\documentclass[11pt]{article}
\pagestyle{empty}
\usepackage{amssymb}
\usepackage{latexsym}
 
\title{UK Leader's Report on the 44th IMO, Tokyo}
\author{Dr Geoff Smith, {\tt http://www.bath.ac.uk/$\sim$masgcs}\\
University of Bath}
\date{July 2003}
\begin{document}
\maketitle

\section*{Introduction}

The International Mathematical Olympiad is an annual competition which 
takes place in a major city each July. Six students from each country 
are allowed to take part in what amounts to the World Championships of 
Secondary School Mathematics. Students are disbarred from the 
competition either by entering full-time tertiary education, or reaching 
the age of 20. At most half the students receive medals, and these are
awarded in the best possible approximation to the ratio 
gold:silver:bronze = 1:2:3.

There are two examination papers sat on consecutive days. Each paper 
lasts 4 hours 30 minutes, and consists of three questions. The first
is hard, the second is unbelievably hard, and the third is more or less 
impossible. Each question is marked out of 7, according to a brutal 
marking scheme. The least imperfection will lose a mark, and an 
incomplete solution will usually be rewarded with no more than 2 marks, 
even for significant progress. This is an event where every mark has to 
be earned.

As well as the six students, teams usually send along at least two 
adults. One is the leader, and his or her role at the IMO is to sit on 
the jury. There is also a deputy leader who stays with the team and 
looks after them. In the UK we are very fortunate in that the current 
deputy leader is Richard Atkins of Oundle School, the director of the 
national mentoring scheme. Richard also helps with training. This year 
he was accompanied by Adrian Sanders of Trinity College, Cambridge, who 
will succeed him as deputy leader.

The UK team is selected from a squad of students who are preparing for 
the event. Students are invited to join the squad by virtue of excellent 
performances in national mathematics competitions, normally BMO1 and 
BMO2. The training schedule is designed to have minimal impact on normal 
school work. 

\section*{The Results of IMO 2003}

The questions chosen by the jury for the 44th IMO were as follows.

\begin{enumerate}
\item Let $A$ be a subset of the set $S = \{1,2,\ldots,1000000\}$ 
containing exactly 101 elements. Prove that there exist numbers
$t_1, t_2, \ldots, t_{100}$ in $S$ such that the sets 
\[ A_j = \{ x + t_j \mid x \in A \} \quad \mbox{ for } \quad j = 1,2, \ldots, 100\]
are pairwise disjoint.

\item Determine all pairs of positive integers $(a,b)$ such that
\[ \frac{a^2}{2ab^2 - b^3 + 1}\]
is a positive integer.

\item A convex hexagon is given in which any two opposite sides have the 
following property: the distance between their midpoints is $\sqrt 3/2$ 
times the sum of their lengths. Prove that all the angles of the hexagon are equal.

(A convex hexagon $ABCDEF$ has three pairs of opposite sides: $AB$ and 
$DE$, $BC$ and $EF$, $CD$ and $FA$.)

\item Let $ABCD$ be a cyclic quadrilateral. Let $P, Q$ and $R$ be the feet
of the perpendiculars from $D$ to the lines $BC$, $CA$ and $AB$ respectively.
Show that $PQ = QR$ if and only if the bisectors of $\angle ABC$ and $\angle
ADC$ meet on $AC$.

\item Let $n$ be a positive integer and $x_1, x_2, \ldots, x_n$ be real
numbers with $x_1 \leq x_2 \leq \cdots \leq x_n$. 
\begin{enumerate}
\item[(a)] Prove that
\[ \left( \sum_{i=1}^n \sum_{j=1}^n |x_i - x_j|\right)^2 \leq
\frac{2(n^2-1)}{3} \sum_{i=1}^n \sum_{j=1}^n  (x_i - x_j)^2.\]
\item[(b)] Show that equality holds if and only if $x_1, x_2, \ldots, x_n$
is an arithmetic sequence.
\end{enumerate}
\item Let $p$ be a prime number. Prove that there exists a prime number
$q$ such that for every integer $n$, the number $n^p -p$ is not divisible
by $q$.
\end{enumerate}
If you manage to solve any of them, especially the tough question 3 or 
6, please send your solution to me (address below).

The breakdown of the UK marks is as follows:
\begin{verbatim}
                  Q1  Q2  Q3  Q4  Q5  Q6      Total

Nathan Bowler      7   3   0   5   1   1       17
David Fidler       7   3   0   7   0   0       17
Jenny Gardner      7   7   0   7   7   0       28 
Paul Jefferys      7   7   0   7   7   1       29
Gavin Johnstone    7   3   0   7   1   0       18
Martin Orr         7   3   0   7   1   1       19
\end{verbatim}

As a result of these marks, the team were all awarded medals.

\begin{verbatim}
Paul Jefferys       29 Gold   (Berkhamsted Collegiate School)
Jenny Gardner       28 Silver (Tiffin Girls School)
Martin Orr          19 Silver (Methodist College, Belfast)
Gavin Johnstone     18 Bronze (Dame Alice Owen's School)
Nathan Bowler       17 Bronze (Knutsford High School)
David Fidler        17 Bronze (Haberdashers' Aske's School)
\end{verbatim}
The cut-offs were 13, 19 and 29.

Although the IMO is an individual competition, just as in the other 
Olympics, it is an unofficial habit to rank the performances of nations
using total team scores. On this basis the United Kingdom came in equal 
10th place, recovering from 27th in 2002 and 31st in 2001. Here are the 
marks of the 44th IMO of those countries scoring 90 marks or more. 

