team uniforms, and these are distributed. 
I pass out Union Flags for the ceremonies. The Panama hats
for 2010 have not arrived, and I suspect sabotage, but say nothing.
The last parent says good-bye, and we are ready to face the IMO.

Things are going very well, until we try to check in.
We are booked on Transaero, a Russian independent airline.
Think Virginski. While two of our four landings will be a touch heavy,
Transaero will come out rather well in this report. They messed up
some of the vegetarian meal bookings, but overall we were treated rather well.
The same cannot be said of BMI, the company which was supposed to handle
the check-in process at Heathrow. I will not go into great detail because
I understand that deputy James Cranch is planning a monograph
on the topic. However, here is a brief summary.

First I should explain what a sensible airline does when a large party
(10 people) turns up to check-in together. The party is told
to ignore the machines, and is ushered to a check-in desk or two, so that they
can be processed in a systematic way. Suitable seating arrangements are worked
out, and there are no weight issues because so many students carry
light suitcases. Not BMI. They have a young lady who gives
instructions which we do not wish to follow, and advice of
no quality. The parrot wants us all to check-in separately.
She has no idea what a bad idea this would be. 
We have several students who cannot be
trusted to handle their own passports, and who could not possibly 
navigate through well-designed check-in software (let alone the maze
which is BMI's entry to the world).
Such a strategy would have the party seated all over
the plane.

We ask for advice, and she gives a long complicated answer which
can be summarized as `I cannot help you'.
At this point I have to let off steam. I explain to her what her
answer means, and that I would be perfectly happy with it if it
were so expressed, but that dressing it up
as if it were a helpful explanation is pointless and annoying.

Jacqui and James go to work, 
and find it is an uphill battle. The interface is awful and 
despite deploying considerable intellectual power, 
it takes an age. They are admonished for not obeying the parrot's
instructions. Joseph Myers and I check-in separately, and that proves
easier. Whenever I get stuck I ask `what button should I press Joseph',
and being a professional computer wizard, he tells me, often correctly. 
I saw no sign
of foreign language options, but in fact an inability to
understand English might have made the process easier.

When we finally get our computer-generated boarding cards, we go to hand in
our luggage, and the nice check-in people explain that it doesn't matter
which seats you select using the check-in machines, because they have
the power to reassign seats. If the parrot had mentioned this, then a lot
of nonsense could have been avoided.

We have managed to arrange the party in a loose clump on the packed aeroplane. 
We have an uneventful flight to Moscow. There we need to enter the 
transit area, which means going through a security check. The relevant chappie
is not to be found, so we wait for while, but eventually we
are admitted to the Moscow Domodedovo playground. This is not a bad place,
but has a parochial policy on cash. They take roubles and nothing else. 
Neither dollars nor euros work, though credit cards will do nicely. 
We find an Italian outfit where the team can grind their way through pizzas,
until at midnight we board a Transaero flight to Astana, Kazakhstan.

I spot the Vietnamese leader H\` a Huy Kho\' ai, but he is presumably en route
to Almaty whereas we are going to Astana.

\noindent \textbf{July 1\ }We arrive in Astana very early in the morning. 
There is no hurry, and we find
ourselves at the back of the immigration queues. At this point a 
whirlwind called Madina arrives. She is a local organizer, and 
marches us to the front of the queue. An IMO 2010 sign is then slapped
on an immigration portal and we are whisked through at speed.
There is a friendly lady from the Ministry of Education and Science
standing there to oversee the process. The immigration officer
is genial, and discards my immigration card as being unnecessary.
I have doubts about this, because I fear that the lady from the Ministry
will not be there on the way out, and I may meet a less generous border guard
who doesn't happen to know how important I am, and may feel
that I ought to be brandishing a doubly stamped immigration card.
In fact I am being quite prescient. Read on.

Then the IMO officials and the lady from the Ministry
attempt to march us through the green channel of customs.
I explain that I have read the Kazakh customs regulations, and that I
wish to declare a large sum of foreign currency (for fear of
piano-wire, gouges, thumb-screws and so on, but I leave out that bit). 
Once again I am informed that 
the regulations do not apply, and we are marched through customs.
I have a sinking feeling.
The local organizers are clearly making a big effort, and this bodes well
for the rest of the IMO. Since we are four days earlier than most of the teams, they are practising the art of greeting on us. Well done, and thank you.

The airport is much like the rest of what we see in Astana. 
It is an excellent building,
but is not supported by the usual clutter of infrastructure -- 
hotels, car hire firms, giant car parks and petrol stations are absent.
This is an airport surrounded by empty space, an asset
with which Kazakhstan is richly endowed. I knew about this in advance,
having failed to find anything when searching the internet using
the keywords: Astana Airport
Hotel. I then tried a map tool and discovered that with Astana Airport, you
get what it says on the tin. Nothing else. 

We have booked a driver to transfer us to the Imperia G hotel. I have
been informed that the driver will carry an `Imperia G' sign. We don't
see him immediately because `Imperia G' is written in the Cyrillic
alphabet. The organizers promise to come to see us soon, and 
we are driven into town.

The Imperia G is in the old city, the naturally occurring place called 
Akmola which has been extended by Presidential fiat into
the new city called Astana.

