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\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
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