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\title{UK IMO team leader's report}
\date{August 2014}
\author{Geoff Smith, University of Bath}

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The International Mathematical Olympiad was held under an 
African sky for the first time.
The event was held in Cape Town, South Africa, during July 2014
and was a great success.
The director, John Webb, is a stalwart of the maths competition circuit, and
bringing the event home to Cape Town is the fulfilment of a long cherished
ambition. The event was held under the auspices of the 
South African Mathematics Foundation,   
but they had help from many sources, including
businesses, charities, foundations, government, private individuals and 
universities.

The IMO is the world championship of secondary school mathematics, and is held 
each July in a host country somewhere in the world. A modern IMO involves
more than 100 countries, representing over 90\% of the world's population.
The competition was founded in 1959. Each participating country
may send up to six team members, who must be under 20 years of age
and not have entered university. 

The University of Cape Town was the students' site, and it nestles
dramatically on the lower slopes of Table Mountain, on land donated by
Cecil Rhodes. It was cold at night, but we were well warned in advance,
and warm jackets were supplied. The organization was very good indeed,
but in an event so large and complex as the IMO, it is impossible
to get everything exactly right (unless you are Dutch). Indeed, that is
the basis of the more cruel aspects of the UK leader's diary which serve to
highlight, dramatise, and if necessary fabricate apparent shortcomings.

The UK Deputy Leader was Dominic Yeo of the University
of Oxford, and our Observer C was Jill Parker, formerly
of the University of Bath.
Here is the UK IMO team of 2014.

\vskip 0.2cm

\begin{tabular}{ll}
Joe Benton  &    St Paul's School, Barnes, London\\
Gabriel Gendler &  Queen Elizabeth's School, Barnet, London\\
Frank Zhenyu Han & Dulwich College, London\\
Freddie Illingworth & Magdalen College School, Oxford\\
Warren Li & Fulford School, York\\ 
Harvey Yau &  Ysgol Dyffryn Taf, Carmarthenshire, Wales
\end{tabular}

\vskip 0.5cm

\noindent First reserve was Neel Nanda of Latymer School, Edmonton, London.

\vskip 0.5cm

For the second year running, each UK student scored at least 20 points.
This has not happened since the run of three
similarly successful years during 1994 -- 1996. We look forward to the
results of IMO
2015 in Chiang Mai, Thailand with interest. Note that Warren Li
has two more years at school, Joe Benton has three, and Harvey Yau
has four. The future prospects of the UK IMO side look healthy.

\vskip 0.5cm

\begin{center}
\begin{tabular}{lcccccccl}
Name &P1&P2&P3&P4&P5&P6& $\Sigma$& award\\
Joe Benton&7&5&1&7&4&0&24&Silver\\
Gabriel Gendler&7&6&0&7&2&0&22&Silver\\
Frank Zhenyu Han&7&4&0&7&2&0&20&Bronze\\
Freddie Illingworth&5&7&0&7&2&0&21&Bronze\\
Warren Li&7&7&0&7&7&0&28&Silver\\
Harvey Yau&7&6&0&7&7&0&27&Silver 
\end{tabular}
\end{center}

\vskip 0.5cm

There are three problems to address on each of two consecutive days.
Each exam lasts 4 hours 30 minutes. The cut-offs were
16 for bronze, 22 for silver and 29 for gold.

There were 101 teams at IMO 2014. Warm congratulations
to China and the USA for excellent performances.
Special celebrations are in order for 3rd placed Taiwan 
because that is their highest ever ranking, and
for Ukraine which equalled its highest ever ranking
of 6th. The 13th
place for the Netherlands is their best rank position since
1983.

Here are a few of the leading scores. 
1 China (201), 2 USA (193), 3 Taiwan (192),
4 Russia (191), 5 Japan (177),
6 Ukraine (175), 7 South Korea (172),
8 Singapore (161), 9 Canada (159),
10 Vietnam (157), 11 Australia (156),
11 Romania (156), 13 Netherlands (155),
14 North Korea (154), 15 Hungary (153),
16 Germany (152), 17 Turkey (147),
18 Hong Kong (143), 18 Israel (143),
20 United Kingdom (142), 
21 Iran (131),
21 Thailand (131),
23 Kazakhstan (129),
23 Malaysia (129),
23 Serbia (129),
26 Italy (128),
26 Mexico (128),
26 Poland (128),
29 Croatia (126),
29 Indonesia (126),
29 (Peru) (126).

Anglophone interest in other scores might include
39 India (110), our neighbours and traditional 
allies 45 France (96),
53 Bangladesh (84), 54 Sri Lanka (82),
60 New Zealand (76), 64 Ireland (67),
64 South Africa (67), 72 Cyprus (53), 
75 Pakistan (50), 82 Trinidad and Tobago (32).
It was simply excellent to see delegations from
Burkina Faso, Gambia, Ghana and Tanzania
competing at an IMO for the first time.