At first sight, Astana seems to be the capital of Mars. It is a showcase
of contemporary architecture. The shapes are a festival of geometry.
Stately pleasure domes erupt from the steppe at startling angles.
Postmodern shopping centres, apartment blocks and sports arenas
do not jostle at all. Giant administrative buildings 
curve and glint, and are rarely simply connected. In non-mathematical
terms, think Gruy\`ere cheese, or the work of Henry Moore,
or polo mints. Kisho Kurokawa and Lord Foster have been set loose.
Norman Foster has supplied the Khan Shatyr Entertainment Centre, the
Palace of Peace and Reconciliation and the Pyramid of Peace.

We check-in, and have breakfast. The Australians are staying at the
same hotel, and we run into them shortly before going upstairs for showers
and bed. 

We get up at lunchtime, to try to force ourselves
into the local timezone. By evening, the Australians have yet
to return, so we explore. A nearby market allows us to change
some cash into the local currency (1 euro = lots of tenge). 
We find a Turkish restaurant, and let the team loose on the
menu. It is very satisfactory, including kebabs, ice-cream and
other health foods. The host doesn't quite understand about 
giving clients space, and proceeds to join us at our table.
I think that we are so far East that Brits remind him of Turkey.

Next day we hook up with the Australians over breakfast. Angelo
and Ivan are their leaders. We plan to have practice exams together.
The hotel quoted us a silly price for the hire of appropriate
facilities, but our excellent IMO guides have obtained the use of a local 
school. Joseph Myers and I must transfer to the jury site today, so
we bid farewell to the students as they leave for their first 
training exam.

The IMO guides persuade us that it is pointless and expensive
to take a taxi to the airport, and a schoolteacher escorts me, 
Joseph, Angelo and our bags 
on the necessary pair of bus journeys. As a grotesque foreigner,
I am a source of entertainment for local toddlers, and I spend quite 
a bit of time with my fingers clenched in tiny Kazakh hands.
The Australian leader Angelo is with us, and given his
unmarried and relatively youthful state, seems to be making
excellent progress with the young women of Central Asia.
His success is probably down to his experience working the Melbourne trams. 

The flight to Almaty is uneventful. There is a lot of cloud, but when we
do get to see the ground we observe that it is empty grassland, 
steppe in all directions.
We are on Air Astana, the only local airline without an explicit
five-skulls health warning at the UK Foreign Office website. Air Astana 
are excellent. We are met at Almaty airport, and the local organizers put
the three of us, and all our bags, into one medium sized car. 
This is not actually possible, but it happened. The driver clearly
had ambitions in Formula One, but was inhibited by being placed
in rush hour traffic. Nonetheless, he managed several inappropriate
and entertaining changes of lane.

After a little more than an hour we reached the sanatorium where
the jury was based. I have some experience of hotels in the former
Soviet Union, and this place is top of the range. The food
is not the universal hotel diet, but is all the better for that.
I have a bedroom on the 7th floor, and to my delight I discover that
I have a floor lady, just as it was in the USSR. A woman sits in
charge of each floor all the time. They have shifts, but she is always there.
She never does anything. The system was part of the full-employment
economics of Soviet la-la-land, and presumably the ladies could also report
if counter-revolution was breaking out on their floor. Why are they
still doing this, twenty years on? Presumably there was no moment
at which it seemed appropriate to change this bizarre arrangement 
(don't knock it, this is the basis of marriage).

Although the food is wholesome, and I am attuned to the former Soviet
obsession with buckwheat, some people clearly find the diet a little strange.
Personally I thought it fine, and in fact providing such an array of
meals must have taken a great deal of effort and thought. Kazakhs are
hippophagists, and when I see an unusual meat on a cocktail stick, I 
cannot resist adding another tick to my list: 
frog, camel, goat, crocodile etc. While it is absolutely fine, I wouldn't
go out of my way to eat dobbin again, for it is a little dry. 
My attempts to find \emph{kumys}, 
fermented mare's milk, end in failure. 
There was also
a tuck shop, where dissolute sanatorium clients could purchase chocolate,
vodka, brandy and beer, or so I was told. 

This is one of the happiest times of the IMO, when you get to see old
friends again, and engage in ruthless teasing and mockery until
settling down to see what events may bring.

After some effort, we get our hands on copies of the short-list, the
proposals from which the problems on the papers will likely be chosen,
and we go to work. I am concerned about $G1$, the easiest geometry
problem, because I can solve it immediately. It turns out to be a lovely 
exercise by my UK colleague Christopher Bradley, but it is a little too
easy, even for a first question. If you work forwards and backwards at
the same time, from where you start and also from what you want to be,
you quickly meet in the middle by the use of cyclic quadrilaterals and 
chasing angles. Fortunately $G2$ 
looks about right for a geometry problem.
It will yield easily if you recall that if two points are
mutually inverse with respect to a circle, then that circle is a circle
of Apollonius with respect to those points. 

There are very many excellent medium level combinatorics problems. We 
are blessed with an excellent choice this year. There are a few
monstrous questions too. Other than a shortage of easy questions, it
is a good selection, but there is no ``grasshopper problem'' as memorable
as the one used in 2009.

\noindent \textbf{July 3\ }We continue to work on the problems. 
I have set up camp 
on my balcony. The view is magnificent. We are on the edge of the
Zailisky Alatau Mountains, and sufficiently high that insects are not
a problem. The air is clean, and towards the end of the day the Kazakh brandy
is delicious.

As we work on the problems, we confront the fact that the IMO 
is happening in parallel with the World Cup. Late at night, the
jury room is darkened, and we can watch matches on a giant screen.
Relevant leaders take a keen interest in proceedings.

A jury meeting excludes $G3$ and $N3$ because of prior art.

