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\title{IMO 2016 UK Team Leader's report}
\author{Geoff Smith, University of Bath}
\date{}
\begin{document}
\maketitle

\section*{ }
The International Mathematical Olympiad is the world championship of secondary
school mathematics. It is a nomadic festival, and is held every July. 
Each country or territory may send up to six students, and these contestants
face two exams sat on consecutive days. Each exam lasts 4 hours 30 minutes,
and each paper consists of three problems. In terms of problems which
students may face as part of a normal secondary education, the three
problems on each paper would be successively rated: 
\emph{very hard indeed}, \emph{completely impossible}
and \emph{surely you are joking}. However, the students who participate
at the IMO are among the strongest young mathematicians of their generation,
so for them we adopt the inaccurate category names 
\emph{easy}, \emph{medium} and \emph{hard} for the problems.

Dominic Yeo has taken the lead role in UK team training this year,
and any reflected glory from the UK team's splendid performance
at IMO 2016 in Hong Kong shines upon him. It makes sense to regard
his report as the principal one for 2016, with mine containing a few
incidental observations.

The British team was found through a selection process based
on performances in test exams. They were

\medskip

\begin{tabular}{ll}
Joe Benton & St Paul's School, Barnes, London\\
Jacob Coxon & Magdalen College School, Oxford\\
Lawrence Hollom & Churcher's College, Petersfield, Hampshire\\
Warren Li & Eton College, Windsor\\
Neel Nanda & The Latymer School, Edmonton, London\\
Harvey Yau & Ysgol Dyffryn Taf, Whitland, Carmarthenshire, Wales
\end{tabular}

\noindent and the reserves were

\begin{tabular}{ll}
Rosie Cates&Hills Road VI Form College, Cambridge\\
Michael Ng&Aylesbury Grammar School\\
Thomas Read&The Perse School, Cambridge\\
Renzhi Zhou&The Perse School, Cambridge
\end{tabular}

\medskip

The IMO of 2016 took place in Hong Kong, a Special Administrative Region
of China. Dominic Yeo (Oxford University) was Deputy Leader and 
principal organizer of team training camps. Jill Parker was Observer 
with students, and I was UK team leader, sitting on the IMO Jury.
As well as the official UK delegation, there were another
couple of UKMT
volunteers in Hong Kong. James Cranch and Joseph Myers are there
as experienced anglophone co-ordinators, and Joseph has an extra brief
to shadow IT specialist Matja\v{z} \v{Z}eljko to build some redundancy into
IMO computer support.

Here is a table showing the performances of the British students:

\medskip

\begin{tabular}{|c|cccccc|c|c|}
\hline
 & P1&	P2&	P3&	P4&	P5&	P6& $\Sigma$ & Medal 
\\
\hline
Joe Benton&7&7&2&7&2&1&26&Silver medal\\
Jacob Coxon&7&1&0&6&7&3&24&Silver medal\\
Lawrence Hollom&7&7&0&7&3&0&24&Silver medal\\
Warren Li&7&7&2&7&7&3&33&Gold medal\\
Neel Nanda&7&7&0&7&2&7&30& Gold medal\\
Harvey Yau&0&7&0&7&7&7&28&Silver medal\\
\hline
        &35&36&4&41&28&21&165& \\
\hline
\end{tabular}

\medskip

The cut offs are 16 for bronze, 22 for silver and 29 for gold.
Here are the problems:

\section*{Day 1}

\noindent \textbf{Problem 1\ }Triangle $BCF$ has a right angle at $B$.  Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen such that $DA=DC$ and $AC$ is the bisector of $\angle DAB$. Point $E$ is chosen such that $EA=ED$ and $AD$ is the bisector of $\angle EAC$.
Let $M$ be the midpoint of $CF$.  Let $X$ be the point such that $AMXE$ is a parallelogram (where $AM \parallel EX$ and $AE \parallel MX$).  
Prove that lines $BD$, $FX$, and $ME$ are concurrent.