1 Bulgaria 227, 2 China 211, 3 USA 188, 4 Vietnam 172, 5 Russia 167, 6 
Korea 157, 7 Romania 143, 8 Turkey 133, 9 Japan 131, 10= Hungary, United 
Kingdom 128, 12= Canada, Kazakhstan 119, 14 Ukraine 118, 15 India 115, 
16 Taiwan 114, 17= Germany, Iran 112, 19= Belarus, Thailand 111, 21 
Israel 103, 22 Poland 102, 23 Serbia \& Montenegro 101, 24 France 95, 25 
Mongolia 93, 26= Australia, Brazil 92, 28= Argentina, Hong Kong 91.

These were the best performances among the 82 nations which 
participated. Observers were also present from Mozambique and Saudi 
Arabia, so those nations may participate in IMO 2004 in Greece.
Those of you who are interested in more IMO statistics may enjoy 
visiting Joseph Myers's site 
{\tt http://www.srcf.ucam.org/$\sim$jsm28/imo-scores/}

Before addressing parochial concerns, it is appropriate to note with 
respect the astonishing performance of Bulgaria. Their lowest mark was 
34, and on the two hardest questions they obtained 11 scores of 6 or 
above. The population of Bulgaria is 7.6 million and its GDP is 1/30th 
that of the United Kingdom and 1/120th that of China (population 1248 
million). Their performance is no flash in the pan; they last slipped 
out of the top 5 in 1997. 

Now for some navel-gazing. The United Kingdom has, at least for one 
year, regained the sort of form it displayed in the early and middle 
1990s. To look on the bright side, the UK managed to get the most marks 
of any country in the European Union or the Commonwealth. We also 
managed to do better than any successor state to the USSR save for 
Russia itself. The tie with Hungary is a particularly happy chance, 
since we train with them every winter and they are therefore our closest 
friends in the competition. 

Every UK student obtained a medal, and the marks of each UK student were 
in the top 30\%. Paul Jefferys obtained our first gold medal since 1997 
(by one mark), having failed to do so last year (by the same margin). 
Jenny Gardner was the 5th placed girl in the world, and missed the gold 
cut by a squeak. She was actually ranked equal 38th. Martin Orr's silver 
medal is the best performance ever by a UK student resident in Northern 
Ireland. No student representing the Republic of Ireland has won a gold 
or silver medal so far, so Martin is top of the all-Ireland list. Our 
three bronze medals were all very strong, and the least bit of good 
fortune could have tipped any of them into the silver zone.

We had three very strong reserves in Bryn Garrod, Alex Davies and Paul 
Smith. Each one of them had every hope of making the team, and moreover 
it is extremely likely that each of them would have obtained a medal if 
he had been selected. Alex is available for selection in 2004.

Note that the foundation of the UK score is 82/84 on the relatively easy 
questions 1 and 4. In fact only China and the United States managed to 
do better on these two problems. On the other hand, our performance on 
questions 3 and 6 was dismal (as it was in 2002). The way forward for 
the team is to try to consolidate our performance on 1 and 4, to try to 
match it on 2 and 5, and to do better on 3 and 6. Note that a perfect 
score on questions 1,2,4 and 5 would alone have produced 6 silver 
medals, and would have beaten Russia into 5th place by one mark. 

The performance in Glasgow 2002 (UK ranking 27th) was better than it 
looked, because we were breathing down the necks of lots of other good 
teams. However, one must worry that the performance in 2003 owes 
something to chance; it is unlikely that all six students will solve (or 
almost solve) all the easy questions every year. Even the mighty 
Bulgarians did not manage it in 2003. 

\section*{The year 2002--2003}

The 43rd IMO in Glasgow in July 2002 put the resources of the United 
Kingdom Mathematics Trust under considerable strain. It is a tribute to 
all concerned in the organization of the event that from the point of 
view of outside observers it seemed a model of planning and efficiency. 
The frantic improvisation and the resort to desperate emergency measures 
(essential for any event on this scale) remained hidden from view. It 
was not a financial disaster, all the kids got home eventually, and one 
or two organizers are still on speaking terms. If future IMO hosts can 
match this, they will be doing well.

While the rest of the British mathematics enrichment community went into 
post-trauma therapy, we had to get the UK preparations for IMO 2003 
under way. We began with a camp in September at the University of Bath. 
This was a gentle event, mixing returning students with newcomers. We 
began our mentoring scheme. There are several levels of this scheme in 
the UK, the one relevant to the IMO being the Advanced Scheme. The 
deputy leader Richard Atkins is the overseer of these schemes, and the 
Advanced Scheme was run by Michael Ching of the Massachusetts Institute 
of Technology. I also send out a couple of mock IMO papers every month 
for the IMO squad to administer to themselves.

At the turn of the year the squad went on tour to Hungary for our annual 
joint camp with Sandor Dobos (the names commute) and the Hungarian 
squad. This is administered by the Bolyai Society. This camp really 
deserves a report all by itself. In the morning we had lectures by 
various Hungarian experts, and in the afternoon the students worked on 
problems and presented their solutions before supper. For the second 
year the camp was held in a rambling boarding school. A minor defect of 
the current arrangement is that some of the Hungarian students who live 
in Budapest tend to slip home in the evenings. By agreement with Sandor 
the next camp will be held in an academic hotel on the Danube in the 
beautiful Hungarian countryside. This will bring the students even 
closer together. We have been approached by the leader of the Luxembourg 
IMO team to ask if he may come and bring one or two students in the 
future. This seems very likely to happen. 

In the Spring the results of our most advanced national mathematics 
competitions (BMO1 and BMO2) came in, and we used the information to
inform our selection of 20 students to attend a camp at Trinity College, 
Cambridge. I rather enjoy the marking week-end for BMO1 which has been 
held at Egham in recent years. 

Hitherto the Easter Trinity College camp has unified, but this time we 
split it into two streams for less and more experienced students (`mint' 
and `used'). We had at least one joint session every day, but otherwise 
the two groups worked most of the time separately. As usual we were 
completely overwhelmed by offers of help, and it is this rich supply of 
talented coaches that gives us the option of holding so many parallel 
sessions. Selecting the final 9 for the intense training period was as 
tough as ever, and we hope that we made the right choices. This is not 
an exact science.