Here are the unusual prizewinners for 2014.
The first country to have its rank higher than its score was Latvia.
The team leader with the most names was from
Brazil (5), only one debatable word ahead of the leaders of
Bolivia, Cuba,
Mexico, Nigeria, Puerto Rico, Vietnam and Uruguay. However, in the 
leaders' nominal accents competition, Vietnam (four with three different)
was a clear winner
since no other leader had more than two accents (but Serbia had
two accents and a crossed D).
Japan was the top constitutional monarchy, 
and Luxembourg managed to retain the
Grand Duchy title. The UK was the 4th Commonwealth
country, which actually indicates excellent performances
from Singapore, Canada and Australia.
Romania topped the European Union and 
Netherlands the euro area.

The second bullet point of General Regulation 2.8
may or may not apply in 2015,
so you might have three IMO teams from the Atlantic Islands
next year.

\section*{The Papers}
Contestants have 4 hours 30 minutes to sit each paper. The three
problems on each paper are each marked out of 7. It is intended that
the three problems should be in increasing order of difficulty on each day.

\subsection*{Day 1}

\noindent \textbf{Problem 1\ }Let
$a_0 < a_1 < a_2 < \cdots$ be an infinite sequence of positive integers.
Prove that there is a unique integer $n \geq 1$ such that
\[ a_n < \frac{a_0 + a_1 + \cdots + a_n}{n} \leq a_{n+1}.\]

\vskip 0.5cm

\noindent \textbf{Problem 2\ }Let $n \geq 2$ be an integer.
Consider an $n \times n$ chessboard consisting of $n^2$ unit squares.
A configuration of $n$ rooks on this board is \emph{peaceful}
if every row and every column contains exactly one rook.
Find the greatest positive integer $k$ such that,
for each peaceful configuration of $n$ rooks,
there is a $k \times k$ square which does not contain a rook on
any of its $k^2$ unit squares.

\vskip 0.5cm

\noindent \textbf{Problem 3\ }Convex quadrilateral $ABCD$ has 
$\angle ABC = \angle CDA = 90^\circ$. Point $H$
is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ 
lie on the sides $AB$ and $AD$, respectively, such that $H$ is inside triangle
$SCT$ and \[\angle CHS - \angle CSB = 90^\circ, \quad \angle THC - \angle DTC 
= 90^\circ.\]
Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.


\subsection*{Day 2}

\noindent \textbf{Problem 4\ }Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$
so that $\angle PAB = \angle BCA$ and $\angle CAQ = \angle ABC$. 
Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, 
such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$.
Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$. 

\vskip 0.5cm

\noindent \textbf{Problem 5\ }For each positive integer $n$, the
Bank of Cape Town issues coins of denomination
$\frac 1n$. Given a finite collection of such coins
(of not necessarily different denominations) with total value at most
$99 + \frac 12$, prove that it is possible to split the collection 
into 100 or fewer groups, such that each group has total value at most 1.

\vskip 0.5cm

\noindent \textbf{Problem 6\ }A set of lines in the plane is in 
\emph{general position} if no two are parallel and no three 
pass through the same point. A finite set of lines in general position 
cuts the plane into regions, some of which have finite area; we call these
its \emph{finite regions}. Prove that for all sufficiently large $n$, in
any set of $n$ lines in general position, it is possible to colour 
at least $\sqrt n$ of the lines blue in such a way
that none of its finite regions has a completely blue boundary.

\vskip 0.5cm

\noindent \emph{Note:} Results with $\sqrt n$ replaced by $c \sqrt n$
will be awarded points depending on the value of the constant $c$.

\vskip 0.5cm

\noindent \emph{Note:} (not in the original) the word finite has been inserted
as the second word of the second sentence of Problem 6 for reasons of mathematical integrity.

\vskip 0.5cm

These questions were submitted to the IMO by 
Austria, Croatia, Iran, Georgia, Luxembourg and Austria respectively.
The American juror Po-Shen Loh suggested this
modified statement of Problem 6.

\section*{Forthcoming International Events}

This is a summary of the events which are relevant for the UK.
Of course there are many other competitions going on in other
parts of the world.

The next few IMOs will be held in Thailand 2015, Hong Kong 2016, 
Brazil 2017,
Romania 2018 and the United Kingdom 2019. Forthcoming editions of 
the European Girls' Mathematical Olympiad will be in
Belarus 2015, Country $X$ in 2016 and Country $Y$ in 2017. In fact 
matters are far more advanced than this cryptic list suggests, and 
EGMO is unlikely to need a new host country until 2018.
The Balkan Mathematical Olympiad will be held in Greece in 2015, 
and the Romanian Master of Mathematics will be in business again
in 2015, now with the Romanian Mathematical Society deeply
involved.