\noindent \textbf{Problem 2\ }Find all positive integers $n$ for which each cell of an $n \times n$ table can be filled with one of the letters \textit{I}, \textit{M} and \textit{O} in such a way that:
\begin{itemize}
\item
in each row and each column, one third of the entries are \textit{I}, one third are \textit{M} and one third are \textit{O}; and
\item
in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are \textit{I}, one third are \textit{M} and one third are \textit{O}.
\end{itemize}

\noindent
\textbf{Note}: The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order.
Thus each cell corresponds to a pair of positive integers $(i,j)$ with $1 \leqslant i,j \leqslant n$. For $n > 1$, the table has $4n - 2$ \emph{diagonals} of two types. A diagonal of the first type consists of all cells $(i,j)$ for which $i+j$ is a constant, and a diagonal of the second type consists of all cells $(i,j)$ for which $i-j$ is a constant.

\noindent \textbf{Problem 3\ }Let  $P = A_1A_2 \dots A_k$ be a convex polygon in the plane. The vertices $A_1$,~$A_2$,~\dots,~$A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$.  An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.


\section*{Day 2}

\noindent \textbf{Problem 4\ }A set of positive integers is called \textit{fragrant} if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n) = n^2 + n + 1$. What is the least possible value of the positive integer $b$ such that there exists a non-negative integer $a$ for which the set
\[
\{P(a+1), P(a+2), \dots, P(a+b) \}
\]
is fragrant? 


\noindent \textbf{Problem 5\ }The equation
\[
(x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016)
\]
is written on the board, with 2016 linear factors on each side. What is the least possible value of~$k$ for which it is possible to erase exactly $k$ of these 4032 linear factors so that at least one factor remains on each side and 
the resulting equation has no real solutions?

\noindent \textbf{Problem 6\ }There are $n \geqslant 2$ line segments in the plane such that every two segments cross, and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it, facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps, each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. 

\begin{enumerate}
\item[(a)]
Prove that Geoff can always fulfil his wish if $n$ is odd.
\item[(b)]
Prove that Geoff can never fulfil his wish if $n$ is even.
\end{enumerate}

\medskip

\section*{Extra Information}
The word ``fragrant'' refers to the meaning of ``Hong Kong'', 
which is ``fragrant harbour''.
The problems were proposed by
Belgium, Australia, Russia, Luxembourg, Russia and the Czech Republic 
respectively. The authors were Art Waeterschoot, 
Trevor Tao, Alexandr Gaifullin, Gerhard Woeginger,
Nazar Agakhanov \& Ilya Bogdanov (so two authors for Problem 5) 
and Josef Tkadlec. 
The answer to the obvious question about the second composer is ``yes''.

\section*{Analysis of Results}

Here are the countries ranked in the top 21 at IMO 2016.
There were a total of 109 countries sending contestants.

\medskip

\begin{tabular}{|r|l|l|l|}
\hline
Rank & Name & Points & Medals\\
\hline
1 & United States of America  & 214 & GGGGGG\\
2 & South Korea & 207 & GGGGSS \\
3 & China & 204 & GGGGSS \\
4 & Singapore & 196 & GGGGSS \\
5 & Taiwan & 175 & GGGSSS \\
6 & North Korea & 168 & GGSSSS \\
7 & Russia & 165 & GGGSB \\
7 & United Kingdom & 165 & GGSSSS \\
9 & Hong Kong &  161 & GGGSSB\\
10 & Japan & 156 & GSSSSB\\
11 & Vietnam & 151 & GSSSSB\\
12 & Canada & 148 & GGSSBH\\
12 & Thailand & 148 & GGSSBH\\
14 &Hungary & 145 & GSSSBB\\
15 &Brazil & 138 & SSSSSB\\
15 &Italy & 138 & GSSSHH\\
17 &Philippines & 133 & GGSSHH\\
18 &Bulgaria & 132 & SSSBBB\\
19 &Germany & 131 & SSSBBB\\
20 &Indonesia & 130 & SSSBBB\\
20 &Romania & 130 & SSSSSB \\
\hline 
\end{tabular}

\medskip

This was the first
ever top ten performance by the host territory, and the first 
top 20 performance by the Philippines. 

The UK team collectively scored 165 points, putting them 
joint top of Europe with Russia, the team with which
we shared 7th place. This is the best UK performance 
since 1996 when the UK finished $5$th. 

Other continental champions were Africa: South Africa,
Asia: Republic of Korea,
Australasia: Australia,
North America: USA and 
South America: Brazil.

Bitter local rivalries include the annual Nordic knife fight:
Sweden (1), Denmark (2), Norway (3), Iceland (4).
The Baltic republic bar brawl finished: Lithuania (1), Estonia (2)
and Latvia (3).

