\documentclass[12pt]{article}
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\title{UK Leader's Report, IMO 2004}
\author{Geoff Smith}
\date{28-vii-2004}
\begin{document}
\maketitle

The 45th International Mathematical Olympiad was held
in July 2004 in Greece. It was a tremendous success.
This was the latest cycle of the annual world championship
of secondary school mathematics. Each participating nation
may send up to six contestants. In common with the sporting
Olympic Games,
this is a competition between individuals, not nations, though
friendly rivalry between states adds spice.

The examinations were held on consecutive days in the University of
Athens. There were three questions on each day, and the students
had 4 hours 30 minutes to address them.
The problems were drawn from the four traditional
subject areas: algebra, combinatorics, geometry and number theory.
The team leaders arrived in Greece on July 6th and formed
themselves into the oracular jury at Delphi. They carefully selected 
questions at this magnificent location, perched
on the side of Mount Parnassus. The air was clean and mercifully
cooler than in the Athenian cauldron. From a height of nearly 1000m
the view was extraordinary; the unspoiled landscape sacred 
to Apollo drew the eye down
to the Mediterranean Sea.

Two new nations were welcomed into the IMO fold: the Kingdom
of Saudi Arabia and Mozambique. Originally 87 nations were
scheduled to participate, but unfortunately in the event
neither Guatemala nor Tajikistan were able to come, so we
had a contest among students from 85 nations. The Arab world
is usually not represented well at IMOs, but I am delighted
to report that four Arab nations attended the 45th IMO: 
the Kingdom of Saudi Arabia, Kuwait, Morocco and Tunisia.

The UK team consisted of UNK1 Giles Coope (Fallibroome High School),
UNK2 David Fidler (Haberdashers' Aske's), UNK3 Paul Jefferys (Berkhamsted
Collegiate School), UNK4 Martin Orr (Methodist College, Belfast), UNK5 Alexander
Shannon (Kings School, Canterbury) and UNK6 Anne Marie Shepherd 
(Ilkley Grammar School). The team leader was Dr~Geoff Smith
of the University of Bath. The deputy leader was Mr~Adrian Sanders 
of Trinity College, Cambridge. The UK Observers were Dr~Jeremy King
of Tonbridge School and Mrs~Patricia King of Benenden School 
and the Executive Committee of the British Mathematical Olympiad.
The three UK reserves were Alexander Davies of Winchester College,
Matthew Lee of Robert Smythe School, Leicestershire and 
Alex Smith of King Edward VI Five Ways, Birmingham. 
Dr~Vin de Silva of Stanford University attended the
pre-IMO camp as a coach.

Here are the questions of the 45th IMO. The wording is the
one actually used in the English language version. The countries
which submitted the problems are indicated. 

\section*{Day 1}
\begin{enumerate}
\item (Romania)
Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle
with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$, 
respectively. Denote by $O$ the midpoint of the side $BC$. The
bisectors of the angles $BAC$ and $MON$ intersect at
$R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$
have a common point lying on the side $BC$.

\item (Korea)
Find all polynomials $P(x)$ with real coefficients 
which satisfy the equality
\[
P(a-b) + P(b-c) + P(c-a) = 2P(a+b+c).
\]
for all real numbers $a,b,c$ such that $ab + bc + ca = 0$.
\item (Estonia)
Define a ``hook'' to be a figure made up of six unit squares as shown
in the diagram 
\begin{center}
\begin{picture}(100,100)
\put(0,0){\line(0,1){90}}
\put(30,0){\line(0,1){90}}
\put(60,30){\line(0,1){60}}
\put(90,30){\line(0,1){60}}
\put(0,0){\line(1,0){30}}
\put(0,30){\line(1,0){30}}
\put(60,30){\line(1,0){30}}
\put(0,60){\line(1,0){90}}
\put(0,90){\line(1,0){90}}
\end{picture}
\end{center}
or any of the figures obtained by applying
rotations and reflections to this figure.

Determine all $m\times n$ rectangles that can be covered
with hooks so that

\begin{itemize}
\item the rectangle is covered without gaps and without overlaps
\item no part of a hook covers area outside the rectangle.
\end{itemize}

\section*{Day 2}
\item (Korea)
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, $\ldots$, $t_n$ be
positive real numbers such that

\[
n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1}
+ \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right) .
\]

Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all
$i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

\item (Poland)
In a convex quadrilateral $ABCD$ the diagonal $BD$ bisects neither the
angle $ABC$ nor the angle $CDA$. 
A point $P$ lies inside $ABCD$ and
satisfies
\[
\angle PBC = \angle DBA \mbox{ and } \angle PDC = \angle BDA.
\]

Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP$.