\section*{Diary}
This diary is a facetious summary of my personal experience at the IMO, and 
is not necessarily fair, balanced or accurate. Bits of it are. Note that
Joseph Myers will appear from time to time in the sequel. 
He is a long-time UKMT activist who
volunteered to help with co-ordination in South Africa at short notice
when someone had to pull out. He clearly enjoyed the experience immensely.
He feels most at home when working with people who have at least two
times as many gold medals as himself. 

\subsection*{June 29th} I start from Bath. The new pastoral specialist, our
Observer C, is Jill Parker. She brought up three diverse but excellent
sons, and so looking after the six UK lads should be easy
enough. Sorry, the UK has an all male team again, but at least we sent
four British girls to the 
excellent European Girls' MO in Turkey in April.
An incomprehensible UK Egmoid 2014 report is available via 
Joseph Myers's 
IMO register, together with documents which make considerably more sense.

\vskip 0.5cm

\begin{center}
{\tt http://www.imo-register.org.uk/reports.html}
\end{center}

\vskip 0.5cm

\noindent Unless you were actually there, the Egmoid report
might as well be written in \emph{Linear A}.
There will be a crackdown of awesome brutality, and henceforth
British student reports will be fewer in number, 
and will be less infused with \emph{magic realism}.

Jill and I travel to Heathrow by train and bus, and arrive
in good time despite the best efforts of British drivers
to slow us up by crashing their cars on the motorway.
Our party assembles in Terminal 5, and items of clothing
are distributed. There are some very natty quartered
sports shirts, but unfortunately not in my size.
Dominic is responsible, if that is the word, for the uniform,
and he has selected some virulent lime green shirts in an effort
to confine the wearing of mathematical clothing to 
the actual competition. All items are emblazoned with the 
name of our generous sponsors, \emph{Oxford Asset Management}.
Anyway, my ersatz DayGlo shirt will enable me to attract the
chair's eye in jury meetings.

Freddie Illingworth burst on the competitions
scene last year with a gold medal in the abnormally late 
Balkan Mathematical Olympiad in 2013. It is extremely likely
that he would have been in the UK team for the Romanian Master of
Mathematics in 2014 had it not been cancelled, and this remark 
applies to some other UK team members of course. Freddie was a
reserve for the IMO team of 2013.

Joe Benton's international debut was in the Balkan MO 2014 
in Bulgaria
where he obtained a bronze medal. Joe will have three more
chances to compete in IMOs.

Harvey Yau is our youngest and most Welsh contestant. He made
his debut in Bulgaria this year, obtaining a bronze medal. The maths
competition community have been aware of Harvey for a long time,
and we have been waiting for him to become old enough that we could
reasonably take him to an overseas competition without risk of
attracting the attention of the social services.
He first reached the 
BMO2, the final round of our national mathematical olympiad, two years ago.
He will have four more opportunities to compete in IMOs.

Warren Li is our Yorkshireman. His debut was in the Romanian Master of
Mathematics in 2013 where he secured a bronze medal, and he 
won another bronze in the Balkan Mathematical Olympiad later
that year held in Cyprus, and a silver medal in IMO 2013 in Colombia.
He won a bronze medal in the Romanian Master of Mathematics in 2013.

Frank Han has the Chinese name Zhenyu. He is being educated at 
a boarding school in the UK. His debut was also in the Balkan
Mathematical Olympiad in Cyprus in 2013 where he won a silver
medal. He was a reserve
for IMO 2013 in Colombia.

Gabriel Gendler won a bronze
medal in the Balkan Mathematical Olympiad of 2012
in Turkey. He was a reserve for IMO 2012 in Argentina,
and secured an Honourable Mention at the Romanian 
Master of Mathematics in 2013. He won a silver
medal at IMO 2013 in Colombia.  


The flight from Heathrow to Cape Town is a single hop on
British Airways. I have acquired a wrap-around travel pillow to assist
sleeping, but unfortunately it is designed for 
Keira Knightley, rather than for Geoff Smith or 
a \emph{Sontaran warrior}, and it renders breathing difficult.

\subsection*{June 30th} We arrive at Cape Town Airport at breakfast time. 
The Australians have beaten us by a few minutes, and
the efficient IMO greeters have already sent them 
off to our common base, the \emph{Little Scotia}.
This is an excellent boutique hotel, close to the 
University of Cape Town campus. The setting is delightful.
You look uphill to see the campus, then the Cecil Rhodes Memorial,
and finally the extraordinary Table Mountain. This must
be one of the most dramatically placed universities
in the world.

The Little Scotia is a friendly place, and is guarded by
giant dogs. It is July, and therefore midwinter, and so the
nights are little chilly. However, the IMO organizers have
warned us very well, and those of us (e.g.\ me) too stupid to pay heed to
this information have only ourselves to blame.