The monarchy roll of honour (taking a liberal view of Grand Dukes,
Princes and elected Kings)
is UK (1), Japan (2), Canada (3), Thailand (4), Australia (5),
Sweden (6), Saudi Arabia (7), Netherlands (8), Spain (9), Belgium (10),
New Zealand (11), Malaysia (12), Morocco (13), Denmark (14), Norway (15),
Luxembourg (16), Cambodia (17), Jamaica (18) and Liechtenstein (19).

Other European countries which performed well were
Hungary [14], Italy [16], Bulgaria [18] and Germany [19]. 
Romania had, by their high standards, a modest year in position [20].

The results of Commonwealth and anglophone countries (taking 
an extremely relaxed interpretation of the word anglophone) were
USA [1], Singapore [5], UK [7], Hong Kong [9], Canada [12], Philippines [17],
Australia [25], India [34], Bangladesh [35],
New Zealand [53], Malaysia [56], South Africa [58], Cyprus [63],
Sri Lanka [66], Ireland [75], Nigeria [88], Trinidad and Tobago [88],
Myanmar [96], Uganda [98], Kenya [99], Jamaica [102], Ghana [104],
and Tanzania [106]. Note that many of the teams ranked lower on this list
did not send a full team of six students.

Personally I received a fair amount of teasing about Brexit, and
its supposed impact on team performance. This teasing
mostly and mysteriously stopped 
after the IMO results were published. Spurred by this insolence,
I can report that the medal count of the UK and that of the
best six students of the rest of the European Union were equal;
two golds and four silvers in each case. However, it must be admitted
that the Rump European Union outscored the UK 178 to 165.
The REU team consisted
of Attila G\'asp\'ar of Hungary (35), Filippo Gianni Baroni (31)
from Italy, and four participants chosen from
a set of two Czechs, a French person, an Italian and two Swedes,
all of whom scored 28 points.

\section*{Diary}
\textbf{July 4\ }
As I walk across the concourse outside Heathrow T5 I hear the 
characteristic tones of Richard Quest, CNN's poster boy for 
business class travel. As you probably know, he sounds 
like a concrete mixer on steroids. I am just wondering why someone has 
parked a TV tuned to CNN outside T5 as I turn my head to the 
right to see Richard Quest in person, doing a piece to camera. 
I muse as to whether I should linger, and offer to give CNN my views
on the global crisis, Brexit, the US election,
the future of the Labour Party
and whether or not 0 is a natural number. Perhaps the world is not yet ready.
The flight from West to East is not happy, and I am not sleepy. 
I watch five films
of progressively inferior quality.

\noindent \textbf{July 5\ }I arrive in Hong Kong and am met by the 
friendly and efficient organizers.
The drive to the Kowloon Harbour Hotel is longer than I had anticipated.
There is a lot of geography in Hong Kong. 

My arrival at the Kowloon Harbour Hotel involves a minor disaster, 
as I succeed in
breaking the bottle of single malt Scotch which I had somehow acquired
in Heathrow duty free. I check in, and the staff attempt to explain
everything at once. I am in shock from the loss of whisky combined
with sleep deprivation, and grasp lots of pieces of paper in the hope that
they may be relevant.

\noindent \textbf{July 6\ }Next day we have the exciting all day meeting of the IMO 
Advisory Board sitting in committee.  
Since Joseph Myers does not have enough to do already
(IT, co-ordination) the IMOAB ropes him in to take notes during
their day long committee meeting. This has the advantage (or disadvantage)
of allowing
IMOAB secretary Gregor Dolinar to spend more time contributing to
the discussions.

First item on the agenda
is the American offer to host IMO 2021. After several attempts,
we establish a decent Skype link to Michael Pearson of the 
Mathematical Association of America in Washington DC, but unfortunately
the US leader Po-Shen, who should be here in Hong Kong at the IMOAB
meeting, is not physically present. This is in contrast
to my state of being not mentally present. It turns 
out that Po-Shen has also underestimated the impressive quantities of
Hong Kong geography, and he arrives a few minutes late. 

Next comes the presentation from Nazar Agakhanov of the offer by the
Russian Federation to host IMO 2020 in St Petersburg. After that 
we process a huge amount of IMOAB business, over which I draw a veil.

In the evening I get hold of the IMO shortlist, and start to 
enjoy the problems. 