\item (Iran)
We call a positive integer \textit{alternating} if every two
consecutive digits in its decimal representation are of different
parity.

Find all positive integers $n$ such that $n$ has a multiple which is
alternating.
\end{enumerate}

I would be interested to read original solutions to these questions,
especially the more demanding problems 3, 5 and 6. Please post them to 
Dr G C Smith, IMO 2004 Solutions, Department of Mathematical Sciences,
University of Bath, Claverton Down, Bath BA2 7AY.

Here are the performances of the UK team. The jury aims
to make the questions of increasing order of difficulty on
each day, and to make the questions on the second day a little
harder than on the first. Each question is marked out of
7 points according to a strict marking scheme.
In 2003 the United Kingdom students cleaned up on the relatively
easy Problems 1 and 4 with 82/84. This time we were not quite so
efficient and secured 78/84.

Here is our table of performance:

\begin{verbatim}
       P1   P2   P3   P4   P5   P6    Total  Medal

UNK1    7    2    2    7    3    2      23   Bronze 

UNK2    7    6    1    7    3    0      24   Silver

UNK3    7    7    2    7    3    6      32   Gold

UNK4    6    7    2    7    0    0      22   Bronze

UNK5    2    6    0    7    1    0      16   Bronze

UNK6    7    1    1    7    1    0      17   Bronze

Total  36   29    8   42   11    8     134
\end{verbatim}

Medals were determined as per the regulations. At most one half the
participants may receive a medal; happily this year the bound was achieved.
The medals are then awarded approximately in the ratio 1:2:3, 
gold:silver:bronze. Paul Jefferys's gold medal and David Fidler's silver
medal were obtained with no margin of error whatever, as was Alexander
Shannon's bronze, the cut-offs being 32 gold, 24 silver and 16 bronze.
Paul Jefferys is the first UK contestant to have secured 4 IMO medals
(B 2001, S 2002, G 2003, G 2004). In 2003 he also scored the gold cut-off
mark and in 2002 he was one mark below it. David Fidler improved on
his bronze medal of 2003. Martin Orr slipped back from silver in 2003
to bronze in 2004. He hails from Belfast so he becomes
the second Irishman to secure two IMO medals, and so far is the only
one to obtain a silver. Martin is the only one of our players
who will be eligible for IMO 2005 in Mexico. This year Martin was not
the all-Ireland champion, since a contestant from the Republic of Ireland
secured a bronze medal with one more mark than Martin. That
young man will also be eligible for IMO 2005, and it is to be
hoped that their friendly rivalry will spur them both on 
to even better performances
next year, and that Martin will come out on top. 

The three newcomers to the UK team were Giles Coope, Alexander Shannon
and Anne Marie Shepherd. Giles was unlucky not to get a silver medal,
falling just one point short. Anne Marie and Alexander 
were our most inexperienced players, both entering training less than
a year ago (Alexander only since April 2004), but they did all that
was asked of them, and pocketed their bronze medals. 

The UK medal haul 1G, 1S, 4B was only slightly inferior to that of
last year (1G, 2S, 3B) and it was heartening that all team members 
received medals. I am confident that each of our three
reserves (Alexander Davies, Matthew Lee and Alex Smith)
would have secured a medal had they participated. Our overall
national ranking slipped from equal 10th to 20th. Several
former Soviet Republics and Far Eastern teams hurtled by us. 
In my opinion the team of 2004 was about as good as that
of 2003, but this time the harder questions on the papers were
not quite so hard as in 2003. Our weakness on the tough problems
was exposed, though it was very pleasing that Paul solved
Problem 6, and David's partial solution to Problem 5 was a joy.

The IMO is a contest between individuals, not nations, but
inevitably the table-makers will have their way, and team
performances are compared by their overall mark. This is
absurd of course. Some nations are prosperous and can afford
to pamper their teams. Other nations are so populous that
a sufficiently diligent trawl through their schools is bound
to produce a strong team. Yet more nations have education
systems which focus whatever resources
are available on the intellectually able. There are also countries
where the pressure on teenagers to perform academically is
more direct than in the more gentle (decadent?) societies of  
Western Europe. In the United Kingdom 
we have a fair sized population (60 million or so) and by world standards
we are a prosperous country. If we are to engage in friendly national rivalry
at the IMO, then our obvious competitors are France and Germany
where standards of living, population sizes and attitudes to
education are comparable with those prevailing in the United Kingdom. 
In terms of national performance, we just maintained our position 
as the leading Western European team, keeping our nose in front
of a resurgent German team. Geopolitical changes outside 
the competence of the IMO jury
mean that the UK is no longer the leading nation in
the European Union; two new members of the Union swept past us.
Our training partners Hungary came a magnificent 1st, and Poland 
a worthy 2nd. In the Commonwealth the UK ranking slipped from 1st to 3rd;
India are the champions this year, with Singapore in 2nd place. 

