\documentclass[12pt]{article}
\title{The 46th International Mathematical Olympiad in Mexico}
\date{}
\author{Geoff Smith, UNK7}
\begin{document}
\maketitle

The preparations culminating in the 46th
International Mathematical Olympiad in M\'erida,
Mexico, reached their climax in July 2005. The IMO is an annual
competition between the most able young mathematicians
in the world. The host nation varies from year to year,
and this time the event was held on the Yucat\'an peninsula,
in the area inhabited by the Mayan people.  

The actual competition consists of two papers, each containing
three very hard questions drawn from algebra, combinatorics,
geometry and number theory. Each question is marked out of 
7, according to an agreed marking scheme. There are no cheap
marks at an IMO. Half the competitors receive medals, and these
are awarded in the ratio gold:silver:bronze $= 1:2:3.$ This
year 93 nations tried to compete, but a couple were unable to 
secure visas so in the event there were 91 nations present. 
Each country may send up to six students, and most do send 
the maximum number. Like the athletic olympic games, the competition
is between individuals, not countries, but inevitably nations
compete unofficially by comparing the sum of the marks obtained 
by their students. Populations of countries, educational 
standards and levels of preparation vary widely between nations.
Some countries have specialist IMO schools which take talented
young mathematicians out of the normal school system at a young
age, and others have training regimes which involve weekly
meeting of the most able students with their trainers.

In the UK we do not allow IMO preparations to interfere with
normal school life, and almost all our events take place 
during school vacations. Our training is amateur, and is conducted
by a small army of volunteers, many of whom were once IMO
competitors themselves. This year a quite exceptional role was
played by the deputy leader, Adrian Sanders. In his final year
in post, he made extraordinary efforts to nurture the mathematical
development of the team, and I am sure that they are grateful
to him. This is not to ignore the efforts of everyone else involved.
Next year Adrian will be replaced by Ceri Fiddes of Millfield School;
she has been shadowing Adrian's role for some time.
 
When trying to measure UK performance at the IMO we
tend to compare ourselves with the other large 
social democracies of Western Europe since this is, broadly speaking, a fair fight.
The annual friendly tussle between Germany and the UK  
adds spice to the competition for both teams. Such rivalries happen at all levels;
the Scandinavian nations compete among themselves, as do the Latin
American countries for example.

As usual the year began with 
a training camp at Bath for students new to IMO preparation.
Two participants of that event, Saul Glasman and Jack Shotton,
were to go on to figure in our IMO team in Mexico. 
During the year our IMO squad of 20 were fed the usual diet of problem sheets
and practice examinations. The next camp was held in Tata, Hungary,
together with Hungarian students over the new year. I think the high point
of that camp was when the Hungarian deputy, Sandor Dobos, illustrated
his lecture on projective geometry by picking up the overhead 
projector, and varying the shape of a triangle by shining the image
at the ceiling and onto the walls. Do not try this at home. 

The camp at Trinity College Cambridge at Easter was also a success, 
and Sandor
flew in as a guest trainer. As always we were blessed with a 
dazzling array of training talent. David Monk returned after a break
of a few years, and in harness with Christopher Bradley, set about
giving our students a serious taste for geometry. A large number of
former team members gave sessions, along with former team leaders
Adam McBride and Tony Gardiner. Following two rounds of the 
First Selection Test, we designated a squad of 8 for the final
push. At the end of May we met at Oundle school for a camp
and more selection tests. Eventually the 6 students of the UK team were
selected. They were
Saul Glasman, Latymer School, London;
Nathan Kettle, Hitchin Boys School, Hertfordshire; 
Andre Kueh, Bromsgrove School, Worcestershire;
Matthew Lee, Robert Smyth School, Market Harborough, Leicestershire;
Martin Orr, Methodist College, Belfast and finally
Jack Shotton, Portsmouth Grammar School.
The two reserves were Tom Eccles 
of St Paul's School, London and Alex Smith
(for the second year running) of King Edward VIth Five Ways, Birmingham.
Here are the questions of the 46th IMO:

\newpage
\centerline{\Large Day 1}
\vskip 1cm
\noindent {\bf Problem 1} Six points are chosen on the sides of an equilateral
triangle $ABC$: $A_1, A_2$ on $BC$; $B_1, B_2$ on $CA$; $C_1, C_2$ on
$AB$. These points are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$
with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and
$C_1A_2$ are concurrent.
\vskip 0.5 cm
                                                                                                                             
\noindent {\bf Problem 2} Let $a_1, a_2, \ldots$ be a sequence of integers
with infinitely many positive terms and infinitely many negative terms.
Suppose that for each positive integer $n$, the numbers
$a_1, a_2, \ldots, a_n$ leave different remainders on division by $n$.
Prove that each integer occurs exactly once in the sequence.
                                                                                                                             