We soon find the Australians, who make us cups of tea.
We all visit the \emph{Hussar} steak house at the end 
of the street, and discover that restaurants in South Africa
are very cheap by European standards, and the quality is often 
excellent. The Hussar offers a menagerie based menu, and
might form the basis for another David Attenborough BBC
series: \emph{Mammals on a Plate}. Carnivorous readers
yet to sample a Kudu steak should do so as soon as possible.

In the evening we find a Thai restaurant which allows you to
tune the level of seasoning of dishes on the scale
mild, medium, hot and Thai. Their Thai level 
\emph{Tom Yum Gai} straddles the divide between delicious and
the threshold of pain, like a Turkish Bath for your insides.

It is quite cold that night, and I keep warm by huddling under
the blankets and thinking about Tom Yum Gai.

\subsection*{July 1st} The first Australia 
versus UK exam is held in the morning.
I visit the exam site by means of the campus bus service called the
Jammie shuttle. The IMO Director John Webb is storing 
a large number of packs of UKMT branded playing cards in his office.
We have had problems with customs authorities when having
IMO gifts delivered from the UK. We sidestepped this problem
by having these packs printed in South Africa, and then using
Prof Webb's office as a storage facility.

The Jammie Shuttle bus service is pronounced like the 
popular biscuit, the Jammie Dodger.
It is named for the controversial colonial statesman 
and drummer, Leander Starr Jameson, rumoured (but only by me)
to be the great-grandfather of Ringo. Jameson Hall is the
site of both opening and closing ceremonies of IMO 2014.

The upper campus has two bus stops facing one another.
Having arrived at one, I try to depart using the other,
using my extensive experience of two-way roads.
After four buses drive past the other way, it becomes
evident that the Jammie Shuttle is not really a shuttle at all, but
a Jammie circular, and the second bus stop is just put there to
waste the time of first-time campus visitors.

In the afternoon I assist Dominic with the marking, and happily 
the UK students do quite well. The Australians have Alex Gunning 
on their team.
He has become so good at IMO mathematics that no-one can remember 
the last time he lost a mark on a practice paper. The forthcoming
Maths Ashes looks daunting with Alex on the Australian side.

In the afternoon I recognize Gordon Lessells 
walking past my room, together with the Irish team.
One of their side is the popular
Luke Gardiner who has been training with 
the British students a couple of times during the year.
Where you find the Irish, expect to find 
Trinidad and Tobago close at hand,
and I know that we can expect a Caribbean presence
in the hotel soon.

In the evening I notice something in my eye, and decide not
to worry about it.

\subsection*{July 2nd} John Webb picks me up in the morning, and I expect to
transfer to the teacher training college in the country which
I had been told was the leaders' site. It turns out that there has
been a change of plan, and that instead the leaders will stay at a hotel
on the other side of Table Mountain. Thus security will
be provided by the mountain itself, and its
savage little inhabitants called Rock Hyraxes (Hyraces?)
which are related to
elephants and dugongs. In case anyone tries to go
round the mountain rather than over it, the muggers
of Cape Town are standing by to impose stiff penalties.

There are very few leaders at the hotel when we arrive, so 
I decide to worry about my vision. The reception people 
of the \emph{Garden Court Hotel} kindly arrange a telephone call
with their tame doctor. I describe my symptom, and ask if I can
wait until I get back to the UK to sort out the black object in
my right eye, or whether I should seek attention immediately.
He is extremely clear about the need for haste, and I go into
controlled panic mode. As Joseph Myers sails in, I am issued with
a minder in the form of Phil Labuschagne. He drives me to the Cape Town
Medi-clinic, and stays with me for an hour or two while they
do the preamble. We spend a happy time filling out forms
and having my blood-pressure taken, until finally the nurse
says that Dr Donelly will try to fit me in. At this point I decide
that poor Phil should be released, and he returns to the IMO.

I get a dose of astringent to dilate the pupil of my right eye
and wait to be seen. In fact it doesn't take too long.
Dr Donelly is admirably robust, and begins by enumerating 
possible causes for my symptom. It is like a list of possible
prizes in an unlucky-dip. Everyone gets to win one, but you don't
know which it will be. There are a few things on the list which
I think I might skip (don't worry, we could easily sort that out with laser
surgery), and finally he comes to the least interesting but
in every way the most attractive option of a very large floater in the 
vitreous humour. Having set the stage for the forthcoming drama,
he then bends forward and has me perform the dance of the eyeballs 
while he peers
and considers. He quickly determined that it was only a floater, and
that there no serious threat to my vision. Thank you for a job
well done Dr Donelly.

I return to the IMO leaders' site by taxi, and by now the
building is starting to fill up with leaders, co-ordinators
(marking police) and organizers. I have a copy of the 
shortlist, and life doesn't get much better than this.

