\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage{amssymb}
\usepackage{latexsym}
\title{IMO 2008 in Madrid}
\author{Leader's Report}
\date{}
\begin{document}
\maketitle

The International Mathematical Olympiad is the
annual world championship of secondary school 
mathematics. It has been running
since 1959 (except for 1980). Teams of six students sit 
two papers on consecutive days. Each 
paper consists of three problems, and each problem is worth 7 marks.  
Thus a perfect score for a student is 42/42. The students are 
ranked according
to their personal scores, and the top half receive medals. These are 
distributed in the ratios gold:silver:bronze = 1:2:3. The host city 
of the IMO varies from year to year. Detailed contemporary and historical 
data can be found at  

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

In 2008 the 49th IMO was held in Madrid. Students from 97 nations 
participated, and the UK team won four silver medals and two bronze medals. 
Our rank improved from 29th in 2007 to 23rd in 2008. However, this 
year we were in a part of the table where the 
rank statistic was very sensitive. If the team had solved even one 
extra problem between them, then they would have shot up the rank order.
The teams ranked 17th were Romania and Peru on 141 points and the team ranked 24th were 
Italy on 132. A mark of 15 was required for a bronze medal, 22 for a 
silver medal and 31 for a gold medal.

The leading three nations at IMO 2008 were China (217), Russia (199) and 
the United States of America (190). The leading country in the EU was 
Hungary (10th, 165) and the top Commonwealth performance was 
Australia (19th, 140).

\clearpage

\begin{verbatim} 
Contestant          P1   P2   P3   P4   P5   P6   Total  Award 
 
UNK1 Tim Hennock     7    3    0    7    7    0    24    Silver 
UNK2 Peter Leach     7    1    0    7    7    1    23    Silver 
UNK3 Tom Lovering    7    1    3    7    7    0    25    Silver 
UNK4 Freddie Manners 7    2    0    4    7    0    20    Bronze 
UNK5 Dominic Yeo     5    1    0    7    3    0    16    Bronze 
UNK6 Alison Zhu      5    7    0    6    7    0    25    Silver

total               38   15    3   38   38    1   133    SSSSBB

\end{verbatim}

This was a very solid performance, following on from the 
United Kingdom team's winning performance at the inaugural 
{\em Romanian Master in Mathematics} competition 

\begin{center}
{\tt http://www.rmm.lbi.ro/}
\end{center} held 
in Bucharest in February 2008. Only two nations performed
better on Problem 5, a gratifying result for the trainers who
helped the students to 
put in a big push to improve our combinatorics this year.

\medskip

The UK Competitors attended the following schools.

\medskip 
\noindent 
Tim Hennock, Christ's Hospital, Horsham, Sussex \newline \noindent
Peter Leach, Monkton Combe School, Bath, Somerset \newline \noindent
Tom Lovering, Bristol Grammar School,  Bristol \newline \noindent
Freddie Manners, Winchester College, Hampshire \newline \noindent
Dominic Yeo, St.\ Paul's School, London \newline \noindent
Alison Zhu, Simon Langton Girls GS, Canterbury, Kent \newline \noindent

\medskip

Our reserves were as follows.
 
\medskip

\noindent Jonathan Lee, Loughborough Grammar School \newline \noindent
Craig Newbold, Whitley Bay High School \newline \noindent
Preeyan Parmar, Eton College \newline \noindent

\medskip

There were two new faces in the side. Freddie Manners has been 
frittering away his 
youth by going to non-mathematical olympiads, and 
Peter Leach is a rower who
is trying to extend his range of interests. 

I can also report that our notional competitors, Max and Min, 
would have respectively
scraped a gold medal with 32, and secured no reward at all 
with 13 points (and a dishonourable
mention for solving no question completely).

\medskip

The adults accompanying the UK team were as follows.