\vskip 0.5 cm
                                                                                                                             
\noindent {\bf Problem 3} Let $x, y$ and $z$ be positive real numbers
such that $xyz \geq 1$. Prove that
\[
\frac{x^5 - x^2}
{x^5 + y^2 + z^2}
+
\frac{y^5 - y^2}
{y^5 + z^2 + x^2}
+
\frac{z^5 - z^2}
{z^5 + x^2 + y^2}
\geq 0.\]

\vskip 0.5cm                                                                                                                              
\centerline{\Large Day 2}

\vskip 1cm

\noindent {\bf Problem 4}
Consider the sequence $a_1, a_2, \ldots$ defined by
\[a_n = 2^n + 3^n + 6^n -1\ \ (n=1,2,\ldots).\]
Determine all positive integers that are relatively prime
to every term of the sequence.
\vskip 0.5 cm
                                                                                                                             
\noindent {\bf Problem 5}  Let $ABCD$ be a given convex quadrilateral
with sides $BC$ and $AD$ equal in length and not parallel.
Let $E$ and $F$ be
interior points of the sides $BC$ and $AD$ respectively
such that $BE=DF$.
The lines
$AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the
lines $EF$ and $AC$ meet at $R$.
Consider all the triangles $PQR$ as $E$ and $F$ vary.
Show that the circumcircles of these triangles have a common point
other than $P$.
\vskip 0.5 cm
                                                                                                                             
\noindent {\bf Problem 6} In a mathematical competition 6 problems
were posed to the contestants. Each pair of problems was solved
by more than $\frac 25$ of the contestants. Nobody solved
all 6 problems. Show that there were at least 2 contestants
who each solved exactly 5 problems.

\newpage

On each day, the students had 4 hours 30 minutes to address the problems.
Here are the results of the UK team. The cut-offs for bronze, silver
and gold medals were 12, 23 and 35 points respectively. They obtained
one gold, three silver and two bronze medals, our best medal harvest since 1996,
according to the 
following table (we keep our traditional UNK codes, though the local
organizers used GBR; the hunks from UNK will not have this):

\[
\begin{array}{lcccccccl}
\mbox{ } & P1 & P2 & P3 & P4 & P5 & P6 & \Sigma & \mbox{Medal}\\
\mbox{UNK1 Saul Glasman} & 2 & 7 & 0 & 7 & 3 & 1 & 20 & \mbox{Bronze}\\
\mbox{UNK2 Nathan Kettle}  & 7 & 7 & 7 & 7 & 3 & 2 & 33 & \mbox{Silver}\\
\mbox{UNK3 Andre Kueh}    & 3 & 7 & 7 & 7 & 1 & 7 & 32 & \mbox{Silver}\\
\mbox{UNK4 Matthew Lee}     & 1 & 7 & 0 & 7 & 7 & 1 & 23 & \mbox{Silver}\\
\mbox{UNK5 Martin Orr}     & 7 & 7 & 6 & 7 & 7 & 1 & 35 & \mbox{Gold}\\
\mbox{UNK6 Jack Shotton} & 2 & 7 & 0 & 7 & 0 & 0 & 16 & \mbox{Bronze}\\
\mbox{ } & 22 & 42 & 20 & 42 & 21 & 12 & 159 & \mbox{1G, 3S, 2B}
\end{array}
\] 

Collectively the team scored 159 points giving
the UK an unofficial ranking of 13/91. This year
the UK retained its 2004 ranking of 3rd in the EU, after 
Hungary and Germany (passing Poland but passed by Germany). 
We improved our position
in the Commonwealth from 3rd in 2004 to 1st place in 2005.
Nations performing very well this year include Italy (120), which
unusually managed to finish ahead of France (83), and Denmark (69) 
which took the Scandinavian championship. Peru (104) have made dramatic
strides this year, and almost snatched the Latin American championship
from Colombia (105). Luxembourg was the leading Grand Duchy and the UK topped
the table of those nations which include a conjunction in their full title.
Hungary get the landlocked crown, and the leading country whose rank (54th)
exceeds its score (50) is Macedonia. Japan (8th) is the leading monarchy.
The dominant nation with a prime score is Romania (191), and the table-topper
among the nations with score which is not
{\em quadratfrei } is Russia $(212 = 2^2 \cdot 53)$.