\subsection*{July 3rd} This is the normal day for leaders to 
arrive, but I spend
most of it in conclave with the other members of the IMO Advisory
Board. We talk a lot, and make some good decisions, and listen
to a presentation from Thailand about their plans for IMO 2015.
I am intimidated. At one point I present the offer to host
IMO 2019 in the UK, and then withdraw from the meeting while the
matter is discussed. Happily, for me and my dear friends in 
the UK Maths Trust, the IMO AB decides to recommend to the jury that the
offer be accepted. So, just the one more hurdle to go, my presentation
at the final jury meeting.

We consider various matters concerning IMO policy, and the development
of the IMO Foundation. Mike Clapper explains the progress that has
been made during the year. 

\subsection*{July 4th} Many leaders work on the problems, but there is some
introductory jury business. I have noticed that in the shortlist,
$G1$ surrenders to a short areal calculation, and point this
out to the jury. I supply the proof to the co-ordinators, who 
promptly and sensibly request a tutorial on areal (barycentric) 
methods. Various problems are turfed out by the jury
in the next couple of days for being
too close to problems which have been used before. 

There is a debate on the problem selection protocol.
Last year we introduced a new one, designed (by me)
to ensure that problems 1, 2, 4 and 5 all come from 
different areas of mathematics. There is substantial
support for this protocol, but it will be tested nearly 
to destruction by the shortlist of IMO 2014. The 
difficulty is that the shortlist has 9 official 
combinatorics problems on it, and additionally many of
the problems in both the algebra and number theory sections
are clearly combinatorics problems dressed up to look like something
else.

This problem is compounded when the jury sets about ejecting
some genuine algebra and number theory problems on grounds
of prior art, so we are left with geometry problems, and 
a sea of combinatorics. Unsurprisingly, we find ourselves
in difficulty when constructing a paper. Some people
claim that this is a shortcoming of the 2013 protocol, but
in my view this is nonsense. We have a defective 
and unbalanced shortlist,
and no voting protocol can remedy that situation by magic.

There are people who do not like the 2013 protocol for
a specific geometric reason. Some people want 
two geometry problems at each IMO, and would like them
to be one easy, and one medium. For people who love
geometry, this does look very attractive, but it would
have the consequence that at least one of the four 
IMO subject areas would not appear in an easy or medium
position in such an IMO. My judgement is that there are more
jurors who are concerned to have a balanced set of accessible 
questions than there are jurors who are determined to have two
accessible geometry problems on each IMO. 

Anyway, this is up to the jury to decide. The 2013 protocol
does what it says on the tin. You get a balanced paper, and that
means at most one geometry problem
in the easy and medium positions. If the jury wishes to have
two geometry problems on the paper, then if using the 2013 
protocol, it follows that one of them must be a problem 3
or a problem 6. This is the real matter at issue. 

The excellent jury chair Dr Sizwe Mabizela
has a genial manner,
and likes to begin jury meetings with a homily related
to the World Cup. In the evening he reflected
on the passage of Germany through to the 
semi-finals.

\subsection*{July 5th} In the morning, Sizwe Mabizela's 
``thought for the day'' concerns the progress of 
Brazil at the World Cup.

The jury is using
the 2013 voting protocol for the second time, and we choose the paper
in the afternoon. Problem 6 has been modified to introduce
a constant $c$, and points will be awarded according to the 
value of $c$ which is attained. Since this is a new departure
for the IMO, I can see that the wording is going to be
hotly contested.

At length the results of the \emph{Mathematics Ashes} appear on the internet,
and Joseph Myers briefs me with the results. Unfortunately the UK has won.
Again. Presumably our students have slipped a sedative into Alex
Gunning's breakfast cereal to even things up a bit. I find the
Australian leader Angelo di Pasquale, and
helpfully offer to set up some kind 
of mathematics training system for Australian
students.

After dinner, the English language committee meets with
the usual suspects as prominent as ever. The wording of Problem
5 is a nightmare. Must coins in a pile all have the same value?
Must coins in a pile all have different values? We invent
the notion of the Bank of Cape Town and strive to be 
as precise as we can. By 10pm local time, five questions 
are done, and I have had enough of trying to referee
this bunch of over-educated pedants, and even more 
importantly, the Netherlands are about to play Costa Rica
in the World Cup. I cede the chair to Po-Shen Loh (who dreamed
up this ``points for the value of $c$'' business) and
suggest that as the expert in the question, he should chair the
meeting. I stroll happily towards the bar and the television, 
whispering to 
the committee that they should not allow him to mess around with
the spelling of the word ``colour''.
 
\subsection*{July 6th} This morning I present the work of the English Language 
Committee to the jury. We get through the first five problems quickly and 
without
incident, but I can see the linguistic vultures circling 
over Problem 6, so when we get there I ask Po-Shen to take over 
the presentation, and return to my seat to enjoy the
bloodbath. After about three hours of debate, the 
jury eventually decides that the original ELC wording was 
optimal. This is normal jury behaviour. 