\medskip

\noindent UNK7 Dr Geoff Smith, Leader, University of Bath \newline \noindent
UNK8 Dr Ceri Fiddes, Deputy Leader, Stowe School \newline \noindent
UNK9 Dr Vesna Kadelburg, Observer B, Sevenoaks School\newline \noindent
UNK10 Ms Jacqui Lewis, Observer C, St Julian's School, Carcavelos, Lisbon
\newline \noindent

\medskip

The schedule for future IMOs is that the event will be held in 
Bremen, Germany in 2009,
in Astana, Kazakhstan in 2010, in the Netherlands in 2011 
and in Argentina in 2012. 

\subsection*{The Problems of the 49th IMO were as follows}

\newcommand{\problem}[1]{\paragraph{Problem #1.}}

\problem{1} An acute-angled triangle $ABC$ has orthocentre $H$. The
circle passing through $H$ with centre the midpoint of $BC$ intersects
the line $BC$ at $A_1$ and $A_2$. Similarly, the circle passing
through $H$ with centre the midpoint of $CA$ intersects the line $CA$
at $B_1$ and $B_2$, and the circle passing through $H$ with centre the
midpoint of $AB$ intersects the line $AB$ at $C_1$ and $C_2$. Show
that $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ lie on a circle.

\problem{2} (a) Prove that
\[
\frac{x^2}{(x-1)^2} + \frac{y^2}{(y-1)^2} + \frac{z^2}{(z-1)^2} \geq 1
\]
for all real numbers $x$, $y$, $z$, each different from $1$, and
satisfying $xyz = 1$.

\vskip\baselineskip

\noindent (b) Prove that equality holds above for infinitely many
triples of rational numbers $x$, $y$, $z$, each different from $1$,
and satisfying $xyz = 1$.

\problem{3} Prove that there exist infinitely many positive integers
$n$ such that $n^2 + 1$ has a prime divisor which is greater than $2n
+ \sqrt{2n}$.

\def\mbig#1{{\hbox{$\left#1\vbox to 10pt{}\right.\nulldelimiterspace=0pt\mathsurround0pt$}}}
\def\mbigl{\mathopen\mbig}
\def\mbigr{\mathclose\mbig}
\problem{4} Find all functions $f : (0,\infty) \to (0,\infty)$ (so,
$f$ is a function from the positive real numbers to the positive real
numbers) such that
\[
\frac{\mbigl(f(w)\mbigr)^2+\mbigl(f(x)\mbigr)^2}{f(y^2)+f(z^2)}
= \frac{w^2+x^2}{y^2+z^2}
\]
for all positive real numbers $w$, $x$, $y$, $z$, satisfying $wx =
yz$.

\problem{5} Let $n$ and $k$ be positive integers with $k \geq n$ and
$k - n$ an even number. Let $2n$ lamps labelled $1$, $2$, \ldots, $2n$
be given, each of which can be either \textit{on} or \textit{off}.
Initially all the lamps are off.  We consider sequences of
\textit{steps}: at each step one of the lamps is switched (from on to
off or from off to on).

Let $N$ be the number of such sequences consisting of $k$ steps and
resulting in the state where lamps $1$ through $n$ are all on, and
lamps $n + 1$ through $2n$ are all off.

Let $M$ be the number of such sequences consisting of $k$ steps,
resulting in the state where lamps $1$ through $n$ are all on, and
lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n
+ 1$ through $2n$ is ever switched on.

Determine the ratio $N/M$.

\problem{6} Let $ABCD$ be a convex quadrilateral with $|BA| \neq
|BC|$. Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_1$
and $\omega_2$ respectively. Suppose that there exists a circle
$\omega$ tangent to the ray $BA$ beyond $A$ and to the ray $BC$ beyond
$C$, which is also tangent to the lines $AD$ and $CD$.  Prove that the
common external tangents of $\omega_1$ and $\omega_2$ intersect on
$\omega$.

\medskip

The problems were proposed by Russia, Austria, Lithuania, South Korea,
France and Russia respectively.

\section*{United Kingdom Leader's Diary}

This annual diary is loosely based on reality, and would 
be unsustainable in a
court of law. Inevitably this diary
tends to focus on things that went wrong (for comic effect), but this should
not detract from the overall truth that this IMO was a triumph. We had really
good exams and they were marked fairly.