Four students secured perfect scores. They came from Canada, Hungary,
and two from Russia. The young man, B\'ela Racz, from Hungary is an old
friend of the UK team from our common training sessions, and it is
delightful to see him doing so well. It was also most pleasing to
see Jacob Tsimerman from a Commonwealth country, Canada, achieving
a perfect score.

Now we once again resort to the annual device which enables
the UK leader to burst the shackles of truth. The tenses may
vary, but a self-serving version of reality remains the
underlying theme. 

\section*{Leader's Diary}

This diary has become read internationally on the web. It is extremely
easy to give accidental offence, especially when trying to portray 
events in a light-hearted manner. Any negative comments about aspects of the
organization must be seen against a background of the overwhelming
success of the 45th IMO, and the hard work of many hundreds of people 
who made it all work so well. Moreover, jibes at other leaders and deputies,
co-ordinators, guides and team members should be regarded as the self-indulgent
ramblings of a sad man.  

\noindent\textbf {July 3:\ }  
Today I meet five of the team, the deputy leader Adrian and
a trainer Vin at Heathrow. Vin has flown in from the States to help.
Alexander Shannon is staying behind in the UK because
of long-standing musical commitments. He will join the team next week
with the two UK observers Jeremy and Patricia. The plane journey to
Athens was uneventful, and I am pleased that Olympic Airlines is at last
trying to live down
its {\em flying ashtray} reputation. The new Athens airport is quite 
a sight for me, since I knew its less swish predecessor very well. The
party fits itself into three taxis and we head for the Armonia hotel.
At least, that is the plan.
The taxis have clearly been upgraded for the Olympic Games. The 
meters work, and the drivers no longer tell you implausible stories about
your hotel having been destroyed by meteorite (but they always knew 
an excellent
alternative which would give you a good deal). 
I am in the lead taxi, and am troubled by the difficulty I have 
convincing the cabbie that we want the Armonia hotel rather than
Omonia. The Armonia is due south of the airport, away from Athens city
centre which, worryingly, contains a district called Omonia. 
After a while the drachma drops and we call the other cabs by mobile
telephone. Happily the third taxi is driving into the sun, but unfortunately
Adrian's vehicle has the sun at its back. We pass the telephones to our
respective taxi drivers, and my driver tells Adrian's driver to hang
a $\pi$. 

When we arrive the receptionist at the Armonia assures me that
I have only booked beds for six people but expresses concern that
there are eight of us. I point out that one can only reserve rooms
of one type using their internet reservation system, and that he may
find that I have made a second reservation. 
All is well. We dine
in the hotel and go to bed early because we have a flight to the island
of Limnos in the middle of the night. We troop to bed passing 
the poolside disco
as it is being set up. 

\noindent \textbf{July 4:\ } A few hours later the bleary party walks past the disco
again as it is being dismantled. Three cabs take us to the airport
where we board a turboprop. We land soon after dawn in Limnos, a quiet
island in the Northern Aegean, close to Lesbos and Turkey. It is famed
for its refreshing winds. We board the usual taxi convoy and head for
the Porto Myrina Palace. This joint is 5 star but out of season.
Now 5 star on a Greek island is not the same thing as 5 star in a capital
city (where you should be able to see your reflection in the marble floor),
but nonetheless it is very agreeable, and I can see that the team members
from state schools are impressed. 
The receptionist reasons, with faultless Greek logic, that the student's surname
`Shepherd' should be pronounced `Sheffard'.

We spend the day swimming in the hotel
pool and recovering from the journey. We stroll towards Myrina looking
for a taverna for lunch, and discover a cheap and pleasant place 
with a friendly waitress who is 
a refugee from Cardiff. She steers me clear of the dodgy grilled octopus.
Given that I am English, this is a remarkable act of kindness.
I have been issuing problem sheets to the team on attractive
University of Bath pastel A4 sheets at random moments 
to add to the gaiety of life, and from time to time 
Adrian's cry of ``trig drill''
means that the students have to face a public grilling concerning
some of the more attractive formulas found towards the back of
{\em La g\'eom\'etrie du triangle}, the masterwork of Lalesco (it is even more
thrilling in the original Romanian). I particularly enjoy post-prandial
trig drill. 