Here is the unofficial table of the top 30 countries (out of 91) in 2005.

\[\begin{array}{llrllrllr}
1.&  \mbox{China} & 235 & \ \ 11. & \mbox{Bulgaria} & 173 & \ \ 21. & \mbox{Moldova} & 130\\
2.&  \mbox{USA} & 213 & \ \ 12. & \mbox{Germany} & 163 & \ \ 21. & \mbox{Turkey} & 130\\
3.&  \mbox{Russia} & 212 & \ \ 13. & \mbox{United Kingdom} & 159 & \ \ 23. & \mbox{Thailand} & 128\\
4.&  \mbox{Iran} & 201 & \ \ 14. & \mbox{Singapore} & 145 & \ \ 24. & \mbox{Italy} & 120\\
5.&  \mbox{Korea} & 200 & \ \ 15. & \mbox{Vietnam} & 143 & \ \ 25. & \mbox{Australia} & 117\\
6.&  \mbox{Romania} & 191 & \ \ 16. & \mbox{Czech Republic} & 139 & \ \ 26. & \mbox{Kazakhstan} & 112\\
7.&  \mbox{Taiwan} & 190 & \ \ 17. & \mbox{Hong Kong} & 138 & \ \ 27. & \mbox{Columbia} & 105\\
8.&  \mbox{Japan} & 188 & \ \ 18. & \mbox{Belarus} & 136 & \ \ 27. & \mbox{Poland} & 105\\
9.&  \mbox{Hungary} & 181 & \ \ 19. & \mbox{Canada} & 132 & \ \ 29. & \mbox{Peru} & 104\\
9.&  \mbox{Ukraine} & 181 & \ \ 20. & \mbox{Slovakia} & 131 & \ \ 30. & \mbox{Israel} & 99+
\end{array}\]
The rankings of selected other countries are France 32nd, India 36th, New Zealand 38th, Ireland 51st, Spain 58th
and South Africa 62nd. 
One question on the Hebrew paper was not printed correctly. This impeded two students
and a small indefinite quantity 
should be added to the Israeli score, so their true position is given by a probability distribution.

The quality of hospitality and organization delivered by our Mexican hosts was
quite exceptional. Throughout the IMO, hurricane Emily bored across the 
Caribbean toward us, and hit the Yucat\'an peninsula on the very day that the 
closing ceremony was scheduled. Shelters were constructed in windowless
function rooms, and events were rescheduled and relocated at very short notice,
but this all happened with remarkable smoothness. Congratulations Mexico, and 
many thanks.

\section*{Leader's Diary 2005}

\noindent{\bf June 23} Ceri Fiddes, UK Observer for IMO 2005 and 
deputy UK leader designate for
2006 has been thrown from and subsequently under a horse. She has a broken 
leg and is ruled out of IMO 2005 by her doctors. This seems an over-reaction.
All she had to do was to say that she didn't want to come.

\noindent{\bf June 24} We have passport problems. I had decided to upgrade my battered
and mangled document for a neat new one designed 
to please the most discriminating
US immigration official. I applied in good time at the start
of June but an unexpected tangle in the process (don't trust the Post Office checking service)
has me starting to get nervous.
However, we now discover that Andre Kueh's passport is not machine readable.
In order to pass through the USA he will need either a visa or an upgraded
passport. We have a week to sort this out.

\noindent{\bf June 28} Joy. My shiny new smooth unbattered passport arrives, and 
Adrian has an appointment on 
June 30th in the afternoon with a view to getting Andre a new passport. 
This will involve
Adrian first driving to Andre's school to pick up his application form, then
to Bath to get my signature, and finally to London to get the new passport
using the only passport interview slot available at any passport office in
the United Kingdom this week.

\noindent{\bf July 1} Adrian picks up Andre's passport with less than 24 hours to spare.

\noindent{\bf July 2-6} The team gather at 10am at Gatwick Airport. The Shotton family
make a surprise donation of a team mascot, an orange cuddly toy in the form
of large Newt who answers to the name Newton. Our Continental flight was on time, and 
we arrived at George Bush International in good spirits and ready to face
the cheerless footsoldiers of the US Department of Homeland 
Security. Their main tactic is to ask you to prove that you plan to leave again.
In the age of ticketless travel this is a good wheeze for inflicting grief on
tired and confused travellers. The new fingerprint machines are pretty useless;
you have to press down with extraordinary force before Mr Sour is what passes
for happy in this context.

Rice University have sent a bus to meet us. It is blessedly air-conditioned
and fit for purpose. We are whisked to campus, where we meet students who helpfully
show us our rooms, give us keys and so on. 
As expected, there is no bedding, so a small party goes to Wal-Mart to 
purchase 8 sets of their cheapest.