The 49th IMO in Spain was a great success, and this is primarily
due to the generosity and energy of our Spanish hosts. 
The vast IMO local organization is invisible to participants. There are 
hundreds of them, but you only get to meet a few. This is a shame. 

\subsection*{July 6th} 
We fly out to Lisbon for a pre-IMO camp. A Portuguese resident, 
Jacqui Lewis (UNK10 and Observer C),
has kindly arranged a camp there for both the UK and Australia. 
We are staying in a private hotel, taking our lunch and examinations 
in St Julian's School, Carcavelos, and dinner at beach restaurants. 
The school, its students and their parents prove to be extremely 
hospitable for which many thanks, especially to our
expert translator Marianne.

\subsection*{July 8th} 
The Australians start to arrive in the morning, in the form 
of the leader Angelo di Pasquale and six students. In the afternoon 
the deputy Norman Do and the Observer B Denise Lin also turn up. 

\subsection*{July 9th} 
The first exam. The Australians win. I am traumatized. 
Last year in Hanoi
Peter Taylor of the Australian Mathematics Trust proposed to donate a trophy
called the {\em Mathematics Ashes} for an annual competition between the 
United Kingdom and Australia. He and I agreed the details. One of 
the later exams at this camp will decide which country is to be
the first holder of the urn. This arrangement is based on that 
which pertains for cricket,
whereby Australia and England compete to hold `The Ashes'. A funeral urn holds 
`The Ashes of English Cricket', the burnt residue of some cricket items which
were set on fire
in the nineteenth century. For our mathematics
competition, we have agreed that the scripts of the 
2008 losing team will
be burned and placed in an urn for all time. Details and photographs are
at \begin{center}{\tt http://www.amt.edu.au/news02.html},\end{center} 
at least for now. 

Angelo has recognized that he is not worthy to transport the urn 
from Australia to 
Europe, and this task is delegated to a courier company. Terry Tao has been
blogging about this competition, and the Australian media have become 
very interested.
This is, of course, the way forward for Australian mathematics. 
Once mathematics
is regarded as a competitive sport rather than an academic subject, 
there will
be no limit to the resources made available to support it in Australia.    
 
\subsection*{July 10th} Today 
I catch a flight from Lisbon to Madrid to join the jury. 
A lady called Linda has fallen from the pages of Hemingway and drives 
me to the airport. She is an American, but a life spent in Portugal has 
given her soft Iberian edges. She has gone native to the extent that 
she warns me of the dangers of travelling to Spain, a place which she 
views with serious suspicion. 

At Lisbon airport Easyjet has laid on a pretty good check-in queue, 
and I set about enjoying it. Eventually 
my bags and their owner shuffle to the front where I am sharply 
scolded for putting my bag on the luggage belt too quickly. 
I must watch that, as I do not wish to cause offence. The check-in 
lady has an animated conversation with other check-in staff before 
eventually she waves me forward to engage in the intimate phase of 
our relationship. It is now clear that she is feeling
guilty for admonishing me for promptness, and starts to explain 
what a difficult day she is having.
It seems politic to ignore the hundreds queueing behind me, and focus on
her troubles. Eventually my bag disappears and I clutch the precious 
boarding pass.

I have lunch while waiting for boarding. I order a black coffee 
and get a coke. Actually in the heat it seems a better idea. I follow 
the instructions and go to the correct gate and settle down.
Time passes and nothing happens. Then suddenly a whole plane-load 
of people join the queue. I check the screen and discover that they 
are going to Paris. Now either Paris has moved, or my 
plane has got lost. I soon discover that my plane has been assigned to another
gate. There has been no announcement to this effect, and foolishly I 
am concerned that time is now short. I rush to the new gate to discover 
hundreds of better informed passengers in a well-developed
queue. We are all off to Madrid but the plane is late. Perhaps you are 
getting the hang of this now.