The correspondence course is fairly intense, but sharing the marking 
with Richard and Adrian lightens our load. In all honesty it doesn't 
help the students very much. In late May we had a camp at Oundle School 
near Peterborough, and selected the final team of six and the three 
reserves Alex Davies, Bryn Garrod and Paul Smith. Leaving out these 
three excellent young mathematicians was not an easy decision. The last 
place was completely open right until the Oundle Camp where David Fidler 
displayed flashes of brilliance which secured him a berth. 

We had hoped to have a pre-IMO camp in Hong Kong jointly with the team 
from the special administrative region en route to Tokyo, but 
unfortunately the SARS epidemic had been at its height at the time when 
it was appropriate to book tickets, and following UK government 
recommendations, we had no choice but to cancel that camp. Our students 
were very disappointed, and I would like to take this opportunity to 
thank the Hong Kong mathematics enrichment community for their help and 
understanding.

The motivation for having a final camp in Hong Kong was longitude 
training. We wanted the team not to be suffering from jet-lag during the 
IMO. We therefore sought somewhere in a time-zone close to that of 
Japan, and somewhere fairly hot to mimic a Tokyo summer. A glimpse at 
the map and a suggestion from Tony Gardiner were enough. We headed for 
tropical Queensland. Our hosts were to be James Cook University, which 
has a small campus in Smithfield (good name) in a northern suburb of 
Cairns. The financing of all this was possible because JAL were offering 
very cheap return tickets from London to Australia, with the option of a 
stay in Japan on the return leg. 

To get the flavour of IMO 2003, I will follow the usual practice and 
present a personal IMO diary. Of course this is only one perspective, 
and the students' experience will have been very different.

\section*{Leader's Diary IMO 2003}

\noindent{\bf June 28th}  Seven of us gather at Heathrow; two more (Adrian and David) 
will fly tomorrow. We all have JAL flights, but my party's tickets have 
been re-routed via Paris. The flight there is easy enough, but the 
flight on to Osaka is horrible. Richard and I have piecewise rigid legs 
(obviously it is easier to fold students). We are unable to obtain seats 
with proper legroom, and by the end of the 12 hour flight neither of us 
is a happy bunny. 

\noindent{\bf June 29th} We arrive in Osaka airport Sunday lunchtime. The team is 
mostly fine but we seniors are used and bruised. Happily the airport
is virtually deserted. Many of us have showers in the executive 
pampering zone, but frugality dictates that we spare ourselves the 
vibro-massage chairs. We eat a decent Japanese lunch after pointing at 
plastic models of the meals. It is now clear that the party will divide 
into two schools, those who embrace and those who are suspicious of 
alien cuisine. Gradually we all turn human. Osaka (near Kobe) airport is 
built on an artificial island. Apart from that there is not a lot to be 
said. 

The cabin staff on the flight from Paris were so anxious about Richard 
and myself that they phoned the check-in desk for our Australia bound 
flight to reserve us seats by the emergency exits. We were very 
grateful, but happily the flight south is on a nearly empty jumbo, so 
everyone is able to stretch out and relax.

\noindent{\bf June 30th} We arrive at dawn in Brisbane. There is a minor crisis when 
immigration go sour on my admittedly very beaten-up passport. They are 
squaring up to bounce me from entry when it dawns on them that this will 
cause an international incident. I grovel, and they let me in `just this 
once'. We take a hotel shuttle bus to our hotel, the Royal on the Park. 
We are only going to be in Brisbane overnight, so I have reserved rooms 
in a fairly luxurious hotel. This is a very popular move as the mangled 
remnants of the team need some serious opulence. The hotel let us into 
our rooms well before they need to, and give us a big discount on a 
hearty breakfast. We do quite a lot of sleeping and showering that day. 
In the evening we get up, and stagger 200 yards to an Irish 
pub-cum-restaurant. This is our first encounter with Xcite pheromone wipes. 
These are sold in packets and apparently enable the user to attract 
members of the opposite (or presumably the same) sex. They carry a dour 
warning message `use responsibly'. I decide to refrain from purchasing 
one of these towels lest I unleash forces that I might be powerless to 
control.  

\noindent{\bf July 1st} Richard decamps to the airport to meet David Fidler and Adrian 
Sanders. They all arrive back in time for breakfast. Their journey was 
more comfortable than ours. We offer to book them into the hotel for the 
day, but they heroically decline. We pack for the evening flight north 
to Cairns, and go out to play for the day. We spend the morning on a 
municipal river taxi seeing the riverside of Brisbane. We see southern 
cormorants with white breasts, and Australian pelicans (which are 
readily distinguishable from Hungarian ones). At lunchtime I lead a 
party to Chinatown to eat Lakhsa, the world's best soup. The less 
adventurous seek more prosaic cuisine. In the afternoon we stroll in the 
botanical gardens, then some of us savour extra-large cappucinos called, 
as they should be in the land of the tortured vowel, `muggucinos'.

In the evening all nine of us make for the domestic airport to take a 
Virgin Blue flight to Cairns. It turns out that brother Branson has his 
own terminal in Brisbane, and can toy with hapless passengers.
We hear an announcement `Ladies and gentlemen; boys and girls' it began, 
in a bouncy tone familiar to those who watch commercial television 
channels on Saturday evenings. At first it was a joke, and then it 
became a major irritant. Everything was prefaced by this phrase. Never 
mind the passengers, how do the staff put up with it?

The flight only lasts a few hours, during which time one or two male 
members of the team fall in love with the help. If you want to be there, 
try an Australian setting of ``A Subaltern's Love Song'' by John 
Betjeman.