Now the jurors get down to serious work, translating
the problems into the other 53 language versions, sometimes
via the translations of the other official IMO languages:
French, German, Russian and Spanish. In the evening
the jury ignores requests to revisit the original wording
of problem 6 on the ground that we have a competition to run.

\subsection*{July 7th} Marking schemes are presented by the extremely
intelligent and co-operative co-ordinators. The jury
recommends some changes and these are adopted. 
The opening ceremony is scheduled for 17:00. The leaders
are dropped next to the Rhodes memorial on campus. 
Cecil Rhodes is another person about whom it is possible
to hold more than one opinion. He donated the land on
which UCT is situated, with the proviso that
it had to be used for a university. Nice one Cecil.
That stuff about colouring the map pink goes down less well.

We are in great danger of meeting contestants, and that is not
allowed because we know the papers. South African heavies in 
green blazers usher us into a holding pen in the maths department, 
and I have 
a very happy time chatting with the delegation from India.

Eventually we are allowed in to Jameson Hall. The
ceremony begins with a moving rendition of 
\emph{Nkosi Sikelel' iAfrika} in its modern form,
with passages in Xhosa, Zulu, Sesotho, Afrikaans and English.
There is a 
great deal of drumming, presumably to honour the connection
with Ringo, and some excellent dancing and clicking of fingers.

The UCT Vice-Chancellor Max Price gave a splendid welcome speech,
and Nazar later finished the event with a few well-chosen words.
In between I was protected by recently developed
auditory technology which is a synthesis
of Zaphod Beeblebrox's
\emph{Joo Janta 200 Super-Chromatic Peril Sensitive Sunglasses} and
the \emph{Babel fish}. This enables me to not hear things (in any language)
which I might not have found interesting.

The parade of nations is splendid. It is cleverly organized, with
extant countries appearing in the order of their first participation.
Thus the UK appears quite early as one of the class of 1967.
The team is beautifully dressed. Joe Benton has taken over from
Sahl Khan as the person on Gabriel's shoulders, and the union flag 
streams from Joe's back. The UK team hurl vast numbers of sweets
at the audience.

Half way through the parade it is South Africa's turn,
but they break the procession, and have a splendid special 
section for the host nation.
When the parade resumes, we see the 1990s realignment of 
geopolitics reflected in new nations participating in the IMO.
Finally, the last few countries are from Africa. It is magnificent.

\subsection*{July 8th} Today is the first day of the contest, and the 
students are allowed to ask questions of clarification for the first 30
minutes. Normally this is a mess, and the local organizers
get things working for the next day. However, the local
organizers have had a practice run, to make sure that everything
will work smoothly. In a triumph of reality over planning,
it is a disaster. Something somewhere is not plugged in, or possibly
the internet connection is using semaphore to get over 
Table Mountain. We spend much of the morning answering questions.
When it is eventually over, people are seriously discussing 
using the internal combustion engine as the communications device
on day 2. It is only a 10 minute drive round the mountain to the 
exam site.

Now is the time scheduled for the IMOAB elections, but 
since it is just before lunch, we just have the 
election for Chair. Nazar and I keep the speeches
short and light. We are off to the Botanical Gardens for
lunch, and while everyone boards the buses,
the votes are quickly counted but kept secret.
The result is announced at lunch, and it is very close.
I have won, but if just a handful of people had voted
the other way, Nazar would have won. Of course we remain 
excellent friends.

There is some horseplay among the more immature leaders, and 
an African-style crown is placed on my head and photographs are
taken. There are pictures on the internet. 
Angelo di Pasquale, team leader of Australia, attempts
to shake my hand. This is, of course, inappropriate, and I
offer him the back of my hand for the normal
osculatory supplication, but I notice that he is unenthusiastic.

We complete our journey with a visit to the
big Cecil Rhodes memorial on the side of Table Mountain. 
The word ostentatious is an understatement. We manage
to use its steps to get a group photo, also available on the internet.

In the evening we have an election for the other members of the
IMO Advisory Board, and this results in a surprisingly close contest.
The new IMO AB is at
{\tt http://www.imo-official.org/advisory.aspx}

\subsection*{July 9th} Contest Day 2 As usual, the communications 
difficulties
have been sorted out, and the Question and Answer session for day 2 is smooth.

We transfer to the students' site by bus, and this is another
opportunity for suboptimal procedure. The road map of 
the UCT campus is complex, and my driver's strategy is try to find the
leaders' hall by means of a random drive. We take three tours round the
campus until we are dropped near the students' hall. The leaders 
have foolishly not memorized the halls in which they are staying,
despite the fact that they had been occasionally projected onto
the video screens at the other site. Most leaders head pointlessly towards the 
students' hall, but I go the other way (a useful tip at IMOs) 
into Baxter Hall. This is not correct either, but it is
not completely incorrect, and I am rewarded with a water
bottle and a woollen hat. I then make my way to
Gra\c{c}a Machel hall where I am staying, thereby beating the pack.