Eventually I board. Every seat on the plane is taken, but the flight is 
short. Spain looks brown.  The landing in Madrid is not bad. I retrieve 
my luggage and walk boldly through the arrivals gate
expecting to be met by a throng of IMO aides. Usually teams of 
enthusiastic helpers usher you to a seat and bring you water. You 
embrace the local organizers, and pump the hands of other newly arrived 
team leaders, and the air fizzes with bonhomie, human warmth and 
the sense of relief that comes at the end of a lonely journey.

Not this time. I look at the handful of sullen taxi-drivers holding up 
placards and check them individually in case I am their prize. 
Unfortunately the answer is no. There is a sign pointing to an information 
desk and a meeting point. After a few minutes I decide to go there.  
The information lady knows nothing about the IMO, and the meeting point 
is deserted. I wait, half-expecting tumbleweed to be blown along the 
corridor. I return to the arrivals area in the hope that the greetings 
party will have arrived. No chance.

By now I was convinced that I had done something very stupid. The 
IMO of 2008 is definitely to be held in Madrid, and that is certainly 
my location. Wearily it dawned that I must have arrived on the wrong day. 
I check my mental calendar, and persuade myself that I have messed up. 
I should have arrived tomorrow. The solution seems obvious; check-in 
to a hotel and return to the airport next day. However, there remains
the remote possibility that the greetings party has been kidnapped 
by space aliens, but that surviving groups of IMO organizers might be 
present elsewhere at the airport.

I return to the information desk, and explain that I wish to visit the 
arrival halls of each of their terminals in sequence. I am given an algorithm, 
and set off in search of these oases of hope. Eventually I find the 
arrivals area of the next terminal, and locate a nice lady seated next 
to an IMO notice. Hooray!  Apparently there was supposed to be someone 
to greet me at my terminal, but for some reason this did not happen. 

I settle down to await a critical mass of leaders. When we have enough 
we are escorted to a taxi. The driver tells us we are going to Segovia, 
and that the journey will take 90 minutes. We cut through Madrid, 
and then head 
for the mountains. Our hotel is in the grounds of La Granja Palace de San 
Ildefonso, Segovia. Just before we arrive, the taxi stops because the
Latvian observer is feeling unwell. Immediately our 
Eastern European fellow passengers light cigarettes, presumably
because they think this will help. After a mint her disposition
is restored, cigarettes are regretfully extinguished, and we
arrive at the hotel five minutes later.

My room is excellent and the facilities are splendid. A very short walk 
to a second site, the convention centre, takes me to supper and first 
sight of the jury. As expected, the leader of New Zealand has lost his 
moustache, but otherwise they seem undamaged. After dinner I get hold 
of the IMO shortlist and set to work.  

\subsection*{July 11th} We are all working on the problems today. 
In the evening 
the solutions are issued. The immediate concern is the geometry problem 
classified easiest by the Problem Selection Committee, G1. This is invariably 
selected, so it is very helpful if it is a good question. It concerns a 
circle manufactured from a triangle, its circumcentre and orthocentre.  
I had solved it immediately by use of a parallelogram trick.  
However, I was not worried that the problem was too easy because it was 
there to test the less experienced students, and they would surely not 
spot that method.  There are other smooth ways to dispose of the question, 
but the bunnies are not going to spot those either. No, they are going to 
(a) identify the centre of this circle and (b) calculate its radius using 
a trigonometric slog. Some will fall by the way and be crushed underfoot. 
Thus the problem will have some value.

We will find out later that this circle was discovered by Droz-Farny, 
I realise and that the parallelogram proof works for any pair of isogonally 
conjugate points instead the ones in the problem. I feel dim for not 
spotting that first time.  I know a thing or two about Droz-Farny lines
\begin{center} {\tt http://forumgeom.fau.edu/FG2007volume7/FG200702index.html}
\end{center}
but until now his circles had been a closed book to me. The jury decided a 
few years ago that they would be happy to relax the originality requirement 
for Problems 1 and 4, because of the difficulty of inventing genuinely 
new easy problems. The word `easy' is being used here in a relative way. 
It means that the top six students from a large well-trained country 
will probably be able to solve such a problem.  