We arrive in deep midwinter in tropical Queensland. It is dark and 8pm. 
The Student Lodge has promised to meet us with transport. Richard and I 
scour the airport for our bus without success, as cicadas mock our 
plight. Telephone calls to the Student Lodge yield invitations to leave 
messages on their telephone answering machine. We are in trouble. We 
decide to gamble and take a maxi-taxi (with all 9 and our luggage) out 
to the suburbs and the Lodge. If it is deserted we will have to go to 
the centre of Cairns and find a hotel. 

We find reception in deepest murk. We leave Adrian with the team and go 
wandering through the student accommodation. We find a resident tutor 
and explain the situation. She rolls her eyes in embarrassment. We 
deduce that this may have happened before. She quickly rounds up other 
tutors and they prepare rooms for us. It is very late, but a pizza 
parlour is nearby. Richard and Adrian obtain an excellent dinner for the 
team, and so to bed.

\noindent{\bf July 2nd} We awake to consider our position. The accommodation is Spartan 
but acceptable. After breakfast I am introduced to the administrator. It 
takes her some time to realise the extent of her mistake. She formally 
apologizes to me in Queensland fashion; this does not involve the use of 
the word `sorry' but she does acknowledge that `I really stuffed up'. 
Well, I remain calm, polite and friendly. From that moment on our host 
became extremely helpful indeed, and made our stay at the Lodge a very 
happy experience.

David Godwin of the Mathematics Department of James Cook University has 
arranged for us to have use of a beautifully cool room for our daily 
exams. At first we are a little surprised at his apparently eccentric 
room reservation methods, which seem to involve being a personal friend 
of every security guard on campus. Later we have reason to have a minor 
tangle with the official procedures of the university, and realise the 
wisdom of David's technique.

\noindent{\bf July 3rd--6th} We spent this time doing mathematics, save for an 
excursion to the Great Barrier Reef by catamaran. 
Richard prepared a laminated geometry problem sheet which the team could 
work on while snorkelling. There were also trips in a glass-sided 
semi-submersible boat, and I was lucky enough to see some turtles from this 
vessel. Gavin and Martin looked a little the worse for wear after their 
snorkelling sessions, but recover quickly.

Mathematical preparations were proceeding apace, but after a couple of 
papers where the team did very badly, we substituted some easier 
questions on the next paper in an effort to build up their 
self-confidence. I had to fly off to Japan on July 7th, leaving the team in 
the capable hands of Richard, and the deputy-in-waiting Adrian Sanders. 
Adrian is a former IMO team member, and very strong mathematically. He 
will be a great deputy leader, though Richard will be a hard act to 
follow.

\noindent{\bf July 6th} I fly from Cairns to Brisbane, and stay in a hotel near the 
airport. It is a last chance to savour Australian Thai cuisine. 

From now on I am out of touch with the team, but I feel that I should 
pass on the details of an incident which happened after I left. Adrian 
and Richard were conducting an algebra session, and while the students 
were working on the problem sheet, Adrian adopted that inscrutable 
reflective pose which Cambridge people use in an attempt to convey the 
impression of wisdom. Time passed, until Richard realised that Adrian 
was in fact asleep. Richard proceeded to round up the session in 
Adrian's intellectual if not actual absence. Richard is an experienced 
Head of Department, so he is used to this sort of thing.

\noindent{\bf July 7th} The flight north to Tokyo is easy enough. Japanese immigration 
are not fussed about my scruffy documentation, and become very friendly 
when they discover the IMO connection. I reach the arrivals area 
mid-afternoon to be greeted by Tokyo IMO apparatchiks.  The next bus will be 
at 6:30pm in a couple of hours. I see old friends the leaders of New 
Zealand, and Spain, and am introduced to some other leaders. The 
secretary of the IMO Advisory Board, John Webb, has arrived from South 
Africa, but his luggage has been lost in transit. I sympathize and 
recount how Austrian Airlines recently managed to lose my luggage three 
times in two weeks (allowing stuff to be nicked in the process). It 
seems that both Dubai and Vienna are luggage transfer black spots. Be 
warned.  

The 90 minute bus journey across Tokyo in the dark is impressive. There 
are towers, Ferris wheels, overpasses, underpasses, middlepasses and 
giant posters of David Beckham. It goes on and on. Eventually we reach 
the National Olympic Memorial Youth Centre, a complex of student 
accommodation and teaching buildings near Shinjuku in the heart of 
Tokyo. At last I get my hands on the problems shortlist, with no answers 
provided. My room is tiny by Western standards, but air-conditioned and 
I have my own bath (rather like a top-loading washing machine without a 
lid). It is fine. I go to bed to enjoy the problems.

\noindent{\bf July 8th--10th} At breakfast I walk in to a sea of friendly faces. The 
leaders are gathered. It is my duty as UK leader to wage psychological 
war on other participants, so I elect to use chopsticks (I am 
proficient). The conversation is all about the shortlist. Of course we 
haven't had time to try all the questions yet, but even at this stage 
some questions have attracted fan clubs and others are clearly headed 
towards oblivion. I work on the problems all day, and late in the 
evening we get the solutions. There are some good geometry questions, 
but the combinatorics section looks weak, and many of its problems are 
very geometric. The algebra and number theory sections are both quite 
attractive. 

I wonder if any of the UK questions have made the shortlist. As a matter 
of policy I deny myself the opportunity to see the UK proposals in case 
it influences my judgement. 

The next day is spent in pleasant discussion of the merits of questions, 
making no apparent progress towards selecting the examination until a 
critical moment passes and we quickly choose the two harder questions. 
The Japanese chairman Prof Yuji Ito is very good, taking us forward by 
gentle coaxing. One is the geometry problem which will become question 
3. The problems committee classify it as hard but it doesn't seem too 
bad to me. Since the UK will subsequently score 0 on this question, it 
shows how much I know. The other choice will become Question 6. This is 
a number theory question which I reckon is simply impossible. The 
solution seems to be applied magic. There is commentary in the shortlist 
explaining how one might come across the correct argument by a rational 
process, but I am far from convinced. The result is pretty, and it seems 
surprising that such a result can be obtained using IMO methods. It 
seems like a good question to sort out the top of the rankings, so it 
gains my support. 