\noindent \textbf{July 4\ }
UK Observer A Joseph Myers has sunk $C7$, the most difficult of
the combinatorics proposals.
He supplies chapter and verse of a paper published by the American
Mathematical Society which includes the result. I ask permission
for Joseph to address the jury, and this is granted. I have taken
out all of his batteries, and only given the clock-work mechanism
a quarter-turn. Joseph normally speaks at the same rate that he thinks,
and for most people this causes serious bandwidth problems, especially
people who are not native speakers of English. Joseph manages to
slow down enough to get his point across, and unfortunately
a very attractive problem is removed.

Eventually we start to select the papers. This is done in the
order easy, hard and finally medium. We choose a functional
equation with an unusual solution set and a nice geometry
problem for the easy problems. For the hard problems we
find an intriguing question which is, in a way, another
functional equation, but it has a peculiar method of solution.
We also select a hard algebra problem. 

Next we turn to the two medium problems. Since we have
not yet chosen a combinatorics problem, and we have
a raft of excellent combinatorics problems of this
level, we do have the option of choosing both medium
problems to be combinatorics, but of course there is the geometry
lobby. They have their way, and we select $G4$. Next we agonize over
the combinatorics question. In the end, when it comes down to a choice
between two problems, and is clear that support for the problems 
is equally divided, we make the choice on the basis
that one problem is likely to much easier to co-ordinate than the other.

There is a notation phase, and eventually the English Language Committee
meetings to sort out the wording. We produce rival versions of questions
for the jury's perusal. 
In fact we offer to recast Problem 3 in terms of sequences, 
because in its current formulation, it is a little too 
close to being another functional equation problem. However, the jury
decides to go with the original formulation.

\noindent \textbf{July 5\ }
Individual language groups proceed with the translations of their papers,
with the official languages done first.

There are elections to the advisory board. Nazar Agakhanov
of Russia will be the new IMOAB chair following two terms by the excellent
J\' ozsef Pelik\' an. There are eight candidates for the two remaining
vacancies on the board. There is a two-stage electoral 
process for the two seats, and I had played a significant role
in the design of this system last year. Joseph Myers seemed very amused when I 
forgot the details of the process and accidentally spoilt my ballot
on the first round. Myung-Hwan Kim of South Korea is re-elected,
and I am elected for the first time. IMOAB elections are a little strange,
because you generally want everyone to win, especially your opponents.

\noindent \textbf{July 6\ } 
The principal value of travelling 
the world is to meet 
and try to understand radically diverse cultures. As much of western and 
central Europe is converging, Kazakh priorities are very different.
This is clear from the amazing effort that they put into ceremonial
activities. The DVD of the IMO will have a few brief shots of young people
doing mathematics or playing sport, but it is dominated by formal public
events; parades, medal ceremonies and middle-aged people making speeches.

It is this attitude to the role of public ceremony which had Aztec children
given as sacrifices to the gods, prisoners fed to wild animals 
in Roman circuses and IMO leaders and Observers A transported from
Almaty to Astana in the middle of the night. This was to allow us to 
attend the IMO opening ceremony. 
Personally, 
if I were running an IMO, I would have sought another solution.
These people have flown in from all over the world, most of them 
suffering from jet-lag. What do they need after a couple of days in 
Kazakhstan? Well, obviously, they need to get up in the middle of 
the night in order to attend an opening ceremony at the other end of the 
country.

We woke in the night at 02:30 in order to be on the buses at 03:00. 
Samson the Nigerian leader is nervous that he will oversleep and asks
me to wake him. After loading our luggage we engage 
in ritual sitting. After a while, we were told we were 
on the wrong bus. The resulting transfer involved getting off the bus, 
digging out 
suitcases from its innards, and moving to another bus. You might
ask why this was necessary, given the law of conservation of seats, but let it pass.
I am reminded of IMO 2003 in Japan when the guides tried to persuade 
leaders to sit on buses in alphabetical order of country.
We have a police escort (which in Kazakhstan means that 
you go more slowly),
but still arrive at the airport in time to catch one of the early flights to
Astana.  

Our buses pass through the Martian metropolis to the Independence Palace.
We walk past what seems to be an honour guard of ladies in blue, 
through
to a magnificent breakfast spread and lots of tables on which to eat,
and the empty set of chairs. I quaff some keffir, the local
dairy drink. After this, it seems that some lounging around on soft furnishings
would be a good idea, but none is to be seen.

Now an IMO guide, a lady in yellow, tells us that we can go upstairs to relax. 
I quickly scoot off to get first go at the upholstery. No chance. The
ladies in blue forbid us to go upstairs. I go back and report this to the lady
in yellow. She looks astonished, and a stand off ensues. The issue was
resolved when yellow triumphed over blue, and we were allowed upstairs.
Once again I was quick off the mark, and at the top of the stairs there
were sofas and a remarkable picture.

I plonked myself down to gaze on this curious and fascinating piece of 
work. I hesitate to make judgements on works of art, and will confine myself
to a simple description. The focus of the painting is President
Nursultan Nazarbayev. He is wearing what is presumably the emblem of a chivalric
order round his neck, and is walking towards the observer. On each side of him
are arranged many of the celebrated world leaders from about 10 years ago, 
mostly looking on the President with warm admiration. The applause is clearly
being lead by Jacques Chirac. While most people (Berlusconi, Yeltsin, George
W. Bush, Jiang Zemin etc) have faces indicating great pleasure in 
the moment, two figures
stand out by showing darker emotions. I detect shrewd admiration from
Vladimir Putin and definite jealousy in the face of Tony Blair.