The accommodation is excellent and
we have our own dedicated  and magnificent
cook in the form of Chef Chenko. This fellow goes through
life apparently unaware of A. C. Milan's European footballer of the year. 
He has clearly been briefed to pander to us, and produces an endless stream
of giant meals fit for gridiron footballers. He never ceases to be disappointed at
the meagre indentations we are able to make in his food piles. I contact
Jason in the Housing Department who has arranged things for us. 
One curiosity about this substantial university is that it does
not have a shop which sells things a student might want, such as soap, shampoo
or a toothbrush. You can purchase baseball caps without difficulty, but they
are of limited value in the context of personal hygiene. I put this problem
to Jason, who avers that there is no such shop nearby. He promptly drives 
me to Wal-Mart himself. Nice one Jason.  

The team soon settle into a daily routine of exams, food piles and
problems sessions.

\noindent{\bf July 7} It is Nathan's 18th birthday, and the worst kept secret of the camp is that
we will be watching the 
Houston Astros play the San Diego Padres at the Minute
Maid baseball stadium tonight. There is the problem of how to get there. We ask
various people on campus, who struggle in turn with the idea that we 
don't have cars. Someone helpfully 
suggests that we take the tram into town. I ask
if we could then take a taxi, and receive the reply that this would probably work.
I hit the internet, and drag up a map of downtown Houston. I turn up a 
winning strategy: take the tram into town, then walk half a mile to the stadium.
This works. It takes us past the Houston Court House, and consequently we
walk past endless bail bond emporia. 

We queue up outside the ground, and our bags are searched. It turns out that
it is not allowed to bring consumables into the stadium unless they are 
sold by the sponsors. Saul has his water confiscated. We are amazed when we get
inside. Imagine a premiership stadium with a roof, and then have it air-conditioned
when outside the weather is fit only for salamanders. 
On one wall is a 19th century style
railroad, and a full size train will run up and down the track to celebrate home runs.

Adrian has been giving us fastidious tutorials on all aspects of baseball, but
I will not show off my new found knowledge here. Suffice it to say that
the game is a mixture of rounders and testosterone. The fans are mainly bluecollar,
i.e. American working class. We sit among many very genial Homer Simpsons.
Manners are exemplary, and little kids are everywhere. Earlier today four bombs
had gone off in London. The match begins with the Union Jack on screen, the British national
anthem and a silence. There is much mutual hushing as this crowd quickly brings itself
to orderly attention. There is not a flicker of doubt, as far as this crowd
is concerned, we are in the early 1940s again. 

The match is delightful. It is more like test cricket than I had imagined. 
Instead
of the tedious and repetitive bland dramas of basketball, 
there are long periods of
inaction when the crowd fetch food and drink, punctuated by moments of extreme 
violence as mighty home runs are struck. There are enough ways
to deliver the ball to rival Shane Warne's repertoire.
 
\noindent{\bf July 8} Today I must leave to join the jury.
In these health and safety rich times, it is unthinkable
that Adrian should look after the team on his own. We have an
excellent fix for the absence of Ceri; Thomas Barnet-Lamb flies down
from Boston where he is doing a Ph.D. at Harvard. We pre-position
senior wranglers throughout the world for use in such emergencies.
I pass Thomas without meeting at George Bush International. Thanks.

In the
departure lounge I meet several IMO friends, including Bogdan who used
to be from Romania but now turns out to be the IMO observer from Pakistan.
When the Irish leadership turn up we lock 
verbal horns in the usual way. The American leader Zuming Feng turns up, and the Azerbaijani
and Slovenian leaderships. The flight south to M\'erida is uneventful, and we are met 
with smiling Mexican faces. There is a non-trivial bus journey to the Yucat\'an Reef Hotel.
Leaders arrive in very substantial numbers, very late. We queue up to register
and get our keys. It turns out that this process is ponderous, and there is only
one guy working the desk. Personally, this was easily 
the worst moment of what was otherwise
a very well organized IMO; if only someone had pointed out the truth -- that 
there was a free bar next door, and we could sit and have a margarita instead of 
waiting in line and in the heat, then there would have been 30 much happier bunnies.
No matter, from that moment on it all got much better.

\noindent{\bf July 9 and 10} I struggle with the questions as usual. There are some 
nice (relatively) easy
number theory problems, and lots of medium geometry ones. The algebra list
is woefully short and there are 
lots of combinatorics questions. We get the solutions 
far too soon, and the jury begins its 
deliberations. We choose questions 1 and 4 first:
a number theory question and a geometry question which is falsely perceived 
to be fairly easy. If we had been given 
more time to struggle with this question, its
subtlety might have become clear. 
It will turn out that there are many false trails
that can be followed in this question, and it would have been better classified
as of medium difficulty. The harder questions are 
selected next, and since there 
is a good combinatorics question, the obvious thing is to select what has been
classified as the hardest algebra question. Many leaders 
don't know the technique
known as Muirhead's inequality, and some of those who do seem to think that 
it is an advanced exotic technique. In fact many of the IMO students are
networking to great effect, partly with the help of the twin internet sites 
{\em Mathlinks} and {\em AoPS -- the Art of Problem Solving},
and they are sharing ideas and techniques independent of their national training
regimes. The IMO problem selection committee and jurors will have to monitor
these sites to see what ideas the students are sharing. It turns out that
Muirhead's inequality might be better known among the students than parts 
of the jury. 