We begin to focus on the easier questions. I am pretty sure that a lot 
of combinatorics questions which are classified `medium' should actually 
be classified `easy'. I know this because I could do them before the 
solutions were handed out. The jury will not hear of it. I produce and 
distribute a fast solution to C4, as do several other leaders, but the 
jury are not persuaded. I quite like the `easy' number theory problem 
N1. It concerns recurrences, and it is possible to do it in your head. 
The solution makes me laugh out loud. There is a trick which you have to 
see or know, otherwise the problem is completely intractable. Apparently 
this trick has been used in another problems competition, and prior 
knowledge would give a candidate a fantastic advantage. For this reason 
this question does not make the final paper. Also, the distribution of 
marks on this question would have been bimodal. If you see the trick you 
would get 7, without it 0 would be certain.  

We carry on choosing. We select the easier questions. One combinatorics 
and one geometry. The leader of Poland makes a learned submission as to 
why G1 and G2 are actually the same question at a deep level, and that 
in turn are equivalent to a question in a recent Austrian-Polish 
competition. Close analysis of his arguments persuades me (and the jury) 
that anyone who makes these connections deserves a gold medal anyway, so 
he is ignored. We now have a difficulty. We have chosen two geometry 
questions, and most of the likely remaining `medium' combinatorics 
questions are highly geometric. We have to pick a `medium' algebra 
question so we do, an inequality, and a number theory question makes it 
as the second `medium' question. 

When the questions have been chosen the national origins are revealed.
They are (in sequence) from Brazil, Bulgaria, Poland, Finland, Ireland 
and France. The Irish leader and observer are delighted to have a 
question on the paper. It turns out that no UK problem has made the 
shortlist this year. 

As UK leader I chair the English language committee. There is the 
problem of the excessive articulacy of the New Zealand leader. I 
neutralize him by making him the secretary of the meeting. He will be 
kept very busy making drafts and keeping notes, so that more timorous 
leaders can have their say. There is also the problem that most leaders 
think that they can speak perfect English, and some get resentful when 
the native speakers meet behind closed doors. I solve this by announcing 
to the jury that only native speakers may attend, but that the 
definition of native speaker is `anyone who would like to be a native 
speaker of English' (apologies for the infinite regression). Once the 
meeting is open to all, attendance ceases to be attractive, and most of 
the non-native speakers drift away. The leader of Luxembourg, Charles 
Leytem, attends, but he is an honorary native speaker of everything. We 
finish quickly, and then I present our proposals to the full jury. I am 
fatalistically relaxed, because I am expecting our wording to come under 
fierce attack (because the jury is ahead of schedule and so there is 
time to fill). In fact we quickly dispose of the English language with 
very few changes, and then enter into a debate about mathematical 
notation. I realise that I should really hand the chair back to jury 
chairman Prof Yuji Ito, but he looks very unenthusiastic and obviously 
wants me to carry on. 
    
The annual agony over the use of summation notation starts spiralling 
round without coming to any conclusion. The discussions ramble in an 
unstructured way. At last I decide that we have to put a lid on this,
and I suggest to the jury that we should proceed formally with motions 
and votes. There is general agreement. I then start applying the rules 
of order inflexibly, and through a sequence of votes we quickly come to 
a conclusion. The leader of Bosnia and Herzegovina wants the jury to 
agree on a standard answer to questions about summation. I suggest to 
the jury that Bosnia and Herzegovina be put in charge of this and they 
agree. That takes the edge off the appetite for further suggestions, and 
the presentation of the English language committee draws to a 
conclusion.

As I return to my seat the leader of Iran makes a point of order as to 
whether the leader of the United Kingdom should have been allowed to 
conduct votes. The jury chairman quickly blows this problem away by 
asserting that he had delegated the responsibility.

Michael Albert of New Zealand types up the English version with me 
riding shotgun. It is then distributed to the leaders responsible for 
the official IMO languages (French, German, Russian and Spanish). It 
turns out that there is a problem. No standard template has been defined 
and everyone is making up their own. It is a nightmare. I meet with the 
leader of Spain and agree a format. We tell the leaders of France, 
Germany and Russia. Everyone agrees but France ignores the agreement 
(plus \c{c}a change). In the interests of world harmony we all swap to the 
French notation (it involves the use of bold type in a certain place). 
The official versions are then duplicated and the translations into all 
the other languages can begin.

\noindent{\bf July 11th} The students are arriving today. The leaders must be taken 
away for security reasons. We are allowed to leave belongings in our 
rooms at the Centre, which is very convenient. We have a two hour drive 
across town and round the bay to the Makuhari hotel. This pleasure 
palace is 25 floors high and fairly swish. I have a room on the 10th 
floor which is far from large by Western standards, but by comparison 
with the previous accommodation it seems huge. We have meal vouchers, 
which is a good idea because the restaurant prices are high.

Finally there is a presentation by the co-ordinators (the marking 
police) of what they plan to use as a marking scheme. Mostly it is fine 
but in a couple of places there are major problems. It is a political 
matter to persuade the co-ordinators that they are wrong, and to allow 
them to make changes without loss of face. This is accomplished without 
too much blood on the carpet, with the chair of the IMO Advisory Board 
and Hungarian leader J\'ozsef Pelik\'an playing the tough cop to Michael 
Albert's soft cop.  

\noindent{\bf July 12th}  We are taken back to the Youth Centre by bus in order to attend 
the opening ceremony which is mercifully short. Together with the other 
leaders I am required to enter the theatre via the stage, passing in 
front of giant drums being frenetically beaten by Japanese dervishes. 
The UK team look great. They are in their uniform and look menacingly 
professional. I hope this has an unsettling effect on some other teams. 
The leaders are taken back to the Makuhari hotel without contact with 
the teams. The final marking scheme is agreed. It is now significantly 
improved. The co-ordinators are doing a great job. I make a speech 
thanking them for their reasonableness in a pathetic attempt to curry 
favour. 