President Nazarbayev has an extensive collection of honours from 
all over the world. My own Queen Elizabeth II has awarded him
a knighthood. 
I often make reference to the Luxembourg leader, Charles Leytem,
and here is another opportunity. The Grand Duke has awarded 
President Nazarbayev a \emph{grand-croix de la Couronne de ch\^ ene}
and by co-incidence, Charles is in the same order, for he
has the \emph{croix de ch\^ ene} (the Oak Cross). I look up chivalric orders
on the internet, to see what is available. While some might favour
the UK's \emph{Order of the Bath}, I am attracted to Denmark's
\emph{Order of the Elephant} and Brittany's 
\emph{Order of the Stoat and the Ear}.

At length the ceremony begins. We leaders are at the front and the students 
are at the back. The students parade across the stage carrying flags, and there
is a significant amout of folkloric activity. This involves formal dancing,
and girls wearing remarkable hats resembling those tall thin desserts
with a vertical axis of symmetry which, in 1960 at least, were 
forever dusted with dessicated coconut and sat on a wafer base. 
At one point hundreds of
children fill the stage and each performs on the traditional Kazakh
two-string lute.

There was also some dramatic drumming. Kazakh percussionists dress 
in shiny olive green clothes, big boots, and hats with
giant ear flaps. Let us be honest, they are camp. 
Presumably the truth is that these flaps cover great wads
of cotton wool which are taped to their ears in order to help them survive
the sonic firestorm of their performance. The programme gently 
morphs into a traditional Kazakh heavy-metal set by \emph{Ulytau}. There
is an associated video involving nomadic horsepersons, wolves and
eagles. It all makes perfect sense, the Kazakh folk-myth is
of Hell's Angels before the invention of the motor-cycle.
I blame the Germans for introducing break-dancing and strippers
to the IMO opening ceremony. The Kazakhs have simply carried the idea a 
stage further. I hesitate to suggest what the Dutch might do next year,
in case they take it seriously. 

At length the Minister for Education and Science makes a splendid
speech of
welcome, and the ceremony comes to a close. 

We adjourn to our hotel about lunchtime, and there is complete
chaos. Hundreds of people all want to check-in at the same time,
and bless 'em, the Kazakhs have retained Soviet era stuff about
filling in cards, handing in passports and so on. The hotel
have laid on no extra staff to assist with this process, nor has anyone 
in the IMO organization given any thought as to how to make this
business easier. The rooms are filled with extra beds, because 
the students had been jammed into the same hotel the previous night.
There are no extra staff, so the hotel is having difficulty reconfiguring
to normal patterns of bedroom use.
Also many bedrooms have no desks, because they have been blagged
for the jury room.

In such circumstances, I normally head for the bar,
and wait for the foyer to clear. However, after the 2:30am start, I 
cannot face a drink. I scout around and find upstairs a nice big sofa outside
the hotel manager's swish office. I reason that this person is 
partly responsible
for the turmoil below, and that he or she would probably be annoyed if
a slob of a guest set up camp outside their office. I remove my shoes and
enjoy a prolonged kip. 

After five hours I wander downstairs to find the queue has finally disappeared.
Apparently one of the guides had eventually taken charge of the situation.
By handing out registration cards, and getting people to fill them
in while in the queue, she had dramatically sped up the process.
She deserves a gold medal. I check-in, and discover (at 18:30) that my room is 
yet to be made up. I don't care. The staff are onto it quickly enough, and
at length I have an extremely comfortable room on the fourth floor, not far 
from Joseph Myers. The TV carries lots of channels, including BBC World,
and Channel 3 which has a strong focus on the presidential calendar.

\noindent \textbf{July 7\ } This is the day of the first exam. The jury sits awaiting
the first message from the students. The students are allowed
to send questions of clarification to the jury during
the first 30 minutes of the exam. IMO internet 
Tsar Matja\v z \v Zeljko is trying to look serene, but the Kazakh
`just in time' philosophy of communications management is not
completely attuned to his more cautious approach. 
Matja\v z is
clearly doubtful that communications between the sites
will work smoothly.
The first student message is from an Australian student.
It is a complaint about the conditions under which
they are living. After that there are some sensible mathematical
questions. 

After the question phase is over, 
text messages start to filter in from the deputies.
At this stage, it would normally not be appropriate for informal
communications to take place between the two sites, but clearly some of
the deputies and observers C are very concerned. They are not at the same 
site as the students, and there is as yet no means of the deputies getting 
to the students. This was bad enough for deputies, but for observers C
this was outrageous. 
The designated purpose of observers C is to stay with the students.
This is particularly important for countries such as the UK where
parents have the expectation that their offspring will be 
protected and cherished by a designated person from their own country.
Our observer C was Ceri Fiddes. She had travelled at great expense 
a quarter of the way round the world in order to look after our students,
and now she was denied access.

I took comfort in the fact that I knew that Ceri had access to funds, and 
if necessary could simply pull the team out and take them home. Since
I was not there, I cannot incorporate events at the student site
into this diary, but I hope that Ceri and James will make a public report.
I will recount one horror story. Ceri opened a public area 
first-aid box at the students'
site, and found it to be full of homeopathic quack remedies
(I deliberately use non-pejorative terminology). For those
of us preferring science-based medicine, this is a little disturbing.
Parents have no fear, for Ceri also had access to a UKMT first-aid kit
containing strange things such as antiseptic ointments, 
bandages, plasters and so on.