\noindent \textbf{July 5:\ } The 
team is up early for they have their first practice IMO exam.
I have a chance to inspect the island which seems idyllic save for
the fancy military equipment bristling on the hilltops. We are precisely
in the area where the Greeks and Turks like to wind one another up by flying
fighter jets 10 metres inside their own airspace. The heat of the day
sets the cicadas off at a terrible rate. I quiz the team as to what is
causing the noise, and receive perfectly genuine ignorant suggestions.
Thank goodness we are not going to the biology olympiad (electricity lines
humming indeed).  
In the evening we watch Greece beat Portugal 1-0 in the European soccer final
on a large outdoor TV near the pool.
The Greeks seem happy about this in a curiously Mediterranean sort of
way (no vomiting or damage to fixtures).

\noindent \textbf{July 6:\ } I depart very early in the morning 
to join the jury on the mainland.
I leave Vin and Adrian my remaining multicoloured pastel problem sheets, and
copies of key works. Following local advice, I ignore STA Travel's injunction
to arrive at the airport 90 minutes early, and turn up 
with barely an hour to spare. The 
place is virtually deserted and the check-in is shut. Happily the Everest food
outlet is open and I enjoy a double espresso with cheese and spinach pie. 
I doze all the way to Athens airport, and am met by a Greek young man
who is very efficient and does sunglasses very well. I am taken to the
international arrivals area where a handful of other leaders have arrived,
including those of Spain, Thailand, Saudi Arabia and Macedonia. We board 
an IMO bus after a short wait and drive to Delphi. It takes about three hours.
You first pick your way out of Athens, then across a cotton growing area, and
finally up into the mountains. The modern town of Delphi clings to the side
of Mount Parnassus. Coaches cannot turn round on its narrow streets, but there
are large rotation zones at each end. This makes for some complicated journeys,
and I am minded of the theorem that you can turn round a needle of zero width
in an arbitrarily small area.
We try to drop off some of the leaders at a hotel, including the
leader of Kuwait. Unfortunately he isn't on the bus, but an obviously
misidentified leader of the Kingdom of Saudi Arabia is. After a while
the driver correctly decides that Kuwait and Saudi are sufficiently similar
and lets the KSA leader off. The driver 
then fails to find my hotel several times.
This involves much threading through narrow streets and embarrassing rotations.
At last the Macedonian leader and I are dropped at our hotel.

Next I wish to demonstrate how to lose friends by making 
light of a serious issue. 
There is a curious language game being played. In Greece it is 
not done to refer to the country called Macedonia by that name. It is
called Fyrom (the Former Yugoslav Republic of Macedonia). Thus
every time the relevant leader is asked who she is, she claims to
be the leader of Macedonia, and her Greek interlocutor simply nods
and says Fyrom. To an outsider it has a surreal air.

I stroll around town, meeting various old and new friends. I meet
one leader whom I will not identify but who has clearly not been
properly briefed. He has brought US dollars as the medium of exchange,
and is under the impression that the native tongue of Greece is English.
Don't ask because I won't tell.

In the evening we finally get copies of the 
problems short-list, about 30 questions
culled from suggestions sent in by problems committees around the world.
I have no idea if any proposals are British, since I don't look at
our six suggestions. This means that I can argue for and against questions
with a clear conscience. There is a delightful geometry question which
looks a certainty for Question 6 until it is sabotaged by the Japanese leader
who gives a precise reference to a very similar question in his problems
literature. That beauty was actually the only British proposal to make 
the short-list, and the squad of 2005 will really enjoy it. The short-list
of IMO $x$ is classified until IMO $x+1$, so one has some very good questions
to tax the students on.

\noindent  \textbf{July 7--10:\ } Over the next few days we pick the questions.
There is a strong temptation to revert to traditional ways. The sibyl of 
Delphi is the voice of Apollo. This woman is always called Pythia, and must
be an older woman who has led a blameless life. Looking around the jury room,
I see that we may be in difficulty (because of the youth of all the female leaders).
It would also have been necessary for a goat to be sacrificed and
its entrails inspected by priests (presumably the Problem Selection Committee).
Mindful of the fact that the UK team contains four vegetarians, and
the difficulty of selecting a sibyl (taking the Pythia), I assent to the
construction of the IMO papers using more modern methods.
 