\noindent{\bf July 13th} The first exam. The leaders meet at 9am in a room near a bank of 
faxes. Students are allowed to ask their leader questions during the 
first thirty minutes. These are transmitted by fax to the leaders' site. 
Each leader presents their question to the jury, and proposes a reply to 
be sent back by fax. The first question arrives. My heart sinks. It is 
from a United Kingdom student. Paul Jefferys wants to know if he can 
assume the truth of the Axiom of Choice. I don't want to cramp his 
style, so I propose the answer `yes'. Paul's question causes general 
hilarity in the jury room. The leader of France, Claude Deschamps, 
rushes up to me in delight ``so the UK has adopted the conventions of 
Bourbaki!''  A happy time is spent addressing more serious questions, 
most of which concern the meaning of the phrase `pairwise disjoint'.

In the evening there is an excellent banquet, and then the scripts 
arrive. This is a telling moment. I read the scripts carefully, and to 
my delight I can't find anything wrong with any answers to Question 1, 
and both Jenny and Paul seemed to have done Question 2. The solutions to 
Question 3 are not very impressive. Paul has some calculations and Jenny 
has some ideas.

\noindent{\bf July 14th} The jury meets at 9am for the questions associated with the 
second exam. I brace myself for another Paul Jefferys special, but it 
doesn't come. Instead we have a predictable time answering questions 
about summation notation. The final question of the morning is from 
Gavin Johnstone. It has two parts. Firstly he asks a nervous question 
about the meaning of `angle bisector' (can the bisector be of an 
external angle), and secondly he points out that his exam started late, 
so he wants to know if lunch will still be available when he comes out. 
By now the jury is demob happy, and some clown suggests that I write 
`read the question again' in response to his final query. I point out 
that `have your breakfast again' would be more helpful.    

The leaders then depart from the hotel, and race across town to the exam 
site. As the students finish their second paper we wait for them to come 
out. All our students are claiming to have solved Question 4, and both 
Jenny and Paul are also staking a claim to 5. If they are right, then 
each of them will be close to a gold medal and the whole team should 
have medals (fingers crossed).

There are some rooms set aside for marking, but we find them excessively 
noisy so the UK seniors set up camp elsewhere. A careful reading of the 
scripts indicates that the students are more or less accurate in their 
claims. Question 1 definitely looks like 42/42 for the UK and Question 4 
looks like 35/35 plus whatever Nathan's script is worth. He has used a 
method so bizarre and baroque that beyond our confidence in him, we are 
not at all sure if he is right. Faced with a problem about a cyclic 
quadrilateral, he begins by considering a non-cyclic quadrilateral, 
inverting with respect to a circle of Apollonius (which he calls a 
circle of Menelaus in order to add colour) and then concludes with an 
argument which views a cyclic quadrilateral as a limit of an infinite  
sequence of non-cyclic quadrilaterals. Each step of the argument is 
justified with a helpful note such as ``by a standard result of 
inversive geometry''. It is a nightmare. Initially Adrian and Richard 
devote themselves solely to this one question in Nathan's script while I 
work on the other 35 solutions. I definitely have the easier job. They 
spend the day trying to understand Nathan's arguments, and get 
progressively more stressed about what happens to directed angles under 
inversion. I discover that all the other Question 4 solutions are fine, 
except possibly for Paul's which is a long and cunning trigonometric 
slog. I push that over to Adrian and Richard, and go hunting for scraps 
among the fragmentary solutions. I get more and more excited about 
Question 2. Unusually up to 3 marks are awarded for noticing various 
specific things. As I dig through the four non-solutions from David, 
Gavin, Nathan and Martin I keep on finding nuggets worth marks. Each one 
of them, tucked away in odd corners of reasoning, has made enough of the 
correct observations to get all 3 marks. I am purring. After last year 
when everything fell to bits I am very nervous, but this time it looks 
as though things are going our way. Provided Nathan's foray into 
inversive fairyland is not complete nonsense, and he picks up at least a 
couple of marks for it, then everyone should get a medal. Question 6 is 
also interesting because a legalistic reading of the marking scheme 
would give us 4 marks distributed across four members of the team. This 
would be absurd, but mathematicians are very literal, so there is hope. 

Sustained reading of Nathan and Paul's solutions to Question 4 is 
clearly a health hazard, so from time to time Adrian and Richard refresh 
themselves by looking at other solutions. We are all cross-checking one 
another, making sure that at least two of us are expert on any given 
page of writing. 

\noindent{\bf July 15th} We begin co-ordination at 9:00am. This is Question 6 and we have 
our legalistic hats firmly on. When we explain our case, the 
co-ordinators laugh (quite rightly). We act serious, and tell them that the 
marking scheme must be adhered to strictly. They laugh again. Right we 
say, if you are not giving us the marks for this, make sure no-one else 
gets them either. Please write this position up on the board (Adrian's 
excellent suggestion). Looking at the board, we see that another notice 
is already there. It says that a certain observation is worth 1 mark. We 
say that this is new and that we have not searched for this observation. 
The co-ordinators have looked, and together we find 3 marks, 1 each for 
Martin, Nathan and Paul. This mark for Paul is crucial, because with 
four relatively accessible questions and two hard ones, the gold cut-off 
may well be 29. The mark for Martin is also vital, because it will later 
tip him over the silver boundary.

Later in the day we co-ordinate Question 4, the `easy' geometry one. 
Four scripts are perfect, and we spend an age justifying Paul's 
trigonometry. In the end they can't find any holes, but they want to 
sleep on it. We agree to meet next day at 5pm to settle it and to 
address Nathan's new age solution. Perhaps it would be easier to follow 
if we all sat under a pyramid. 