I have told John Webb about the chronic knee pain which I
suffer bravely and without complaint, so it comes as no
surprise that he has arranged for me to have a room on the top
floor of Gra\c{c}a Machel hall, with no lift. I return
to reception and arrange for a change of room.
I bump into the Hungarian leader J\'ozsef Pelik\'an, and he shares my delight 
in the mountainous setting, and the amusingly steep slopes.
Like me, J\'ozsef is in excellent physical condition, so we play some games
of leapfrog in the hotel car park. There are cartwheels and handstands.
What joy it is to be alive!

It is a delight to see the UK students again, 
and to catch up with Dominic and Jill and listen
to their experiences.

There was supposed to be a video screen supplying information
to the leaders, but it was not to be found in the leaders'
residence hall.
The method of
disseminating official information was therefore a curious mixture
of gossip and osmosis. The leaders know that co-ordination is next
day, but have only a hazy idea of where it will be. Rumours spread that it
was in a building called ``Kramer''. At first it seemed that we had
no map, but I learned that if you prized open your lanyard, hidden inside
was an incorrect map of the layout of the campus at an inappropriate
scale. The word in the dining room was that you could cross the big scary
road between us and the Kramer building by means of a pedestrian bridge.
Said bridge did not appear on the
map. I asked Gra\c{c}a Machel folk to explain the route, and they
told me to go via the footbridge. I explained that I was troubled
by the lack of a footbridge on the map. They then explained
an alternative and much longer route which did not go via the
footbridge. I observed that since they were clearly of the
opinion that the bridge existed, it would be more helpful
to tell me where the bridge was, rather than
tell me where it wasn't. Both parties clearly thought the other mad,
and we quietly terminated the conversation.
  
Dominic and I shared the co-ordinating load. Our students
had done little on the hard questions, but scored very well
on the easy questions, and had mixed performances
on the medium problems.

We heard that one of our students was unwell, so Jill and 
Dominic went into pastoral overdrive. At one point an ambulance 
would be called, but in the event no transfer to hospital was necessary. 

\subsection*{July 10th \& 11th} Co-ordination went quite well. I made sure that
our rich harvest of marks on problems 1 and 4 was safe, and picked
up a scrap on Problem 3. Dominic worked hard to understand
and make good cases for the marks on the medium problems. 
He did a fine job, and we completed co-ordination very amicably.
There had been an issue with Problem 2 for a while, where 
a UK student had solved the problem correctly (with an agreed 
2 point flaw), but had then simply written down the wrong formula.
Dominic felt that it would be unjust to punish this, and 
we were in an unlikely alliance with France who had a student
who had solved the problem completely, but did not describe 
the answer by means of a formula. In the end, the problem 
captain sorted it out to our mutual satisfaction.

In the evening of July 11th there was a jury meeting. 
An intruder was ejected.
First there was the
joint meeting of the IMO Advisory Board and the jury. 
Mike Clapper explained about the progress with 
the IMO Foundation. Rachaya Srisurichan presented
her plans for IMO 2015, and the facilities looked so attractive
that the jury burst spontaneously into flames or possibly applause.
I presented the offer to host IMO 2019
in the UK. This was accepted, but I think that if the
jury had a free choice, they would have IMO Thailand every year.

After the joint IMOAB and jury meeting, we focus on the serious
matter at hand, the setting of medal boundaries. I was not completely
impressed by the levels of disinterest in some of the speeches
advocating particular distributions of thresholds. I would not be
surprised if there were some procedural changes next year.  

When the smoke clears, our own narrow concern is that we have four silver
medals and two bronze medals. Dominic, Jill and I are extremely
pleased with this performance, since our lowest score is 20/42.
Moreover, the three highest scores are from our three youngest students.
They can usefully spend the winter trying to close the gap on Alex Gunning
for the Mathematical Ashes just before IMO 2015. 

Our Irish colleague Luke Gardiner has just missed a bronze
medal, and has the rare distinction of a double 
honourable mention, having solved two problems completely,
but not won a medal. Fortunately Luke has another chance
next year.

We still have one UK student who is indisposed, so our attempts to celebrate
the results are a little muted.

\subsection*{July 12th} Today is the major IMO excursion. 
I am the sole UK person to arrive on time.
An outing to a shopping mall is not ideal, but the infrastructure
simply would not support a whole IMO visiting
either Table Mountain or Robben Island simultaneously.
The V\&A Waterfront contains various delights, including a museum of 
rugby. There is a harbour, and so I go on a boat trip. There are over
1000 harbour seals, and when I see over-quota fish being thrown into the 
water, it becomes clear what sustains this huge population. There
are other treats, such as a Ferris wheel and an aquarium. I have
a slow lunch with the Romanian leader Radu Gologan, and we solve
many of the world's problems, at least those connected
with mathematics competitions. Radu is being extremely helpful
regarding plans for future editions of EGMO.