We decide that there
are really only three sufficiently straightforward questions to be
Problems 1 or 4, so we choose those first. Problem 1 will be a geometry
question containing a booby-trap, and Problem 4 is a neat
but relatively undemanding inequality. It is cute though. Try it.

We then move to choose Problems 3 and 6. We have lots of choice, and there
is a consensus that we should have one on combinatorics and
one on number theory. I am not that impressed with the pair selected;
the difficulty may be about right, but neither problem is civilized
mathematically. Whisper it softly, but they both have the aura of a puzzle. 

Finally we select the supposedly medium hard Problems 2 and 5. This
time the aesthetics are better; we get a sweet polynomial functional 
equation and a geometry problem with teeth. The geometry involves demonstrating
that one condition holds if and only if another does. One way is merely taxing.
The other way will really sort them out.

When the questions have been selected the task is then to get them translated
into all relevant languages (about 50). This process begins with the English
Language Committee (ELC), which I chair.
I invite everyone to join the committee 
since exclusion so easily causes offence. Now that English is Orwellian
worldspeak, it is nonsensical for the native speakers to try to claim
exclusive rights. This is the {\em quid pro quo} for 
English being the {\em lingua franca}
of the mathematical {\em cognoscenti} and 
IMO {\em apparatchiks}. I had long since
spotted an error of English in the existing draft of Problem 5, and 
curious to see if it would be picked up, kept silent
until the ELC met. It was then appropriate
to reveal the howler 
and suggest a fix. The ELC takes about an
hour and a half to come up with its proposals. The jury then
reconvenes, and goes through the traditional ritual of
criticizing the proposals, debating changes to the
wording, and usually deciding that the ELC wording is actually fine.
Secretarial assistance in the ELC is kindly supplied 
by the Leader of Canada, Christopher
Small. 

Then there is the frenzy associated with typing the English version,
and then getting it translated into the other official languages
of the IMO: French, German, Russian and Spanish. Next we needed
to make versions in 
all the languages that the students wanted. For almost all languages 
this was done in LaTeX. For reasons which were
never explained, the organizers were planning to automatically convert
LaTeX to WORD. This caused chaos. The software tried to improve
the spelling of languages it was never designed to meet. The Afrikaans 
was particularly badly mangled. Even the English version had
`nor' changed to `not'. The diagram of little squares
was changed into a diagram of rectangles. At length it was decided
not to use WORD after all (a wise move).

We are having lunch every day at the jury site, but most evenings 
we enjoy the hospitality of the mayor of a nearby town. This is a splendid arrangement,
and gives the opportunity to both individual and massed South American singers
to entertain the rest of us. I take the opportunity to have the 
Cuban anthem {\em Guantanamera} translated (loosely) into English. 
It is
a boy-meets-girl poem, drastically shortened for the song. I decide that
in future there must be a British response. I will be working all
winter on my rendition of that profound exploration of our common
fate
in the context of missing headgear
{\em On Ilkley Moor bar t'at}. 

\noindent \textbf{July 11:\ } The opening ceremony. This involves two
journeys of three hours each to get to and from Athens. On the way one of
the buses goes on the blink, but is swiftly fixed by the drivers. We drive
past the new Olympic Stadium en route to the hall. Then we have the
opening ceremony.
Politicians never quite get the IMO
(but we are all very grateful because they are bankrolling the whole
show).
We get to listen to a 
lot of stuff about football and the Olympic Games which
is no doubt supposed to flatter us by putting the IMO in an heroic
context. I wager that almost everyone concerned with the IMO
regards it as far more important than 
quadriennial lowbrow jockfests. The best thing about the
opening ceremony was a recital on a {\em hydraulis}, a recreation of
a recently excavated instrument from classical Greece. It is a pipe
organ where an assistant has to pump away to keep 
the air going. Henceforth I shall think of the Scottish piper at the 
opening ceremony of 2002 as using a portable hydraulis.

\noindent \textbf{July 12:\  } The leaders gather at 9am in the Delphi jury
room as the students sit down to the first paper in Athens. For the first half-hour
the students are allowed to ask questions about the paper. These are then faxed to Delphi.
The relevant leader then suggests a reply to the jury. If this is agreed, the
reply is faxed back. This session was very quiet, with only 21 questions. 
The examination arrangements are never seen by the jury, but as time goes
by stories about the invigilation system start to filter back. Invigilators
are always on call when a student needs them.