\noindent \textbf{July 7\ }Over breakfast I can now start to 
make social contact with large numbers
of other team leaders. This is a very happy time of the year,
when we meet again after a year apart (the ideal basis for
a social relationship).

There is one small disadvantage for me, because I am the 
elected chair or president of the IMO advisory board.
This means that whenever someone has a problem, then it is very tempting
to share it with me. This is because I have magical powers, 
and can make problems go away instantly. Also some people have
ideas (a dangerous side effect of airline travel), and it is 
well known that meal times will be improved if they share 
these excellent
suggestions with me. 

We begin with a jury meeting in the well appointed 
jury facility. The meetings are chaired by Kar-Ping Shum, who often
delegates matters of detail to his assistant Andy Loo. 
It turns out that Andy enjoys decorating the proceedings with 
spurious formality, swinging freely between vague allusion to
Robert's \emph{Rules of Order}, and light pastiche of
Erskine May's \emph{A Treatise upon the Law, Privileges, 
Proceedings and Usage of Parliament}. 

There is some initial 
discussion concerning the schedule for the day, and eventually Kar-Ping Shum
endorses the jury's wish to work on the shortlist all day. In the evening
we receive the solutions.

\noindent \textbf{July 8\ }We discuss the merits of various questions, and 
eventually select a paper. The problem selection protocol I suggested
in 2013 continues to enjoy wide support in the jury. I am slightly surprised
that only one geometry problem makes it on to the paper, since 
most years there are two such problems. I think that 
the \emph{zeitgeist} is that we should choose problems primarily on
merit, and not be so constrained by the problem area. This may have
happened because the 2013 protocol guarantees a wide range of problem
areas will appear in positions 1, 2, 4 and 5, so that issue can be 
left to the algorithm, and jurors can think more about suitability, elegance 
and beauty.

\noindent \textbf{July 9\ }The English language committee (i.e.\ those members of the
jury who have nothing better to do) meets after breakfast. I have prepared 
draft versions of a possible paper but I do far too much talking
at the IMO, and my beloved colleagues Michael Albert of New Zealand and 
Angelo di Pasquale of Australia kindly agree to conduct the meeting. Problem 6
has my name in it, but we suggest that other countries use whatever
name suits them best. The Dutch leader Julian Lyczak seems to have made
the best use of the opportunity, and the person involved in
Problem 6 for Dutch students is Amalia. This honours
HRH Catharina-Amalia, Princess of Orange. Thus she becomes the princess with 
the frogs. However, I am pleased to discover that ``Geoff'' is a suitable 
Chinese name.

There are substantial and fruitful discussions 
between the jury and the co-ordinators concerning mark schemes.

We also have the annual joint meeting of the IMOAB and the jury.
The jury accepts the recommendations that
they take up the generous offers by the Russian Federation
and the United States of America to host IMO 2020 and IMO 2021
respectively. These offers are greeted by warm applause.
Changes to regulations are discussed and in some cases approved. 

There is an announcement by the Ethics Committee that a historical
case had been discovered where a student competed at IMOs
while in breach of educational status regulations. The student's
name has been removed from the official record. It is clearly
the responsibility of team leaders to ensure that contestants
adhere strictly to all IMO regulations, and if in doubt, consult the
secretary of the IMO Advisory Board.

Before the IMO Ethics committee was introduced, there were
occasional irregular incidents at IMOs. Let everyone be aware
that the Ethics Committee has teeth. 

\noindent \textbf{July 10\ }Today we have the opening ceremony so I put on
a suit. When I arrive I see the celebrated algebraist
Efim Zelmanov who I knew slightly a long time ago. It is a 
pleasure to catch up with his news. I try to catch a glimpse
of the UK team, but fail until they walk across the stage
in their stylish and elegant uniforms.

There are many speeches, interspersed (thank goodness)
with some live contemporary
music composed for the IMO by the Hong Kong composer Dr Mui Kwong Chiu.
The composer has donated the music to the IMO, so future
editions of the competition are free to use it.
There is a splendid orchestra, and various guest musicians
and drummers.
I try to keep my own speech light and short, and then
I administer the IMO Oath to two members of the Hong Kong team.

After the ceremony I am interviewed by Reuters TV so I pretend 
to be important and say positive things about
IMO 2016 and Hong Kong. 
I try to
sound dynamic and look thrusting.
My alter ego Richard Quest often copies this technique. 