\subsection*{July 12th--13th} The jury meets and 
begins to discuss the shortlist. 
We are blessed with another very good jury chair, one Carlos Andradas Heranz. 
He seems thoughtful and very fair. The jury is used to Maria Gaspar as 
the face of Spain, and her presence is reassuring.

Several question proposals are thrown out because they are recognized as 
known. This is very irritating because of the time people have wasted working 
on the questions, and because our choice is now more restricted. I 
make the first speech in which I parrot some ideas which I stole from 
the absent Angelo di Pasquale, the leader of Australia.  I acknowledge his 
contribution, but the speech goes down very well and I appear to be getting 
the credit. I make a note of this and decide to pinch more of Angelo's 
ideas in the future.

Discussions and problem selection proceeds apace. I am seated at the back.
Political developments mean that I am no longer inserted between the Ukraine 
and the United States. This is because of the arrival of the leader of the 
United Arab Emirates. Thus Ukraine's Sergiy Torba no longer sits on my left, 
but instead I have a new friend called
Juma Rashed Ali Al Shamisi, or Juma for short. His name is so much more 
impressive than mine that I feel distinctly inferior. I will look into 
ways to upgrade my name. 

We first select an algebra question as the relatively easy problem to go with
the inevitable G1. It is a functional equation question which falls naturally 
into two parts. This will help to fragment the
marks. We then pick a distinctly classy number theory hard problem
and a corresponding geometry problem. Neither seems completely impossible.
The Italian leader Roberto Dvornicich is a number theorist, and he shows us
a quick and elegant proof to that question. The geometry question is an 
extension of the result that a line drawn through a triangle vertex and 
the `top' of its incircle hits the opposite side at the contact point 
of the excircle. The trick with the toy version is to enlarge from the vertex. 
An extension of that idea will solve this IMO problem. 

It remains to discuss the medium questions. At least one must be combinatorial,
and neither can be from the geometry list. We move a question from the 
geometry list to the combinatorics list because (a) that is where it 
belongs and (b) it gives us more scope.  There is a combinatorics 
question involving switching lamps on and off. There have been no 
lamp manipulation problems set in recent years, so it gets selected, 
along with an inequality which, like the other algebra question, falls 
into two parts.

Along with all this happy progress, there is the bad news that the students 
and deputy from Pakistan are having difficulties getting visas to attend 
the IMO. The Pakistan leader is present, but the youngsters are not. The 
Spanish organizers are working the phones frenetically to Islamabad and 
to relevant Spanish ministries, but they are given little reason for optimism. 
This is a most unwelcome development. It seems that the visa applications 
for Pakistan's IMO team have become embroiled in ponderous 
Schengenland protocols. Not allowing students to travel to participate in 
the IMO must give a terrible impression to the students, their families, 
teachers and supporters. The Spanish organizers offer to invite the 
six students from Pakistan to visit Spain later in the year. Whatever 
the ultimate cause of this incident, it was no fault of the IMO organizers. 

An election for the advisory board is very closely fought. Three 
excellent candidates are standing for one position, and it therefore 
comes as no surprise that the winning margin of the victorious candidate 
is small: one vote allows Gregor Dolinar a seat on the IMOAB. In some sense 
this must be a reward for the excellent way in which he chaired IMO 2006 
in Slovenia.  

\subsection*{July 14th} Today the students are scheduled to arrive
in Spain. The IMO really starts when the youngsters land. A high-tech 
display is projected in the coffee area in order to worry the leaders. 
It displays inaccurate information for most of the day. Two of our 
party were supposed to arrive at breakfast time by train, but there is 
no acknowledgement that this has happened even by late afternoon.  
This concerns me until I notice that implausibly few students have arrived, 
and that the information given in the display is worthless.  