\noindent{\bf July 11} The students are arriving as we finalize the paper. This
has been a swifter process than usual. Could this be related in any way to the
almost always open free bar? In the evening I prepare for the English
language committee which I will chair next day. This means I slope off to
my room early and try to cast the questions in perfect English myself, in order 
to have something to start with.

The committee meets first thing in the morning. These
days everyone is welcome in the ELC, including its most important member, the leader
of France. We like to have simple sentences in IMO questions; ones which ideally can be 
translated almost word for word into as many languages as possible. French is 
rather special, and does not allow the rather free word order and grammatical
latitude of English. Therefore the English language version has to be designed
so that it can be easily translated into French. As each English sentence is
suggested, we turn to FRA7, Claude Deschamps, to receive either a blessing
(a shrug which indicates that all is well) or a sad shaking of the head 
which indicates that a particular piece of Anglo-Saxon thuggery simply cannot
be expressed in French. 

The leader of Ireland IRE7 is full of, well, suggestions. It seems best
to install him as committee secretary, a position always reserved for
a person likely to cause trouble if not kept busy.
IND7 comes up with a suggestion for rewording Problem 5. I punish
him by making him write it out in full. This keeps him busy while we discuss
Problem 6. In the end he produces something rather good. This is all rather 
good fun. One advantage of coming from multicultural UK is that one is used to
listening to English expressed in diverse accents from all over the world. Now,
these accents are not necessarily mutually intelligible, and one or two
members of the ELC speak English in such a way that few people understand them.
Part of my job is to repeat their sometimes excellent suggestions in standard
English so that everyone
can follow, but also without drawing attention to the problem. 
This has its comic side.

When all this is done, we do it all again but in front of the full jury.
I carry on in the chair.
I now have the advantage that the ELC should be on board,
so that they can't really stand up and start making radical suggestions 
to reword the questions because they have signed up to the ELC version.
Nonetheless, there is still much fun to be had, especially with suggestions
flowing freely from NZL7, a native Russian speaker. We finish in time
for lunch, and we try to type up the definitive English version 
as quickly as possible. 

In the afternoon the versions of the other official languages are approved;
French, German, Russian and Spanish, and overnight the full babel of papers
is produced. There is a banquet which gets to stop early because of high
drama; KOR7 has discovered that Problem 5 is broken in an extremal 
configuration. The jury fixes the language by inserting the word ``interior''
so that children can sleep more easily in their beds.

\noindent{\bf July 12} More drama in the morning: non-verbatim 
translations of wording
into two languages have produced versions which give some students a minor advantage.
This is picked up by at least two polyglots, and these versions are fixed. 

The opening ceremony includes only finite quantities of
folkloric dancing, for which we are all grateful. One part of it is actually rather
good, with spinning dancers balancing drinks in their sombreros. That is a real skill.
Also the speeches are in Spanish, and are mercifully translated into English on screen
rather than being repeated. Good choice Mexican organizers! Our team looks neat
and impressive in their uniforms. The South African leader 
remarks to me that they look very
academic. I seek to give the impression that they are dilettantes and
dissolute playboys without
a hope in hell; there is no need to raise expectations. 

We listen to the co-ordinators' proposals 
for marking schemes. These are mainly fine
but there are difficulties with Problem 3. 
When the jury explain the complications
of the suggested system, the co-ordinators go away and produce a much clearer
(and fairer) scheme. 

\noindent{\bf July 13} The exams begin. For the first half hour the 
students are allowed to ask questions of clarification. These should
be faxed to the leaders' site and answers to each question should be
approved by the jury. Unfortunately there is a communications failure,
and we don't receive the questions until after a significant but not
disastrous delay. Mercifully there are no outright comedy questions from
the UK this year.