In the afternoon we co-ordinate Question 2 on number theory. This is the 
one where I have been burrowing for marks in the rambling pages. The 
co-ordinators begin by giving us a form, asking us what marks we want and 
why. We ask for two 7s and four 3s. First we look at the partial 
answers. On each occasion we are asked to show exactly where the key 
observations were made. I have the list of locations in front of me so 
we quickly gather up all the part marks. Next we look at Jenny's script, 
which is (as is often the case with Jenny) absolutely perfect. She gets 
her 7. Now for Paul. The story behind this is that Paul had only solved 
the problem close to the end of the time, and wrote it up in a frenzy. 
Our position is that the solution is all there, and that even though 
some of the bends are taken at speed, he deserves 7 marks. This takes 
some debating, and Adrian is particularly good at explaining why it was 
worth 7. He was a co-ordinator in 2002 and had seen similar scripts get 
7 then. The Japanese co-ordinators are finally persuaded, and offer 7. 
At this moment Titu Andreescu steps in, the ex-US leader and advisor to 
the jury chair. He tries to persuade any Japanese he meets that he is in 
Tokyo to study Sumo. He suggests Paul's script is only worth 6 with a 
twinkle in his eye (quite a terrifying sight in fact). The problem 
captain over-rules him and Titu backs down smiling (equally disturbing). 
Paul has scraped a 7. 

\noindent{\bf July 16th} The morning co-ordination is Question 5, the inequality. By now 
intelligence reports have reached us that a clever re-arrangement of the 
left hand side is worth 1. Excellent, for that is more free marks. The 
only person to score 0 here is therefore the unlucky David Fidler. As 
usual Jenny's solution is beautifully clean. Paul's answer is a little 
grubby but it is all there. Once again he gets a 7.

The lunchtime co-ordination is of Question 3, the hard geometry. We try 
to sell Paul's first line and Jenny's ideas as being serious progress in 
the problem. The co-ordinators smile and say no, very politely, and we 
get six 0s.

By now the agreed marks of the various teams are starting to appear on 
display, and it is tempting to do furtive calculations. I am delighted 
since it looks as though we will come in the top 15 countries even if 
everything goes against us in the remaining decisions, and our rivals do 
very well. The Irish deputy, Gordon Lessells, is unofficial keeper of 
the data. He is always the first to know where the cut-offs will be. 

I look forward to the afternoon co-ordination of Question 1 because we 
are sure that we have six 7s. I take the lead in co-ordination, and am 
surprised that the co-ordinators have some questions about Paul's 
script. There is a nasty sinking feeling in my stomach as I worry that 
they have found a glitch which we missed. Happily Paul's solution is 
correct, and the reason for the problem is that they have not understood 
the precise import of the word `until'. All of our solutions are 
different, but they are all perfect.

It remains to have the final session on the scripts of Nathan and Paul 
for the Question 4. One (and possibly more) of the co-ordinators has 
clearly stayed up half the night with these scripts because they now 
know almost as much about them as we do. We deal with Paul's first. We 
deal with a couple of easy points and then they throw us a curve ball. 
Surely he has only dealt with one configuration when two are possible? 
I have a speech ready about this, but Adrian nips in with ``yes, but he 
deals with directed lengths as he explains on page 19''. The 
co-ordinators think, and finally give in. I don't have to use my prepared 
tantrum about how when students use a more geometric approach we know 
that the co-ordinators have let through solutions which do not cover all 
possible configurations. ``Why should the United Kingdom be singled out 
for punishment when other countries have got away with it? Either give 
us 7 or mark the question again for everyone!''. Anyway, once again Paul 
squeaks home with a 7.

As for Nathan's script, I am hoping that if we address enough of their 
questions the co-ordinators will give in and deliver a 7. Unfortunately the 
co-ordinators are too good, and have found the two weak spots in his 
argument. We put our hands up, and ask for 6. They smile and offer 5.
Adrian has long since spread his fingers under the table forecasting 5.
We put up resistance, but they are firm. We take the 5, and our 
co-ordination is complete. 

We rush back to the room where information is being displayed. Gordon 
Lessells pronounces that the gold cut-off will be 29. Paul is incredibly 
lucky to have crawled over the line, and Jenny
is very unlucky to have fallen just short with four immaculate 
solutions. I think we have 1 gold, 1 silver and 4 bronzes. Our score of 
128 marks might put us in about 12th place.

As the results come in, our rivals falter, and in the end we achieve 
10th place equal with Hungary. I go to supper, but Richard runs in to 
say that the silver cut-off is 19. I don't twig at first, but it means 
that Martin's medal is silver not bronze.

Before the final jury meeting the students come back from a trip, and we 
break the news. Everyone is happy, and even Jenny shows no sign of 
disappointment. Martin does not seem to believe that he got silver. The 
leader of the USA, Zuming Feng, is very keen that Jenny should get a 
gold medal, and starts lobbying to move the gold boundary down to 28. 
However, the rules are clear and this is quite impossible. I am touched 
by his thoughtfulness.

The jury meeting is delayed because Taiwan and Turkey are still arguing 
the toss. Finally they agree their scores and in the end the boundaries 
are as predicted. The game is over for another year. 

\noindent{\bf July 17th} Today the students go to Disneyland, and the leaders go to a 
temple by bus. I know about buses and the IMO, so I stay in my room and 
read a PhD thesis to recover from adrenalin poisoning. A sequence of 
extremely hot baths is just the ticket.  

\noindent {\bf July 18th} The day of the closing ceremony. We all wear uniforms, and the 
students have Union Jacks to display if they wish. Martin declines to be 
seen with the flag (he is in the Irish team for the Informatics 
Olympiad, and walks a careful diplomatic path). The ceremony takes place 
in the presence of the Crown Prince of Japan. The audience is instructed 
to remain in its seats. This is a disaster from a photographic point of 
view, so all three UK seniors get up and walk to the front (very 
disobedient). The ceremony is fine save for some moments of confusion. 
The students getting bronze and silver medals are required to stand 
behind the people presenting the medals, so seeing them is hard. 