\noindent \textbf{July 13:\ } On the second day of the examination the jury
receive a very large number of questions, mostly concerning the meaning of the
term {\em parity}. In the afternoon the leaders embark on their favourite bus journey
to Athens. We are taken to the Marriott, a classy $N$-star outfit about a 15
minute cab ride from the students' hotel. Late in the evening the UK students
join us at the Marriott for a small celebration. As they chat with us the
second days' scripts arrive. Adrian and I make our excuses and leave to
burn midnight oil. Patricia and Jeremy will shepherd them home. 

\noindent \textbf{July 14:\ } Co-ordination 
begins. In the hotel foyer after breakfast I find one of the Problem
Selection Committee in melt-down. Apparently large numbers
of co-ordinators have not turned up, and a problem captain
is still asleep. This is going to be interesting.

Since there are four adults working on the UK
scripts, we need space. I go scouting round the hotel and discover that
the jury room is not in use. We decamp to this impressive
space, a room for over a hundred people, and move the furniture
to make a base. We carve up the questions so that 
we will have a complete expert on any one script. Our observers
Jeremy and Patricia are keen to be involved. They are allotted Problem 6.
Only Paul has made a serious fist of this question; their job is to
understand every word he has written, and have a rational explanation 
for every rough jotting, smudge and crossing out.

We are to co-ordinate Problems 2, 3 and 4 today, then 5, 6 and 1 
tomorrow, in that order. The schedule is such that the co-ordinators
have not had time to study the scripts in advance, and this rather slows
up the co-ordination phase. The difficult issue with Problem 2
is David Fidler's script.
The co-ordinators believe that it is worth doodly squat, whereas we
know that he has an almost complete solution. Admittedly there is a lot
of misleading padding, but putting the first page together with the
last and ignoring the false trails in between takes you almost home.
We have to appeal to the problem captain because the co-ordinators don't
follow our explanation. He sees it straight away, and we pocket 
David's well-deserved 6/7. Adrian led on all this, and did a first-rate job.

I lead on Problem 3. We are blessed with very quick-thinking and fluent
co-ordinators for this question, and it is easy to sort out who
deserves scraps and who doesn't. There is an interesting cultural
clash when we are discussing whether or not some work in
rough of Anne Marie deserves a mark. We think that it
clearly does, but the co-ordinators have 
to persuade themselves. After an exhaustive analysis of the
script, the co-ordinators are on the edge of conceding the mark.
Then one of them drops the bomb:
``Is this student a girl?''. This massively inappropriate question
(in UK terms) 
fires off alarm-bells in my head, but of course now
might not be the right moment to engage in discussions concerning
feminism, anonymous examination procedures and Anglo-Greek cultural
differences. I venture the reply `yes' and Anne Marie gets her mark. 

I also lead on Problem 4. Now we are really lucky, because we are
being co-ordinated by a member of the Problem Selection Committee, and he
really knows his stuff. This is a very straightforward co-ordination
because all of our students have supplied perfect solutions. The co-ordinator
carefully checks every line. He is happy with five of the six 
scripts, but Martin
Orr's highly analytical solution could not be properly 
read in the time available. The co-ordinator says that
he will read it in 
the evening (there must have been five
or six pages of detailed calculation) but that he expects it to
be worth 7/7. We casually walk out, and then I do a very careful rereading 
of Martin's script just to be sure. I find two completely inconsequential
transcription errors. When I see the co-ordinator later I ask him how
many glitches he found; he says two. Martin and indeed everyone else
gets their 7/7s.
 
\noindent \textbf{July 15:} Adrian 
leads on Problem 5, a tough geometry question. This is
an extremely laborious co-ordination. We are not asking for more than 
3/7 for any of our students. The default method is an  
arduous and complex angle chase. The co-ordinators
correctly insist on checking every line, but it does take a very long time.
Then we find that we have a difference of opinion concerning David Fidler's script.
The co-ordinators do not believe that he has given a proof at all, whereas
we believe that he has given the best proof of any of our students. 
It is worth explaining what David has done. He took the geometrical
configuration in question, and drew some extra lines. The effect of the
extra lines is to make a figure which has an axis of symmetry (the
original configuration was not symmetrical). The existence of this
symmetry enables one to read off the required result. This was
geometry in the spirit of Felix Klein and Henri Poincar\'e.
I thought we would all stand up and hug for the joy of it all, but
the co-ordinators take another view.
This is not proper geometry they say. 
No triangles are mentioned. No angles
are calculated. One of them says `I am here to defend Euclid'.
We appeal to the problem captain who, fortunately, is more sympathetic
to methods popularized after 330 BC, and David gets his 3/7.