At the welcome dinner, a Chinese Banquet,
The Hong Kong elite is out in force:
Chief Executive C Y Leung (i.e.\ the PM) and 
Minister of Education Eddie Ng head the
cast of officials, along with many other 
important figures, including the well known
mathematician Tony Chan of HKUST which is 
the students' site.
The speeches emphasize the huge state support for
STEM in Hong Kong.

There is an initial toast for the cameras, followed by a 
large number of delicious courses. I had not previously
realised the extent of the influence of the temperance movement in 
South East Asia.

This is my natural milieu of course. Everyone is very positive and
extremely friendly. Towards the end of the evening I am called
outside to do a short piece to camera to encourage the 
contestants who will sit IMO Paper 1 tomorrow. I imagine 
six hundred contestants scanning \emph{Facebook} in the hope that some
foreigner in a suit will make encouraging noises about the exam. 
This is obviously the best form of preparation for an IMO. 
I have some fun by making a cryptic reference to Problem 6:
``I'll be with you in a special way''. I will later learn that
my mischief was successful, and many contestants wasted their time
looking for the meaning of this remark. In reality, it was simply
an joke about the frog problem.

\noindent \textbf{July 11\ }The exam is happening at the students' site, and 
there is 30 minutes at the start of the exam for students to ask 
questions of clarification. There are one or two teething problems
getting the Q\&A session going.
After the chaos, I discuss the matter with various specialists, and
we modify the procedures to make things run more smoothly the next day.

After Q\&A we go on a excursion. 
This starts with lunch.
You have to board a boat to get 
to the Jumbo Floating Restaurant.
Once again I admire the iron grip
of the temperance movement.
The Czech and Bosnian leaders Martin Pan\'ak 
and Jelena Radovi\'c are paraded on stage
in traditional costumes. Perhaps this is 
a prelude to human sacrifice?

Next on the agenda a trip to 
to see the panoramic view from Victoria Peak.
I wander off in search of shade, and discover to my
surprise that I have a minder, Queenie Lee.
She is there to make sure I do not get into trouble.
This is perhaps the hardest job at IMO 2016.

The coach descends from the summit, giving us
magnificent views. We soon arrive at
Hong Kong Polytechnic University, handily 
positioned close to the leaders' hotel. We start 
with tours of various research projects.
I visit an anechoic chamber and am disappointed to hear 
myself when I speak.
After that we listen to a panel of 
local experts discuss olympiad mathematics
as well as aspects of mathematics 
in the modern world.

When we return home in the evening, we get access to 
our students' scripts for the Day 1 paper.
It looks quite good, except that poor Harvey Yau
has melted in the face of Problem 1.

\noindent \textbf{July 12\ }We implement the improved Q\&A procedures.
It is now possible to send standard pictures rather
than redraw them for every student, and there are modifications
to the queuing rules. Also, instead of the leaders 
quietly and confidently concentrating on the matter in
hand, thereby causing complete chaos as they did on Day 1, 
we install Geoff at the front to hector, encourage and 
bully the jury into some kind of order. This works surprisingly
well, but after a while I get a bit fed up with shouting
at people, and my voice starts to deteriorate.
We really do need to find a better way to do this.

Next stop HKUST, the Hong Kong University of Science and 
Technology. I have an excellent student room 
in a flat occupied by several good friends. There is
vigorous air-conditioning, and I am not far from the
UK team. 

I have a meeting with the UK delegation. They are in good spirits,
and are quite optimistic about their performances. They all 
seem to enjoy one another's company, and get on well with 
Dominic and Jill. I am given a briefing on the pre-IMO camp in
the Philippines, their adventures, and the Mathematical Ashes Competition.
The Ashes have been retained (again). I am feeling quite sorry for our 
Australian friends.

\noindent \textbf{July 13 and 14\ }These are the co-ordination days. Dominic and I
have studied the scripts carefully, and we think we know what they deserve.
There is a glitch at the start, and things start a little late. Also
there are only a handful of chairs outside the co-ordination room.
Extra seating is obtained in short order, and when co-ordination starts,
it proves very efficient. Our scripts have been carefully read, and there
are very few matters to discuss in detail. The only things which hold
us up are when students have done things which are not quite covered by the
marking scheme, and the co-ordinators are very sensible and natural justice 
prevails.