The gold medallists go up about three or four at a time to receive 
medals from the Japanese answer to Charles Clarke and the Chief 
Executive of Fujitsu. There is much milling about as all the gold 
medallists gather on stage, but Paul manages to get to the front of the 
pack and switches on the smile. 

The Crown Prince leaves, and we have to be penned up while he does this, 
so we go into daytime TV mode. The presenters walk among the audience 
and ask them questions of limited interest. Each country has a local 
guide, and they all walked on stage to be clapped. Our guide, Junko is 
there. She is very skilled at booking hotels and making mobile telephone 
calls. Her English is very good.   

After the ceremony we all mill around outside. Most members of the UK 
team wear our flag as shawls, and it looks very attractive. Japanese 
television takes a great interest in us, and Paul is interviewed 
concerning his opinion of the quality of Japanese girls. He gives the 
correct answers. Martin is photographed surrounded by the Irish team, 
and there is a very happy mood. To avoid standing around in uniform 
Junko leads us across town by metro to the Centre. We change into 
relaxed clothes for the banquet. The seniors are separated from the 
teams at the banquet which we don't like at all.

After the banquet we are led by Junko to a karaoke bar. We have our 
private room for the UK delegation, Junko and the Swedish guide who has 
joined us for no apparent reason but she is very welcome. Beer is 
brought for the seniors. It turns out that Adrian croons a mean ballad. 
I discharge a long-held ambition and become Grace Slick for 3 minutes. 
Whether it is Jefferson Airplane or Starship I cannot tell, but the team 
suffer my rendition of ``Somebody to Love''. Most people chip in, but we 
can't compete with the duet from Paul and Jenny. Paul has acquired a 
sash from the banquet which reads ``United Kingdom''. Jenny wears this 
in beauty queen style, diagonally. They sing a perfect rendition of 
Aqua's classic ``Barbie Girl''. The original is available at 
{\tt http://user.itl.net/$\sim$bluecdr/barbiegirl.htm}

\begin{verbatim}
Barbie Girl

Hi Ken
Do you wanna go for a ride?
Sure Ken.
Jump in.

I'm a barbie girl, in a barbie world
Life in plastic, it's fantastic.
You can brush my hair, undress me everywhere.
Imagination, life is your creation.
Come on Barbie, let's go party!

CHORUS

I'm a blond single girl, in a fantasy world,
Dress me up, make it tight, I'm your dolly.
You're my doll, rock'n'roll, feel the glamour and pain,
Kiss me here, touch me there, hanky panky.
You can touch, you can play, if you say: "I'm always yours"

CHORUS

Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!

Make me walk, make me talk, do whatever you please,
I can act like a star, I can beg on my knees.
Come jump in, be my friend, let us do it again,
Hit the town, fool around, let's go party
You can touch, you can play, if you say: "I'm always yours"
You can touch, you can play, if you say: "I'm always yours"

Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!

CHORUS

CHORUS

Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!
Come on Barbie, let's go party!

Oh, I'm having so much fun!
Well Barbie, we are just getting started.
Oh, I love you Ken.

[Oh, I love you Ken!]
\end{verbatim}
 
\noindent{\bf July 19th} We all depart for the airport and say farewell to Adrian and 
David. The rest of us kill a day in a hotel at the edge of the airport,
with an interesting adventure in the evening involving an endless bus 
trip (I think the driver must be Kafka) and a meal in a restaurant where 
no-one speaks any English at all.

\noindent{\bf July 20th} The flight home is grim of course, but we are buoyed up by 
thoughts of home. There is a greeting party at the airport, and suddenly 
it is all over for another year.

\section*{Thank you to \ldots}

Everyone. To all the young people who came to our camps to put pressure 
on the team to get better. All the students who sat mathematics 
competitions in the UK, and their teachers, and the people who invent, 
administer and mark the competitions. To Brian Wilson and Christine 
Farmer for great backroom work. To Adam McBride and friends at the BMOS 
for unswerving encouragement. To Peter Neumann, Angela Gould and all at 
UKMT, DfES and ARM for funding and much else. Springer for books. Oundle 
School, Trinity College Cambridge and the University of Bath for 
donating facilities. Christopher Bradley for sessions and wonderful 
training materials. Our Hungarian friends especially the deputy leader 
Sandor Dobos and Cili Kulcsar of the Bolyai Society. Ceri Fiddes for 
looking after the shadow IMO team back in the UK and much else. The 
small army of volunteers prepared to lay on coaching sessions in camps 
all over the country and not otherwise mentioned in this list, including 
Thomas Barnet-Lamb, Robin Bhattacharyya, Matt Fayers, Mary Teresa Fyfe, 
Tony Gardiner, Tim Gowers, Ben Green, Imre Leader, Paul Russell, John 
Silvester and John Slater. 
David Godwin and James Cook University for warm hospitality. 
Junko our guide and all the organizers and helpers at Tokyo IMO 2003. 
All the other leaders and deputies who make IMO such a pleasant 
experience, especially the Irish leader Donal Hurley and observer James 
Cruickshank who had to endure merciless teasing about Martin Orr. Adrian 
Sanders and Richard Atkins for the fantastic job they did on tour. 
Finally the UK team itself, which did such a great job building on the 
firm foundations laid down by its predecessors. I also compliment the 
complement of these people, and assure you that I will never complement 
these compliments. 
\vskip 0.5cm
Geoff
\vskip 0.5cm
For more information contact the UK IMO team leader,
Dr Geoff Smith, Department of Mathematics, University of Bath, Claverton 
Down, Bath BA2 7AY. He can also
be reached at \tt{G.C.Smith@bath.ac.uk}
\end{document}