In the evening Joseph and I get access to the UK solution scripts, and go to
work. I take the geometry problem 2, and the scripts are simply dreadful. Joseph
reports, by way of contrast, that they all seem to have solved problem 1.
Sergei seems to have done problem 3, and a couple of students each have 
some useful scraps which should be worth a mark. I wonder if 
other teams experienced the same dismal performance at problem 2, 
and asking around, it seems that almost every team has done better than us.
Our old geometric weakness has resurfaced. Shudder.

\noindent \textbf{July 8\ } The second exam begins, and there are a decent collection of 
questions from the students. The set-square issue comes up again,
left over from 2009.
After the exam starts, it becomes completely 
legitimate for deputies and observers at or near the student site to 
communicate with us, and text messages pour in.

The UK has won the Mathematical Ashes again. This is a private 
competition held just before the IMO between the UK and Australia.
The urn can stay in the northern hemisphere for another year.
We find out that it is now possible for deputies and observers
to get access to the student site, but that there are security guards 
trying to discourage this process.
I rest easier, since
it seems that now we probably won't need to make an early departure.

I reflect and observe that the excessive zeal of the local organizers in trying
to isolate students from the world actually generated a security hazard.
Unofficial communication between the two sites during the exam 
period is forbidden, but inevitably people who are
\emph{in loco parentis} will put the safety of the 
students above all else, and if they are not allowed access
to the students, this is sure to cause massive problems.
Imagine you went on an organized holiday, and you had arranged for your
family to stay to together in adjacent rooms, and paid extra to ensure that
this happened. How would you react when it turned out that parents
and children had accommodation in different places, and that there
were security guards trying to limit access to the children by their parents? 

In the afternoon we hear that an emergency jury meeting is being
called in the evening, but no agenda is supplied. Rumours of
an irregularity allegation circulate. My heart sinks.

For several days I have been waging a campaign to get hold of a large
number of boxes posted to Astana from UKMT. These will contain all sorts
of presents for IMO people, including the much sought after BMO solutions
booklet by James Cranch. There should also be a plentiful
supply of David Monk's master-work ``New Problems in Euclidean Geometry''.
David has been supporting the UK IMO effort since 1968, and he is the
author of more IMO problems than anyone else. Now he has produced this
lovely book, full of really interesting problems. The boxes will no
doubt also contain whatever trinkets are lying around the UKMT office.
I have asked repeatedly, and been told every time that these boxes 
have not arrived. 

Prompted by the ever-helpful Mary Wimbury in the UKMT office, we get 
the courier company to confirm that
the materials were received, and supply the details as to 
where and when the delivery took place. Then I present this information
to the friendly Kazakh IMO leader Almaz Kungozhin. He goes into
overdrive, with the help of IMO organizer Evgeniya Kalchevskaya 
they track down
the consignment. Soon a van appears at the hotel entrance, and 
a dozen boxes are taken to my room. When I open them, I discover that
the booklets and books are all present, and that UKMT had tidied 
its office by clearing out pens and key-fobs. I set about distributing
the goodies to the rest of the IMO.

In the early evening the deputy leaders turn up after a long journey.
They have moved to our hotel to assist with co-ordination.
They tell stories which are quite disturbing, but also reassuring.
When you have very limited information, you tend to worry that 
conditions might be simply awful. The reality is rarely quite so
bad, and even James Cranch's story about getting into the students' site
by pushing his way through security guards sounds quite normal.
The team can rely on Ceri now that James is here. Gordon Lessells, 
the Irish deputy, had hired a van and driver to provide a bus service
for other deputies while there was no IMO one provided. Well done Gordon!

I don't propose to give a blow-by-blow account of
the jury meeting which concerned the allegations against DPRK. 
The regulations were followed, and North Korea were 
disqualified for this year. On reflection, the procedures
could have been more transparent, and I hope that this unpleasant
experience will lead to the introduction of a more refined 
process to deal with allegations of impropriety.

What has gone wrong? I have attended every IMO since 2001, and I 
know of several allegations of irregular conduct, with
evidence of varying degrees of credibility. A pattern has become
established, and the leaders of many nations clearly believe,
rightly or wrongly, that some nations engage in systematic cheating.
This suspicion is corrosive, because it leads otherwise good
people to behave in a tawdry fashion. 

I recall a recent conversation with someone who should know better.
He is a leader of a country about which I have never heard any allegations.
He told me that he always asks for one mark more than his scripts are worth,
as part of the negotiating process. I was horrified, and asked him why he
did this. He told me that `everyone else does it'. Well, of course
this is not true. There are plenty of honest leaders who
would never ask for an undeserved mark. However, it seems that
the atmosphere of suspicion is leading some people astray, and causing
some otherwise moral leaders to step off the path of virtue.
This is depressing, because it undermines the IMO spirit of which
J\' ozsef Pelik\' an speaks so fondly. 