That evening the students' solutions arrive at the leaders'
hotel. A first reading has me feeling rather pleased. UNK2
may well have had a perfect day, and UNK5 is not far off.
Everyone has got a question out. Late at night, just before 
the bar shuts at 2am, I slip down for a celebratory rum and coke
wearing my best poker face. I expected to find the bar full, but 
all the other leaders are either asleep or working
on scripts. The only other IMO person there is a well-known
figure of firm opinions, exemplary intellectual standards
and a delicious manner which may well give him the role of
a Bond villain in an 007 movie one day. 
``How did your students do on Problem 3?'' he asks.
I reply that we have at most three full solutions.
``Three!!?'' he replies, lifting an eyebrow somewhere
close to Polaris. ``At most three, I can't
be certain yet'' I reply. There is a pause. He goes on,
``I have looked at the scripts of a {\em good} country, and 
they only have two correct solutions. When you say at most three, does this
this include zero?'' (he manages to suppress the implicit ``Mr Bond?''). 
``Of course'', I reply. 

\noindent{\bf July 14} Today the communications are working and 
the jury disposes of the students' questions in short order.
We then  decamp to M\'erida in a fleet of buses, and meet 
our students as they leave the exam. The students are all 
pretty pleased. They have all solved at least two problems
so we should have six medals again, and this is excellent for morale.
Three students, Nathan, Andre and Martin, have all done the best part of 
five problems so we will be in the hunt for gold medals we hope. 
It is delightful to see Adrian again. He has been ill for the past 24
hours and apparently
almost vomited over the students after the first paper. 

Adrian and I are in the leaders' hotel, 
which is disappointingly only a little superior to 
the hotel which houses our students and their splendid guide, Sandra.
Our place has only one indoor waterfall and therefore compares unfavourably
with the leaders' hotel at the Washington IMO of 2001. Still, sometimes
one must suffer for mathematics, and I try out my treble bed.
 
The students can now relax and start to make friends with the other teams.
They will have many adventures, including Jack Shotton having a fish land 
on his head while he was swimming, and Matthew giving free rein to his
humour by cheating at all possible games. Andre and Nathan seem to be popular
with a very non-empty subset of the girls. Andre uses boyish charm, whereas Nathan
deploys his impressive height. Saul is busy setting new records for losing 
hotel keys, and Martin is quietly confident.

\noindent{\bf July 15 and 16} The co-ordination is brilliant. The co-ordinators
have clearly read our scripts in detail, and for the most part are suggesting 
correct marks. In our co-ordination for Problem 3, the co-ordinators incorrectly imagine
that there is a hole in Nathan's solution. After careful analysis, they back down
and he gets full marks. The biggest difficulty concerns Andre's solution for 
Problem 1. It is a non-standard attempt, and involves an unusual clinch in that
he deploys the fact that a triangle with Brocard angle of 30 degrees must
be equilateral. We arrive brandishing a text-book in which this 
fact is stated, and pronounced well-known. However, his solution
is flawed elsewhere, and it is a matter of working out what
it is worth since it is outside the marking scheme. We are hoping for 5/7
but in fact, after consultation with the problem captain, it gets 3/7. 
We cannot complain because there is a logical case for this mark.

There is also minor grief with Problem 6. Andre has a solution which
is superficially similar to a flawed solution that the co-ordinators
have seen earlier,
so the co-ordinators are very suspicious. Adrian is taking the lead on
this question. We go through it line by line. Time after time they 
home in on a particular line and ask how it follows from the
preceding analysis. Time and again Adrian quietly explains how it works.
After half an hour, they begin to realise that Adrian knows exactly what he is 
talking about, and following the detailed textual analysis they eventually
acknowledge that it is worth 7/7.

Usually, during co-ordination, marks are put up as they become known. 
Ideally this should happen both at both the students' site and on a wall
outside co-ordination. This adds to the drama, and winds the tension up
as leaders and deputies try to work out medal boundaries, and whether 
or not they will finish above their traditional rivals in the 
unofficial ranking. On the first day, no marks go up at all which is
very frustrating, but on the second day marks start to appear.
Traditionally Gordon Lessells, the Irish deputy, 
uses his bookmaking skills to forecast
medal boundaries with astonishing accuracy by means of his little book and
bespoke undecipherable notation. This year he makes an uncharacteristic
slip (for which he has a predictable raft of excuses) and forecasts 
the bronze boundary well above the correct answer. In the evening
the jury meets very late to set the medal thresholds. There are
12, 23 and 35. Martin has an IMO gold to sit beside his Informatics Olympiad
gold,
Nathan and Andre have strong silvers, and Matthew has squeaked a silver
medal. Saul and Jack have solid bronzes. 

It is sad news that the Irish star Fiachra Knox has missed a gold medal
by just one mark. Still, he obtained the first ever silver medal for
the Irish Republic, and both he and the rest of the Irish team
made friends throughout the IMO.