Finally we have a problem with Paul's script. We are asking for 
3/7. His argument is long and trigonometric. The co-ordinators
are clearly unimpressed by Paul's admittedly convoluted wizardry
(even though he does use triangles).
Eventually they accept that the main thrust of Paul's argument
is correct, but argue that he has not dealt accurately with 
degenerate cases. Adrian argues that he has, and after involving
the problem captain again, Paul finally gets his 3/7.

The co-ordination of Problem 6 is led by our observer, Jeremy King.
Only Paul had a serious attempt at a solution. Initially the 
co-ordinators seem sceptical, but as they work through all
the details and side cases, I can see that they are becoming
progressively more impressed with Paul's analysis. Jeremy's complete
mastery of the argument is a great help of course. Paul has overlooked
one case. His work is clearly nearly a full solution, but he must 
lose at least one mark for his oversight. The omission is easy to fix by
using a method which Paul had already invented,
and by now the co-ordinators are sympathetic to his solution. We have
two pieces of intelligent work in rough which Jeremy hopes will convince
the co-ordinators that Paul should only lose 1 mark. 
He only has to show them the
first piece of evidence, and they award 6/7.

Finally we move to the co-ordination of Problem 1 led by me. 
This proves a fraught and difficult co-ordination. We are
first kept
waiting for two and a half hours because Australia pinch our slot
(thanks Angelo). Many of the co-ordinators for this
question knock off early and go home because they do not
live in Athens. We have a single co-ordinator, who is clearly over-worked.
We are asking for 7/7 for Giles, David and Paul, but we know
of minor flaws in the other three scripts, so are expecting 
5/7 or 6/7 for each of those. He immediately offers 7/7 for Giles
and 6/7 for Martin, which we accept (I had been worried that
Martin might only get 5/7 which would have extinguished all hope
that he might just scrape a silver medal). 
After careful reading, he
gives us 7/7 for David and Paul. He explains that there is a problem
with Alexander Shannon's script, so we move on to Anne Marie's paper.
We are asking for 6/7. He says that although it isn't perfect, she may
have done just enough to get 7/7 because of her (incomplete) analysis
of degenerate cases. We sit bemused while he convinces
himself that her script was worth 7/7 according to precedent set throughout
the previous two days. Here we were getting the benefit of going last;
some hard-nosed leaders have clearly been arguing that black is white,
and the co-ordinators have given ground somewhere on the way. In order
to be consistent it must be the case that black is white for us too.
Lucky Anne Marie I say.

Next comes Alexander Shannon's script. His solution has two imperfections.
The first is a hole, and the second is that his punchline is too
slick. We had not really noticed this second fault, and still 
think the criticism was unduly harsh (he did not explicitly spell out 
an obvious step). The hole takes the following form.
Alexander wants to show that four points are concyclic. His method
is indirect, cunning and ridiculous. He reduces the problem
to verifying a trigonometric identity. He makes a transcription
error in writing down the expression which forms
one side of the target equation but it does not matter
because he makes no attempt
to demonstrate this trigonometric truth. As soon 
as we had received Alexander's script, Adrian and I raced to fill
the gap. 
We quickly found the necessary trigonometric argument;  
it takes three lines, but contains a cute algebraic trick.
This is a shame because it indicates that filling
the hole is not a triviality. We have the trigonometric argument
written out for the co-ordinator. 
The co-ordinator is dismissive of our trigonometric patch, 
but not for the correct reason. He 
points out that we have used cotangents (which he apparently
regards as obscure). This is absurd.  
Our point is not that Alexander could have found this
argument, but merely that in principle his method
was correct and could be made to work. We hoped
for 5/7 but feared 4/7. We are astonished to be offered 2/7.
The breezy way in which Alex had closed the argument is 
really counting against him. It is the end of an exhausting co-ordination,
but Alexander is going to get a bronze medal (we hope).

As usual Gordon Lessells, the Irish Deputy, is the man with the numbers.
and he correctly predicts the cut-offs and we breathe more easily as all
our students have got medals, three of them with no margin of error.
The jury meeting confirms the cut-offs. The UK medal haul is almost as
good as last year, but our position in the unofficial national rankings has
slipped to 20th. Three fewer marks in the wrong places and our performance
would have looked rather sour (a silver and four bronzes). We were very
lucky this time.