I fear that something will have to be done. Current IMO procedures must
change, and opportunities for improper behaviour must be minimized. Only then
will the IMO spirit be restored to full health. Three things spring to my mind,
and to the minds of many other people I am sure.

\begin{enumerate}
\item The long process of exam paper construction involves the problems being 
seen by a hundred leaders, and plenty of co-ordinators, members of the 
problem selection committee and observers A. All this happens over a 
lengthy period, so that any corrupt people have ample time to transmit 
problems, and to prepare alternative model answers. Maybe it is not
really happening, but in fact sufficiently many leaders believe that this
sometimes happens that the atmosphere is soured. We must develop
a streamlined method of paper construction which involves only a 
very few trusted, experienced and expert eyes seeing the paper in advance.
\item There must be sanctions against leaders who exaggerate the ingenuity 
and accuracy of a student's script. It is a relatively easy thing
to give a slightly false account, especially if student has written 
in a language which is not widely understood. I would favour an ethics
committee which would examine the scans of scripts during the
period between IMOs. If a student has secured high marks on a 
script which is worth little, then the leader of that country 
can be asked to explain 
how this happened to the jury of a subsequent IMO.
\item Host countries must take their pastoral responsibilities seriously.
Deputies, and especially observers C, must have ready access to the students.
When this happens, there will be no need for any unsupervised
communications at inappropriate times.
\end{enumerate}

\noindent \textbf{July 9--10\ }These are the co-ordination days. 
We divide the work between us, and I take all the geometry.
I will address the problems in numerical order, though that is not how they
were scheduled. Sometimes we called for breaks and had sessions re-scheduled.

I was to lead on Problem 1, though Joseph was also very familiar with the
scripts. The co-ordinators conceded three 7s immediately, but wanted to
talk about the other papers. In the time available we were able to
go through two of the remaining scripts line by line, and collected two
more 7s. There remained the script of Nathan `seven lemmas' Brown. 
We knew his script to be correct, and he had broken the argument up
into many small pieces, and had correctly explained how to put the parts
together. The difficulty was that he had written his solution in tiny 
handwriting, and it was very difficult to read, especially for co-ordinators 
who were used to Cyrillic script. 
I volunteered to write it all out again, with associated notes of explanation,
and so we adjourned for a second meeting. When we resumed, the co-ordinators
first convinced themselves that my version was a fair representation of
the original, and then Joseph Myers took them through it, correcting
a couple of gratuitous errors that I had inserted. The co-ordinators
finally agreed and we had six 7s.

Problem 2 was the geometry problem on which we had bombed badly. Luke
Betts got a single mark for accidentally observing a correct triangle 
similarity which was relevant to what had become known as the `Iranian 
Solution'.

Problem 3 was of more interesting. Two candidates got a mark for proving
that the function must be injective. Sergei Patiakin had solved the
problem, but lost a mark for giving a dodgy justification
of one of the steps. 

Problem 4 was the second geometry problem, and so I led. We agreed 
four 7s straight away. The disputes were that I wanted a 7 for Luke
Betts, and the co-ordinators were only offering 3 because they thought
that there was a hole in his argument. On the other hand, they wanted
to give Andrew Carlotti a 7 but I was only prepared to take 3. 
I explained to them the weakness in Carlotti's argument, and they looked
a little surprised. I asked them about the alleged weakness in Betts's
script, and I couldn't understand what they were worried about. I suggested an
overnight adjournment so that we could both make detailed preparations, and
we met again the next day.

When we met again, I explained that Luke had quoted the result
that when two points are inverse with respect to a circle, then 
that circle is an Apollonius circle with respect to these points.
Fortunately, when the marking scheme was being designed, I spoke in
the jury and asked if an educated student quoted this result
(or an equivalent one about orthogonal circles), whether it would be
allowed. I was worried that a student might do this since I had
explained it in a geometry session a couple of months before.
The problem captain had accepted that this was allowed. In our
second meeting, the co-ordinators rapidly agreed that Luke was right.

Finally we came to Andrew Carlotti's script. 
He had used an inversion map which,
unfortunately, did not exist. If it existed, all would be well, and he had
supplied a persuasive argument that it ought to exist. My difficulty was
that the map didn't exist. However, since our previous meeting
I had realised that you could dodge the problem by composing with 
a homothety with scale factor $-1$ about the attempted centre of
inversion, and that would fix the problem. I now thought the script
was worth about 5 or 6. The co-ordinators had found the fix as well,
but insisted that this man\oe uvre was simply a `negative inversion', and 
that other scripts had got full marks for this argument, and so we
had to accept full marks too. I put up some resistance, but finally we
conceded for the sake of the IMO.

James Cranch led on Problem 5. When we walked into
co-ordination we were pleased to see G\' eza from Hungary who is 
an excellent mathematician and speaks wonderful English. 
James quickly agreed two 7s for Sergei and Nathan. He then went
into grovelling mode in an attempt to get a mark for Luke Betts.
In the course of trying to prove the wrong thing, Luke had 
accidentally invented the Ackermann function. James thought that
this was such a wonderful thing to do that it might be worth a mark.
Indeed, it turned out to be in the mark scheme, and the mark was banked.

For Problem 6, only Andrew Carlotti had solved the problem. 
James began by 
remarking that he had translated the first part of 
Andrew Carlotti's problem 6 script ``into English from Idiot'', to which the 
Kazakh coordinators did not react but which caused Ilya Bogdanov at the 
next table to burst out laughing. Eventually James and Joseph
managed to extract the correct score of 6 marks for Andrew.

We have given UKMT presents to all the co-ordinators, and now an interesting
thing happens. Word has got out about David Monk's book, and we receive a
stream of supplicants asking for extra copies. Fortunately we have enough 
copies to keep everyone happy.

\noindent \textbf{July 11\ }
There are various excursions today, but I decide to rest. For me the high point of the day will be a meeting concerning the launch in 2012 of a European
Girls' Event. Murray Edwards College Cambridge (formerly New Hall) and UKMT
are trying to construct an annual event starting in Easter 2012.