The English language committee meets. We work hard, and produce 
proposals and options for the full jury to consider. 
The English language committee is my annual opportunity to strut the boards. 
The protocol is that Michael Albert and I ponder the wording overnight, then
we meet at breakfast to agree a common position. We take our common 
document to the English language committee as a basis for discussion.  
Michael Albert works double time managing the data projected on the screens, 
while simultaneously producing new versions from time to time. We try to 
encourage all troublemakers to attend the ELC, to tie them in to an 
agreed formulation. The idea is that when later we take proposals to the 
full jury, all the firebrands and demagogues will have become tied 
to the ELC's wording, and anyway they should be getting tired of 
complaining about the wording by then, and they will be wanting 
to complain about other things instead.  

This does not work in 2008, because the Dutch leader, dear Quintijn Puite, 
views the jury session on the English wording as an opportunity to revisit 
every issue where he didn't get his way in the ELC (and some where he did). 
I slap him down brutally, and teams of Spanish psychotherapists are 
attempting to reconstruct his personality even as I type. 

The ELC recommendations are mostly accepted in the end, but the 
ELC's alternative formulation of Problem 6 is not adopted, and 
this will have consequences.  

The jury piles into buses, and we set off for Madrid and the opening ceremony.
The event is held in a circus. I am able to exchange deranged waving
with the rest of the United Kingdom Team. Tom has his understated look, and 
wears both a Panama hat and a Union Flag. The parade of the nations 
is attenuated, which had both advantages and disadvantages. Certainly it kept 
things shorter than usual.  The officials and politicians were also admirably 
restrained. Then the circus entertainment could begin. It had several good 
acts and one weak one. The clowns were not to my taste. A nice lady with 
thunder thighs lays with her legs in the air, using her feet to spin and 
juggle with an axle which has flaming wheels on its ends. Then there is
some acrobatic totty which dangles from a great height with only feet 
tangled in net curtains for support. To finish off, a bunch of 
male strippers gets half way through their act before morphing into 
a strongman act.  Then they stand on one another a lot, and finish 
by making a human tower out of the transitive closure of the one-armed 
handstand. The blokes present are confronted with a rather stark 
demonstration of their own physical limitations. We had better stick 
to mathematics then.  

\subsection*{July 15th} Today we complete the business of translating
the examination papers into over 50 languages. We also have meetings with
the six Problem Captains who propose marking schemes for the problems. 
The jury suggests modifications, many of which are accepted and 
included into the official marking schemes. 

\subsection*{July 16th} On the first morning of the exam the Spanish organizers 
demonstrate their 
fancy internet software for handling students' queries. All information 
is projected for the jury to see and consider. There are just seven 
questions from the students, and the system worked beautifully. As usual, 
a few students asked `what is the orthocentre of a triangle?'.  The jury 
wisely did not use my proposed answer `the isogonal conjugate of the 
circumcentre'. 

Most of the jury go off on a tour, and I repair to my executive suite 
for a sequence of baths and siestas. In the evening the more robust 
leaders return from their excellent trip, and the students' scripts arrive. 
I am not happy, because two of my students have thrown away marks on 
Problem 1 by carelessness. Also another student has made a slip on Problem 3 
which means he will get only part marks, but he has all the ideas for a 
full solution. I remind myself that since I don't have to try to solve 
these problems under time pressure, it is not fair to criticize those who do.  
Then I chew some more cutlery.

\subsection*{July 17th} On the second morning of the exams the jury gathers once again to 
field the questions.  It quickly becomes apparent that the quiet times 
are over. We are deluged with questions from the students.  There are 
really just two different questions, asked over and over again in 
all the languages of the world. Although the questions superficially 
take many forms, they are almost all rephrasings of the following two 
possibilities:

\medskip

\noindent (a) Problem 5: when considering a sequence, does order matter?

\noindent (b) Problem 6: I cannot draw the diagram. Can you help me?

\medskip

The answers are, of course, yes and no respectively.

The software and jury protocols which worked so well on day 1 now become 
overwhelmed. I had the pleasure of spending several hours in Heathrow 
Terminal 5 on its opening day, and there were parallels. We managed to 
clear the backlog of students' questions after a couple of hours. 
That is rather slow.  