Having scored fairly well on the geometry questions, now is the time to 
peddle the new UKMT publication ``Plane Euclidean Geometry'' by Christopher
Bradley and Tony Gardiner. A box of 20 steaming copies has arrived from the
UK. I put them on sale, and am almost trampled in the rush. We sell
out immediately. Readers interested in this competitively
priced thick volume of geometric delights, written with the needs of IMO
students in mind, should visit {\tt http://www.ukmt.org.uk/}

All this time, CNN has been reporting Hurricane Emily as she makes
her way toward us, gathering strength en route. The Yucat\'an
minister of tourism visits the leaders' hotel to assure us that
everything will be fine, so we know that we are in deep trouble.
There is no `our man in M\'erida', but there is a consul in Canc\'un.
Since Canc\'un may become matchwood, Adrian goes straight for the Embassy 
in Mexico City. They are splendid, give sound advice, and promise 
to send a rescue mission in the event that the whole province
is demolished. We activate Joseph Myers on the IMO hotline in 
Cambridge, and he phones round the parents of the team members to reassure
them that we have a plan to ride out the hurricane.

\noindent{\bf July 17} The governor of Quintana Roo has 
ordered the evacuation of the neighbouring
coastal city of Canc\'un, but we are in a better situation.
The IMO city of M\'erida is inland, so will be 
proof against the storm surge. Winds of about 200 mph are 
expected from about 2am tomorrow. Country people living in wooden 
houses have every reason to be very afraid, but we are staying
in robust hurricane resistant buildings, so if anyone in the town
is safe, we are.
The leaders' hotel is next to the students' hotel, but it will
not be possible to pass between these buildings in hurricane force
winds. 

The team and Adrian shoot off in the early morning to visit a Mayan 
pyramid, and I head to Wal-Mart to get hurricane supplies. I manage to
get plenty of bottled water, biscuits and chocolate 
but the shops have sold out of torches and lamps. The emergency 
generators in the hotels should kick in if necessary. 

The excursion returns just after lunch, just as the weather is starting
to deteriorate. After supper Adrian and I decamp to the students' hotel
for the night. It is not that we think that the team will be in 
immediate danger, but we prefer not to be separated during the storm.
Adrian has more helpful chats on the phone with the British Embassy.
Don't begrudge your passport charges; the money is well spent.

When we join the team they are in a very large windowless room with
hundreds of other youngsters. This is going to be a hurricane slumber 
party, and I can already tell that it will be awful. Photographs
are available on the net via {\em Mathlinks}. Valentin is pushing his luck
(yoga position indeed).

\noindent{\bf July 18} It is midnight.
There are table tennis
tables, table football machines and hockey tables in the shelter. When the lights
are turned out the students carry on playing the games in the murk.
At first this is funny, and then very funny, and finally it gets annoying,
and eventually very annoying indeed.
The whole room turns against these clowns, and the room is filled with
a hiss as we try to `shhhh' these gamers off the tables. Most desist, but 
a hardened few carry on. Finally, our team member Nathan Kettle jumps up
and heads purposefully across the room toward the moral dwarves in 
the corner. It seems likely that Nathan will thump one or two of the 
miscreants, so I leap up and follow him. While I know that it is my duty
to restrain him, a voice is whispering in my ear that if I arrive 
a few seconds too late,
perhaps it will be no bad thing.
However, Nathan is a gentleman, and when he arrives at the tables he merely give
an impromptu lesson in loud Anglo-Saxon. He retires, and the scum at the tables
just carry on. I arrive, and without raising my voice I put on my best
psychopathic look, and ask the nearest table footballer which nation he
is from. He tells me, but I cannot know for sure that he is telling
the truth. I tell them to stop and fold my arms. They back off, and 
I stand at the tables for about 5 minutes. It is clear what will happen.
As soon as I retire to the far end of the room the noise starts again.
I give in, and move to the corridor in the expectation that the forces 
of natural selection will operate in the function room, and that 
the table-footballers will eventually be lynched or beaten to a paste. Outside I find
one of the local organizers. I tell her the problem and she sends in guides
to take care of the gamers. 

I find a chair to sleep in near the atrium. Finally the winds arrive and 
a sound like the rushing of a train comes from the roof. However, the noise
starts to abate fairly quickly. It turns out that at the last moment 
Emily has changed direction, and she has only clipped us rather than 
rolling straight through us. Midmorning we are allowed
out, and the students are allowed back to their rooms. I am delighted to
find that M\'erida has got off very lightly.

It must have been heart-breaking for the local organizers to have to cancel
the outdoor farewell banquet that evening, and instead we have an 
improvised medal ceremony moved to the exam hall. This
is a great success, and a Mexican band walks through 
the hall playing stirring music. Perhaps the high point of the
ceremony is when a Moldovan student is awarded a special prize 
for his stunning solution to Problem 3. I am standing next to
the Moldovan leader, and to say that he is pleased does not really
capture his mood. 