The original plan was to have a Girls' Mathematical Olympiad, just as they do in
China. Indeed Ceri Fiddes and Alison Zhu will lead a UK team to the China
Girls' Mathematical Olympiad in August 2010, in order to gain experience.
The USA IMO leader Zuming Feng has been encouraging us to do this
for a couple years, and the Americans now send a couple of teams
to this event.
In the UK, the USA and the Commonwealth, separate girls' education
is seen as quite normal. However, in continental Europe it is not widespread.
I have had informal discussions with a few Europeans, and the reaction
is mixed. Some people do not like the idea of a separate
competition at all, seeing it as insulting. Others are in favour,
thinking that it might help develop girls' interest in mathematics.

Having grown up during the early phase of the feminist movement, I
am well used to being denounced for male shortcomings, hegemony
and oppression, usually while I am making the tea.
I can't face any more of that, and so I am hoping to hide behind some sturdy
females. Unfortunately local arrangements mean that Ceri won't be around
until the very end of the IMO, and my expected female deputy Vesna
has had to withdraw from this IMO because, happily, she plans to bifurcate.

It seems that rebranding the event, and incorporating an education
programme might make it more palatable. However, I am not altogether
convinced that this is a good idea. Anyway, I presented the proposal
in a vague and incoherent way to a group of countries, and there
is some support (and some skepticism). There is clearly enough support
to proceed, and I look forward to passing on the discussion to women
mathematicians.

One interesting feature of the meeting is that Morocco, Nigeria and
South Africa are all in attendance, which is stretching the
definition of Europe a bit. However, they
are all share time-zones with Europe, so this is not such a
strange arrangement after all.

After dinner we have the final jury meeting, and the medal 
boundaries are determined. There are fulsome tributes
to the work of J\' ozsef Pelik\' an, and he gets a standing
ovation. 

The incoming IMOAB chair Nazar Agakhanov makes a short speech in which he 
proposes some IMO reform. The decisions must be taken next year,
and we have 12 months to reflect on the best way forward.
Secretary John Webb explains that there is a problem in 2013,
and we currently have no host country for that year. This will
obviously be the first item on the agenda of the incoming Advisory Board.

That night, the World Cup Final was on a large screen in the leaders'
hotel. Now that Germany was out, Dieter was no longer wearing full fan
uniform. As you know, the Dutch tried to kick the Spanish off the park,
but in the end the had to start behaving better when they had 11 yellow cards.
Then a Dutchman was sent off, and the Spanish finally pressed home.
It seems that the people in Spain were quite happy about this.

\noindent \textbf{July 12\ }Today is the day that we get to see the students
again, but the procedure is to go on an excursion in order to meet them. The main event 
is an equine festival out of town. Remember, a horse isn't just for dinner.
It is also a means of transport.

We are so keen to see the students again, that I agree to go on the
excursion. We sit on a bus at the appointed time, and wait for 30 minutes.
Then the bus drives us to the back of the hotel and we get out again, because
this is where the first stage of the excursion takes place. This 
is a model of Kazakhstan, with all the most important mausoleums, 
power stations, palaces and railway stations so that you could
experience the treasures of the country without having to visit it.
As the universe is to a planetarium, as the ocean is to an aquarium,
as the Old Course at St Andrews is to a mini-golf hole, 
Kazakhstan is to this display.

At last we paid a lingering farewell to this
homage to Brouwer's fixed-point theorem, and made our way to the buses.
Less than an hour later, we were at the equine sports
centre. We sat in the stadium and after a while the students
arrived from the other site, after a long and tiring journey.
It was delightful to see Ceri and the team again, but
for reasons connected with who was supposed to eat where,
student delegations and leaders were ordered to sit in
different parts of the stadium.

We watched some excellent Kazakh horsepersonship, including
some `riding two horses at once', which takes some doing,
and some magnificent `pick something up from the ground while
riding at full pelt' which was also impressive.
Then there was an event of dubious political correctness,
where a Kazakh girl on horseback was chased down by a 
male rider who wrestled her from her horse, and pulled her to the
ground in a rather robust form of kiss-chase. The fact that
the programme insisted that this event was called
``come up your girl'' was intriguing. By now the music on the
tannoy was getting very loud indeed, and I started to sneak out when
they started a game of the traditional Kazakh polo. It is quite like
the Afghan game where you use a dead goat instead of a ball. In
Kazakhstan you use a calf, beheaded so that it rolls better.
In fact they sanitized this event and used a bag instead of a calf.

Soon everyone poured out, and the local organizers tried to usher leaders
into luxurious yurts (tents) for lunch. I resisted at first, because I was far
more interested in talking to the team than having lunch. It was
delightful to chat to them again after our long separation, and to hear
edited highlights of their adventures.

At length the UK leaders adjourned to a yurt. This had been 
magnificently prepared by primary school children. We were serenaded
by musicians, and we shown the unveiling part of the
pre-nuptual ceremony. This was delightful. We ate salads for a while,
and then a dish of rice and various sheep parts was presented. I 
was given the head. We shared food and had a wonderful time. We thanked
the artists for all the trouble that they 
had taken, and asked them to pass on our
thanks to the schoolchildren. 

The students would now move their base to Astana city, and stay in
a hotel near to where the leaders are based.


\noindent \textbf{July 13\ } The closing ceremony is held mid-afternoon. As I am now nearly
on the IMOAB, I get to go in the van with the grown-ups (J\' ozsef Pelik\' an,
Secretary John Webb, Slovenian power-broker Gregor Dolinar etc). 

There is a vast pedestrian concourse in front of the Palace of 
Independence, and a long red carpet runs out of the Palace doors,
down the steps, and all the way across the concourse to the road.
There is a guard of honour on each side of the carpet, and 
musicians are waiting outside the palace.