As you know, I am completely immune to human vice, and a smug thought 
never crosses my mind. However, the English language committee did 
point out to the jury that (b) was an accident waiting to happen, and suggested
an alternative formulation of the problem which was phrased so as to constitute 
a kit for drawing the diagram. The jury decided that people trying for 7/7 on 
Problem 6 would be able to draw the diagram using the original formulation.
This was correct, but I suspect that the jury seriously underestimated 
the volume of whimpering, dull moaning, and grovelling that 
would be generated from students who were never going to get 7/7 for this.

Finally the question session finishes, and the jury checks out from its 
luxurious premises.  We move to a Madrid hotel. Now, Madrid is a seriously 
hot place, and one can see the advantages of reducing the area of a 
hotel's windows with a view to diminishing the need for air-conditioning. 
This excellent idea has been taken perhaps a little too far, and the 
architectural influence of Josef Fritzl is apparent. In future we will 
have vitamin D tablets in our first-aid kit.

I meet the UK deputy Ceri Fiddes and our Observer B Vesna Kadelburg.  
They have been elsewhere in Madrid for the past few days. We settle 
in for the co-ordination phase, and divide up responsibilities. I 
take the geometry problems. Ceri and Vesna have the rest. We operate 
a system whereby we have two people expert on any one problem. Ceri 
is definitely the lady with the lamps (Problem 5) because of her 
Champollionesque abilities to decipher combinatorial arguments written 
in the style of Finnegans Wake, the preferred medium of British students. 
In fact our students have done very well on this question, and it is only 
UNK6 Alison Zhu's solution which may need close textual analysis. We enlist 
Michael Albert of NZ to assist, since he is a professional combinatorialist. 
He slices through her script in a few minutes, and explains to us that 
she is actually quite correct. Ceri writes out a careful clean version 
just in case, but this year the vast majority of the co-ordinators are 
excellent, and Ceri will not have to make a detailed case to collect 
Alison's 7/7.  

\subsection*{July 18th--22nd}
Vesna has some uncontroversial work to do. My geometry is all straightforward
except that I get into a tangle with the co-ordinators of Problem 6, of
which more later. I am scheduled to co-ordinate the relatively easy
geometry question, Problem 1, on the second day of co-ordination, but 
the process rapidly gathers pace, and the co-ordinator from our
Problem 1 table approaches me a day early. She asks if we are ready to 
deal with Problem 1 at short notice. I reply 777755. She nods. I go off to to
fetch V \& C to regularize the position.  

Problem 6 presents more difficulties. Four of our students have handed in 
worthless scripts. UNK2 Peter Leach gets a grudging mark for 
giving a correct geometrical characterization of a key point
in the diagram, though he uses an interrogative form of 
delivery designed to provoke suspicion in co-ordinators. Instead
of saying `X is true', he writes (following the style of
our trainer Kevin Buzzard) `Can X be true? Wow, yes, I think it
is true. Incredible. This is amazing.' [Translated from the original Leach.]

More difficult still is the script of UNK3 Tom Lovering. He has spotted
how to do the problem. However, because he spent so long on Problem 5, 
it turns out that he has only addressed Problem 6 for a few moments. 
He has quickly scribbled down in one sentence a correct 
method to solve the problem,
describing a couple of hometheties (enlargements) which, if deployed
correctly, tell you everything. Of course he has not had time to write out a 
proper solution, but he has presented a kit for solving the problem. 
It will eventually earn 0/7, after appeal to the chief co-ordinator and 
several meetings. The co-ordinators are defending a marking scheme
which I regard as unsuited to this script. However, they will not budge.
We find ourselves in alliance with the leader of Montenegro which is 
in a similar position. Vesna's language skills come to the fore, and
her fluency in what she calls Serbian opens up a new social dynamic for 
UK leaders. Other people call very similar languages Croatian, Montenegrin
and Bosnian.

The real problem goes back to the phase when the marking schemes
were presented to the jury by the Problem Captains. For some reason, the
scheme for Problem 6 was not presented until very late. The shortcomings
of its proposed marking scheme were pointed out forcefully, but it was 
not modified.  In fact two countries
were eventually so unhappy with the marks that they were 
offered that they took the matter to the final jury.