\noindent \textbf{July 16:\ } The enthusiasts went on an excursion today,
but Adrian and I were exhausted and spent the day sleeping. In the evening we
went over to the students' hotel and joined a party which went
for a meal in a taverna beneath the Acropolis. This involved tangling
with the Athens metro. This has undergone the same gentrification 
as the airport. The Swiss team were kind enough to show us the way,
and we had a fine time. Our student Giles is in a very good mood having
done so well, and seems keen to improve Britano-Helvetian relations.
He threatens to lead an insurrection when Adrian and I call time.
He is a fine fellow and quickly falls back into line. 

\noindent \textbf{July 17:\ } Today we had a fine closing ceremony. 
By now the European Football
is starting to fade, and the forthcoming sporting Olympic Games are
starting to dominate the politicians' thoughts.
In the evening we have a magnificent alfresco banquet at a 
golf club by the sea.
We are entertained by Greek dancers. Inspired by this,
hundreds of students take part 
in Greek dancing, and the Dutch leader seems to lose about 30 years.
Rumours circulate that the Advisory Board Chairman is about to take the stage, 
but in fact he comes over to see me to discuss future training arrangements.
He has a constructive suggestion as to how to improve morale.

\noindent \textbf{July 18: \ } We spent the day at Athens airport,  because
Olympic Airlines rescheduled our flight. This is annoying but is not enough to allow 
Olympic to join my all time list of truly awful airlines
(Austrian Airlines, British Airways and Malev). By the end of the day,
Olympic will have made it though. Our student from Belfast, Martin Orr, is
going to miss his connecting flight. The helpful Olympic Airlines receptionist
at Athens airport assured me after tapping at her screen that (a) 
there was plenty of availability on Heathrow--Belfast flights that evening and
(b) Olympic Airlines in London would arrange for a ticket for Martin on one
of those flights. When we arrived back in London, both of these
statements proved false, and Martin was stuck in London for the night. 
So well done Olympic Airlines, you have made the list.
Detailed explanations of the crimes of the other tatty outfits is available
on application. Don't use them if you can avoid it.

\section*{Acknowledgements}

I would like to thank the Greek organizers of IMO 2004 for
putting on such a splendid event. I must also express warm thanks
to the UK IMO team and reserves for making my job an unalloyed
pleasure. Leaders, Deputies and Observers of the countries
which participated in IMO 2004 made the experience a joy.
I would like especially to thank the Leadership of France, Ireland
and Luxembourg for showing me how to use the Euro. I have retained
various denominations as I expect my grandchildren to be
curious about what will (by then) presumably be an extinct currency.

The UK IMO effort is the work of so many people that it is not 
possible to name them all. There are all the students who participate in the
various mathematics challenges and olympiad competitions, and their teachers.
There is the United Kingdom Mathematics Trust, its Council, its 
Executive Director Angela Gould and
the staff of the Leeds Office. There is the British Mathematical Olympiad
Subtrust under the redoubtable chairmanship
of Adam McBride. There are the markers for all the competitions. There is now
a very large national mentoring scheme to help bring on strong students
of various ages under the direction of the previous Deputy Leader, Richard
Atkins. There are the students of the UK squad who have made the various
training camps such fun, and the small army of trainers who have helped
at camps, including Jeremy King, Patricia King and Vin de Silva in Greece. 

Our continuing co-operation with the Hungarian squad is of
considerable benefit to both countries, and I thank Sandor Dobos
and Cecilia Kulcsar for making all this possible. Mircea Becheanu
of Romania was our guest trainer at Trinity College, Cambridge 
this Easter, and we thank him for his help.

I must single out Christopher Bradley for his geometry coaching. I have
to keep a wary eye on my deputy
Adrian Sanders who does his job and mine much of the time. 
The ex-olympians have helped
in diverse ways, including Joseph Myers's web expertise. As I write this,
UKMT's founding Chair of Council has finally been allowed a graceful 
retirement. I am sure that we all wish Peter Neumann a complete
restoration of his good health. His successor Bernard
Silverman FRS (an IMO Gold Medallist) has a hard act to follow.

Finally it is appropriate to thank our sponsors: the Department for Education
and Skills, the United Kingdom Mathematics Trust, Trinity College Cambridge,
the microelectronics company
ARM and the publishing house Springer. 
\end{document} 