In a wonderful ceremonial gesture, it turns out that the red carpet is
for the teams. Each of them marches along behind a guide carrying
their country's name. The students carry flags, and as they get close to the 
palace, the incessant Kazakh music becomes overwhelming. The jaunty
firemen
have giant trumpets, in the spirit of a vuvuzela, but three times the length.
Stacks of speakers bring the music to the threshold of pain.  

I and my new friends sit in the front row. 
We have no idea if we will be
asked to assist in the medal ceremony, so we sit there and hope to
be left alone. The teams parade across the stage once more, in a
reprise of the opening ceremony. Then Prime Minister
Karim Massimov comes in, and presents a prize to the 
only student who acheived a perfect score, Zipei Nie of China.

The astonishing Lisa Sauermann of Germany scored 36/42, and so
earned another gold medal. She now has one silver and three
golds, and will soon surpass the German deputy Christian
Reiher who retired from competition in 2003 with one bronze and
four golds. I think Lisa has a couple more IMO chances, and no
doubt she will enjoy climbing above Christian.

Past Kazakh IMO competitors present the bronze medals. Various 
academic luminaries hand out the silver ones, and the Minister
of Education and Science bestows the gold medals. These
are accompanied
by laptops, thanks to Exxon Mobil, one of the principal 
sponsors of this IMO. At this point UK gold medallist Sergei Patiakin
is holding his
Union Jack inverted, a traditional distress signal in the Royal Navy.
Sergei is in trouble, and I surmise that
his crew have yellow fever or perhaps he has been captured by Barbary 
pirates. I gesture to him, and he quickly rectifies the situation.
False alarm.

J\' ozsef Pelik\' an made his farewell speech as chair, and made
very gracious remarks wishing good luck to the new Advisory Board.
We owe J\' ozsef an enormous debt of gratitude.

After the ceremony we return to the leaders' hotel. The closing banquet
is held in a large room round the back of the complex.
It is a splendid affair, and adults even get a little wine.
There is the presentation of the Golden Microphone. 
This has been won by the leader of the Netherlands,
Johan Konter. As usual the comical overblown ceremony
goes very well.

We go to bed early because we have an early flight the next day. 
The IMO organization has laid on a bus for us at 03:00. They wanted
to pick up our students first at 02:00 but Ceri and James refused,
and arranged a taxi for 03:30 instead, and told the organization about this.
Nonetheless, an attempt was made to drag our team out of bed for a bus,
and poor Sergei Patiakin had to mediate because of his Russian language
skills. Sergei is a very well-mannered young man, not versed in the art
of telling people, very firmly, to go away. 

Joseph, James and I go by bus from our hotel, and meet up with the main party at
the airport. As expected, the border guard takes a very dim view of my lack
of an immigration card stamped in two places until I mutter the magic words
`International Mathematical Olympiad', and he lets me out. Joseph
Myers had a similar problem.

The flight to Moscow is fine, and we have only a couple of hours in
transit in Domodedovo. We have a light breakfast, and then find that we
are very fortunate for our final leg. We have a giant plane with very 
few passengers. We spread out, and doze.

Arrival is easy enough, and we are met by friendly families.

\section*{Conclusion}
A great many people worked very hard to create IMO 2010, and
I would like to single out the guides for special praise. It seems
that they were often acting without full information, and were
under pressure to control the activities of the teams. Of course
it would have been more sensible if the guides had been better informed,
and then they could have taken on their proper role as facilitators and
assistants with greater ease. Even so, all reports I had were that
the guides were trying their best to be helpful and kind.

The Kazakh organization DARYN put in an immense effort to run this event,
a sustained project which lasts many years of course, and considerable
resources were expended. The Kazakh state was clearly committed to the
project, and the warm and wise words of two Ministers were very welcome.

Thanks to the students, their families, fellow UKMT volunteeers and
employees who have been involved this year. 
We now have a rich cycle of events, involving more and more students in various national and international competitions. 

As I step back from the task of leading the UK IMO team, I would like
to make a couple of points. First I must thank my successor James Cranch
for agreeing to take on the role, and the various deputies, observers and
trainers who have helped with IMO participation in recent years. 
I would explicitly like to thank Vesna Kadelburg. She wisely decided
not to go to Kazakhstan while her health was vulnerable, but continued
to support the enterprise in many ways.

UKMT continues to prosper. It is a large and 
multi-faceted organization, engaging in various aspects of secondary school mathematics enrichment. All of these activities are 
successful, and moreover the financial structure which underpins
UKMT and the British Mathematical Olympiad continues to function well. BMOS welcomes financial support from elsewhere, for example the
generous support from Winton Capital Management, but 
we are not dependent upon it, but we do rely on the 
continuing goodwill of UKMT
and BMOS volunteers.

Mathematics in the UK is enjoying a boom both in terms of secondary school participation, and in terms of recruitment to undergraduate
mathematics courses. It would be gross hubris to attribute this marvellous
circumstance solely to the activities of UKMT, but 
surely the complex and sustained mathematics 
enrichment programme that has been built
under the UKMT flag in recent years has played at least a part
in the revival of interest in mathematics in this country.

We must stay together, working for the good of the subject, 
and nurturing each generation of mathematical talent as it passes through.
We must build UKMT, both in terms of volunteer effort and financial clout, 
until it sits as a permanent and powerful force behind 
mathematics at school level, ready to protect mathematics education
of quality against the never ending threat posed by fools and charlatans,
and those who would seek to divide us.
\end{document}