In my view the jury will have to take more responsibility in future. There
is typically far more experience in designing IMO marking schemes 
among the jury 
than among the co-ordinators. I can think of several questions which
have had poor marking schemes since I became UK leader in 2002.
This is not because of any ill will or malice on behalf of the Problem
Captains (not in Spain certainly), but rather a lack of experience, or perhaps
a determination to impose uniformity at the expense of natural justice.
 
The jury must take responsibility, and when we are asked to
approve a defective marking scheme, or even a scheme which needs more study,
the jury must say no. I can feel an opening speech at IMO 2009 coming on.  

As the marking phase ends, I get to see more and more of our students.
This is a happy time. Concerning the physical arrangements for sitting the IMO,
our student UNK1 Tim Hennock remarks that the sloping desks are 
an unnecessary additional hazard, and suggests that in future
choosing $\mu \geq \tan \theta$ would make things easier. 

The UK students have been accompanied throughout by a dedicated minder in 
the form of UNK10 Jacqui Lewis of St Julian's School. The teams of students 
have been scattered across central Madrid at seven halls of residence, and 
the facilities vary slightly. 

Our students seem to be in good shape, and Jacqui has clearly cared for 
them well. The British students are full of robust opinions on 
IMO organization which I will not echo in this report.
From what I hear they will not have been very popular with the hosts. As
you move around the world, attitudes towards (and expectations of) 
young adults vary widely. 

At this point we organize the creation of grey residue to be placed
in the {\em Peter Taylor Urn,} the trophy of the Mathematics Ashes. This
involves burning some scripts, and of course we take the matter of safety
extremely seriously. It turns out that by some statistical fluke, Australia
have won the inaugural Mathematics Ashes. In an act of characteristic
duplicity, and in the spirit of Douglas Jardine, the great proponent of {\em Leg Theory}, 
the United Kingdom arranged
that Australian mathematics scripts were also burned and so became 
part of the 
grey residue which will inspire generations to come.

The closing ceremony is held in university buildings 
outside Madrid. Just as the gongs were handed out by the Princess Royal
in the UK, and by the Crown Prince in Japan, we have their Royal Highnesses
the Prince and Princess of Asturias. The Prince's English accent is immaculate,
and our student UNK6 Alison Zhu collects her silver medal from him, thanks 
to some quick counting and repositioning by the ever chivalrous 
UNK5 Dominic Yeo.

A short walk takes us to an alfresco banquet enjoyed by all. As usual, 
the official jury scorekeeper, Rafael S\'anchez Lamoneda of Venezuela
has kept an accurate record of jury speeches, and  
the coveted Microphone d'Or is presented to the most garrulous juror.
My victory in 2006 is but a fading dream, and the Romanian Leader
Radu Gologan's win of 2007 is also a distant memory. The new
champion is the leader of the Netherlands, Quentijn Puite. 

\subsection*{Acknowledgements} Thanks are due to the army of trainers, 
helpers and administrators who sustain
the UK effort at this competition. I also acknowledge financial support from 
Her Majesty's Government, and the close and fruitful
co-operation between the
British Mathematical Olympiad and its parent body, the United Kingdom
Mathematics Trust.

The British Mathematical Olympiad now engages 
in a wide range of regular activities
in addition to our national training camps. We have a joint camp with
Hungary over the new year and a pre-IMO camp with Australia. We compete
as a guest nation in the Balkan Mathematical Olympiad, and were 
one of the nations taking part in the inaugural Romanian Master in Mathematics
competition. These activities enable a large number of students to
participate, not just the six students in the IMO
team. We have also invited overseas students and trainers at our
camps in recent years.
I hope that all of these projects will flourish, and
that more will be created. 

Dr Fiddes hands the Deputy Leadership to Dr Kadelburg. Ceri will now
take on the relatively trivial task of becoming Head of Mathematics
at Millfield School, and so running the largest mathematics staff
of any school in the UK. Thank you Ceri, and to the team, reserves
and Observers of 2008.

\ \ \hfill Geoff Smith UNK7
\end{document}