I study the helicopter rides on offer. For a serious amount of money
you can go on a 20 minute ride around Table Mountain. However, for a stupid
amount of money you can take a genuine Vietnam War era US army surplus
Huey helicopter, and enjoy a simulated combat mission screaming low
over the trees on the west coast. I see that commercial possibilities
are being missed here. It should be that for a ridiculous amount
of money you can use the Huey copter to search rural South Africa 
for big game. You would then strafe a kudu, land the chopper close
by, and the crew would get the barbecue going 
while you had a profound
discussions with the \emph{sommelier}.

Eventually Dominic and the walking members of the UK IMO team
show up, and experience what passes for fast food in South Africa.
\emph{Hint: restaurants are quicker}.

In the evening we have the closing ceremony. This initially involves
being locked out of Jameson Hall while listening to, or perhaps
feeling, a drumming exhibition. At last we are allowed in, and the
other members of the IMOAB all promptly sit in the wrong
place. The head of protocol looks worried, and I negotiate a deal whereby
Nazar stays where he is, and the rest of us move to the end of the front 
row. This puts us in a place where the sound system is not really 
functioning. We can follow the music, but speech is completely mangled.

Students from all over the world collect their medals, and our 
silver medallist Gabriel Gendler carefully steps out of line to get
his gong from me.

When the ceremony is over, we follow a train of fires to find the reception,
generously equipped to address my needs. The leader of Venezuela,
Rafael S\'anchez, is the chair of the Microphone d'Or committee, which
enumerates jury interventions and awards a trophy to the most
frequent speaker. Note that this a scientific process, and is
completely linguametric. No arbitrary personal judgements concerning
quality or intelligence are involved. It is only quantity which matters. 
I win again, following my earlier triumphs in Slovenia 2006
and Germany 2009, and am presented with a culturally appropriate
golden vuvuzela.
The party continues, and Frank Han is looking very well refreshed.

Harvey, Jill and I seek an early exit via the unannounced and 
unexplained late night bus shuttle. We ask for local help and 
are sent in exactly the wrong direction. 
We are very pleased to get back to our residences in the end.
Dominic sweeps up the rest, and they return soon after.

\subsection*{July 13th} Dominic and Frank Han have non-standard
flights out of the IMO, and they leave in the morning.
The balance of the party, including Joseph Myers, have a flight home 
in the evening, so we spend a
gentle day doing very little, and this is most welcome. Our indisposed
student is now disposed. In the late afternoon we transfer to the airport,
and have a relaxed journey home. For once I manage to sleep on the plane,
something I can rarely do in cattle class.  

\subsection*{July 14th} When leaving the aircraft at Heathrow,
Joseph and Jill manage to pass the main 
group while we are waiting for them, and our more socially incontinent
students prove elusive at baggage collection. However, we are soon
reunited, and have a certificate presentation ceremony
in the arrivals hall.

\section*{Girls}
The UKMT effort to promote interest in maths competitions amongst girls
continues. It is pleasing to observe 
that many female IMO 2014 medallists, including the unique girl
to win a gold medal, Michelle Sweering of the Netherlands,
were former EGMO medallists. Two medal winners in the Ukrainian IMO team
were members of their triumphant EGMO 2014 team in Antalya, Turkey. 

We now have a dedicated girls' summer camp in Oxford to 
supplement our other camps,
and a talent search exam, the UK \emph{Mathematical Olympiad for Girls}
to try to locate promising female students. I urge other nations to develop
their own structures with a similar purpose.

\section*{Acknowledgements}
There is a vast competitions, mentoring, camps and publications apparatus
which underpins the IMO team. Almost all young mathematicians in our country
make contact with UKMT via the excellent Mathematics Challenges which form
the heart of our national effort. These Challenges are the foundation of a
structure which tries to stimulate, develop and encourage enthusiastic
young mathematicians, both inside and outside the classroom. I pay tribute
to everyone involved in helping to make this possible: the hundreds of UKMT
volunteers, the small band of professional UKMT administrators, and 
the families whose lives we disrupt with our camps and competitions. 

Dominic Yeo was a brilliant deputy leader, and our observer 
Jill Parker managed our
crises with discretion and calm aplomb. Many thanks to them both. It was also
a pleasure to have Joseph Myers around, though he was not part of the
UK delegation.

The team acquitted themselves perfectly throughout, both mathematically
and in terms of behaviour, and it has been a pleasure to work
with them.
The reserves were all splendid.

I must specifically thank \emph{Oxford Asset Management} for their generous
sponsorship of the UK IMO team, and the other donors, both individual 
and corporate,
who give so generously. Why not join in?

\begin{center}
{\tt http://www.ukmt.org.uk/about-us/}
\end{center}

\vfill

\hfill {\tt G.C.Smith@bath.ac.uk\ \ \ \ @GeoffBath}
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