There is one instance where Dominic notices that we have been treated 
a shade more generously than the Australians, and we know that
the same issue will arise for at least one Polish script. Dominic alerts
everyone to the issue, and the co-ordinators make a minor adjustment
to ensure that we are all treated fairly. 

The final jury meeting is in the evening of July 14. 
There is an election to the IMOAB and
D\'avid Kunszenti-Kov\'acs, team leader of Norway
is elected. As a contestant, D\'avid competed seven times
for Norway. His breadth of experience includes obtaining all
possible results, from gold medal through to nothing at all.

We remember the late Geoff Ball
by standing in silence. Geoff was the IMO Deputy Leader
of Australia
nine times in the 1980s.

Some amicable
disputes between leaders and co-ordinators are resolved by the jury,
and we set the medal boundaries according to the procedure
introduced last year. The jury sees the gold, silver, bronze
pie charts of the 8 most plausible medal boundary selections. Four of them
violate the ``no more than 50\% get a medal'' rule, and some members
of the jury immediately move that we should break our rule. A 2/3 vote of the
jury is required to do this, and this cannot be mustered. Therefore
we focus on the four remaining medal boundary selections and 
make our choice. Then it is revealed what we have chosen.
The cut offs are 16 for bronze, 22 for silver and 29 for gold.

\noindent \textbf{July 15\ }
The closing ceremony is in a hall that we have not visited
before. The event begins with high-voltage Oom Pah Pah 
involving robust Hong Kong youth jumping around like cheerleaders
while playing large brass instruments.

A leading Hong Kong politician disappeared as the medal presentation
ceremony began. I suspect this was a precaution in case he was to see 
a flag to which he would have been obliged to take offence. In fact 
the people who might have caused the offence behaved with
perfect restraint, and no such flag was shown. The politician
surfaced again when there was no danger of seeing anything controversial.
Good work all round. 

The final banquet was in a huge hall, with (approximately) 
each country having
its own dedicated round table. Note to other host countries: this
is the perfect way to run a final banquet. Once more I sit with other
people almost as important as myself including Carrie Lam.
Assiduous readers of these reports will recall that Josh Lam competed
for the UK at IMO 2011 and IMO 2012 while he was being educated
in England. Carrie Lam is Josh's mum and,
incidentally, Chief Secretary of Hong Kong, head of the civil service.

The audience is treated to my surprise speech when I am told,
with no notice whatever, that I will say a few words of appreciation.

After the dinner there is finally time to relax, and to enjoy banter with both
the team and old friends. 

\noindent \textbf{July 16\ }I fly home relaxed and so tired that I sleep
the whole way across China. Unfortunately I am on a plane a couple
of hours ahead of the main party. My only regret about IMO 2016 was that
I did not spend more time with the UK team. Thank goodness that
Dominic and Jill do such a good job. We lose two students to university
this year, but the other four are available for selection for IMO 2017
in Brazil, and Harvey Yau will also be available for IMO 2018.
He will spend the next 12 months drawing neat geometry diagrams I am sure.

I muse on the plane how sad it is that no girl won a gold medal at IMO 2016.
The only two years with no female IMO gold medallist this millennium
are 2013 and 2016. Work is being done all over the world, including at
EGMO, to address this matter. Clearly it is very much work in progress.

A happier thought is that it has become clear 
during IMO 2016 that Norway will soon offer to host IMO 2022 and 
Japan to host 2023. These countries have well funded and 
well organized maths olympiad organizations, so the medium term future
of the IMO looks secure. I have been continuing my campaign to 
get an Australian or New Zealand IMO in 2028 when the
total eclipse of the sun of 22 July ($\pi$ day) will 
pass through Wyndham, Kununurra, Tennant Creek, Birdsville, Bourke,
Dubbo, Sydney and Dunedin. Perhaps the obvious thing to do would
be to put the students in Kununurra and the jury in Tennant Creek.

\section*{Thanks}
This was a remarkably strong performance by a wonderful
team. Sitting behind these young people is huge support,
both in intellectual terms but also in terms of
support from families and schools. There is also the 
UKMT infrastructure, and the army of volunteers and staff
who create the competitions, publications, camps, mentoring
and other activities which underpin the UK contribution to
the IMO and other mathematics competitions.

I must specifically thank \emph{Oxford Asset Management} for their
financial support for the IMO team. Their natty logo adorns the team
blazer. 


\end{document}