The local organizers announce that there will be an all-night party at the 
student hotel, and that teams catching early flights might as well
forget about sleep. This is all very well in principle, but we are 
all feeling mangled from sleeping on the floor the night before.

Blofeld softens for a moment, and congratulates me on the performance of the
British team. He singles out the excellent performance of `the English girl'.
Now, astute readers may have noticed that for once we don't have one.
I later realise that 
he is probably confusing the provenance of the impressive NZL4.

\noindent{\bf July 19}
Our team sensibly snatch a few hours bedrest before the flight back.
This involves a long wait between flights in Houston (where we
say farewell to Andre who makes his own way to Singapore). When
we arrive at Gatwick I approach immigration last. The officer won't
let me in and asks me to take a seat. I assume that my new passport
has been incorrectly activated or some such thing, but then a couple
of detectives move in. It seems that someone with the same name 
as me is wanted for murder. After I prove where I live, everyone relaxes
and they let me in. Such are the perils of being called Smith. We are
met by happy families. The Lee family is extraordinarily pleased to see
Matthew (perhaps they have forgotten what he is like). 
Mrs Kettle is stuck in traffic so Nathan, Martin and 
I go for breakfast and after a while Mrs Kettle joins us.
Finally Martin and I are left alone. 
He has to catch a flight to Belfast in a few hours, and I bid him 
farewell. I think he is a bit pleased that his mark is higher than 
Paul Jefferys's gold medal score in 2004.


Full team scores:
Albania 44,
Argentina                65,
Armenia                  82,
Australia               117,
Austria                  74,
Azerbaijan               59,          
Bangladesh                3, 
Belarus                 136,
Belgium                  74,
Bolivia    (2)            0,  
Bosnia \& Herzegovina   49,  
Brazil   82,
Bulgaria 173,
Canada  132,
China   235,
Columbia 105,
Costa Rica 37,
Croatia 82,
Cuba  (4) 54,
Cyprus 14,
Czech Republic 139,
Denmark  69,
Ecuador  17,
El Salvador  25,
Estonia 68,
Finland 49,
France 83,
Georgia 80,
Germany   163,  
Greece 58,
Guatemala (3) 6,
Hong Kong 138,
Hungary 181,
Iceland    23,
India   81,
Indonesia 70,
Iran 201,
Ireland 55,
Israel 99+,
Italy 120,
Japan 188,
Kazakhstan 112,
Korea  200,
Kuwait (5) 3
Kyrgyzstan 46,
Latvia 62,
Liechtenstein (3) 4, 
Lithuania 53,
Luxembourg (2) 3, 
Macau  38, 
Macedonia 50,
Malaysia 15,
Mexico 91,
Moldova 130,
Morocco 18,
Mozambique (5) 2,
Netherlands 62,
New Zealand 77,
Norway 38,
Pakistan 11,
Paraguay 12,
Peru 104,
Phillipines 30, 
Poland 105,
Portugal 27,
Puerto Rico 8,
Romania 191,
Russia 212,
Saudi Arabia 3,
Serbia and Montenegro 75,
Singapore 145,
Slovakia 131,
Slovenia 49,
South Africa 39,
Spain 46,
Sri Lanka 32,
Sweden 42,
Switzerland 70,
Taiwan 190,
Tajikistan (3) 3
Thailand 128,
Trinidad and Tobago 13,
Tunisia (3)  9
Turkey 130,
Turkmenistan (3) 18
Ukraine 181,
United Kingdom  159,
United States 213, 
Uruguay (5) 37,
Venezuela (2) 15,
Vietnam 143.

I thank everyone who helped in the vast collective effort which goes into
UK IMO training and support. The team delivered the goods, 
very near to the upper limits of our
most optimistic expectations.

As well as thanking everyone who helped train our team, I would like to thank
the reserves who played a vital role. We are also grateful for financial
and other support from the Department for Education and Skills,  
the United Kingdom Mathematics Trust, the microelectronics company ARM,
The Royal Society, the Bolyai Society of Hungary and 
the publishing house Springer Verlag. We are also grateful to all the
administrators and secretaries who have helped us in unseen ways, in 
particular Miss Angela Gould who is now leaving UKMT having
helped IMO preparation in many ways in recent years. 

Finally, I am sure that all recent
UK IMO squad members and trainers would 
wish to congratulate the Hungarian deputy Sandor Dobos
and Miss T\'{\i}mea T\'oth on their recent marriage.
\vskip 1cm 
\hfill {\tt G.C.Smith@bath.ac.uk\ \ \ \ http://www.bath.ac.uk/$\sim$masgcs/}

\end{document}

