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\title{IMO 2007 in Vietnam}
\author{UK Leader's Report by Geoff Smith}
\date{}
\begin{document}
\maketitle
The International Mathematical Olympiad is the annual world championship of
secondary school mathematics. This year
the UK team comprised 5 boys and one girl.
They were selected on the basis of their performances in
many competitions and practice examinations.
The team representing the UK this year was as follows:
\begin{center}
\begin{tabular}{p{3cm}p{9.6cm}}
Tim Hennock & Christ's Hospital \\
Tom Lovering & Bristol Grammar School \\
Takaki Oshima & Westminster School, London \\
Jack Shotton & Portsmouth Grammar School \\
Dominic Yeo & St.\ Paul's School, London \\
Alison Zhu & Simon Langton Girls' Grammar School, Canterbury \\
\end{tabular}
\end{center}
The reserves were:
\begin{center}
\begin{tabular}{p{3cm}p{9.5cm}}
Jonathan Lee & Loughborough Grammar School \\
Preeyan Parmar & Eton College \\
Julia Robson & Perse School for Girls \\
\end{tabular}
\end{center}
An army of volunteers run the mentoring schemes and
camps which are the basis of the British effort. We are
truly fortunate to have such a generous and talented community,
supported by the professional administrative staff in the
Leeds office of the United Kingdom Mathematics Trust.
These days nearly three-quarters of a million students
participate in our competitions, and I thank them all
for supporting the organization.
The IMO has finally adjusted to the existence of the internet.
This has consequences for this annual report, for there is
little point carefully printing information which is
freely available at a well designed web site. We must
thank the organizers of IMO 2006 in Slovenia who took on
the task of producing an official IMO site. They have
done a tremendous job.
Here are the problems of IMO 2007:
\newcommand{\problem}[1]{\paragraph{Problem #1.}}
\problem{1}
Real numbers $a_1, a_2, \ldots, a_n$ are given. For
each~$i$ ($1 \leq i \leq n$) define
\[ d_i = \max \{ a_j : 1 \leq j \leq i\} - \min \{ a_j : i \leq j \leq n\}\]
and let
\[ d = \max \{d_i : 1 \leq i \leq n\}.\]
\begin{enumerate}
\item[(a)] Prove that, for any real numbers
$x_1 \leq x_2 \leq \cdots \leq x_n$,
\[
\max \{ |x_i - a_i | : 1 \leq i \leq n \} \geq \frac d2. \rule{2cm}{0cm} (\ast)
\]
\item[(b)] Show that there are real numbers $x_1 \leq x_2 \leq \cdots \leq
x_n$ such that equality holds in~$(\ast)$.
\end{enumerate}
\problem{2}
Consider five points $A, B, C, D$ and $E$ such that $ABCD$ is a parallelogram
and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through
$A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose
also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.
\problem{3}
In a mathematical competition some competitors are friends. Friendship
is always mutual. Call a group of competitors a \emph{clique} if each
two of them are friends. (In particular, any group of fewer than two
competitors is a clique.) The number of members of a clique is called
its \emph{size}.
Given that, in this competition, the largest size of a clique is even,
prove that the competitors can be arranged in two rooms such that the
largest size of a clique contained in one room is the same as the
largest size of a clique contained in the other room.
\problem{4}
In triangle $ABC$ the bisector of angle $BCA$ intersects
the circumcircle again at $R$, the perpendicular bisector of $BC$ at
$P$, and the perpendicular bisector of $AC$
at $Q$. The midpoint of $BC$ is $K$
and the midpoint of $AC$ is $L$. Prove that the triangles
$RPK$ and $RQL$ have the same area.
\problem{5}
Let $a$ and $b$ be positive integers. Show that
if $4ab -1$ divides $(4a^2 -1)^2$, then $a = b$.
\problem{6}
Let $n$ be a positive integer. Consider
\[ S = \left\{ (x, y, z)\ :\ x, y, z \in \{0,1, \ldots, n\},\ x+y+z >0
\right\}\]
as a set of $(n+1)^3-1$ points in three-dimensional space.
Determine the smallest possible number of planes,
the union of which contains $S$ but does not include $(0,0,0)$.
\vskip 0.5cm
The UK performances
were:
\begin{verbatim}
UNK1 Tim Hennock 7 0 0 7 0 0 14 Bronze
UNK2 Tom Lovering 7 2 0 7 0 0 16 Bronze
UNK3 Takaki Oshima 0 0 0 7 0 0 7 Hon Mention
UNK4 Jack Shotton 7 7 2 7 7 1 31 Gold
UNK5 Dominic Yeo 4 1 0 7 0 1 13 Hon Mention
UNK6 Alison Zhu 7 0 0 7 0 0 14 Bronze
\end{verbatim}
The UK team was supported at the IMO by Deputy Leader Ceri Fiddes,
Observer~A Joseph Myers and Observer~C Pam Hunt.
Two UK questions made the IMO shortlist this year. The excellent
unselected G5 was by Christopher Bradley. Problem N6 was on the paper
as Problem 5. The history of the problem is complicated, and
perhaps it is best to attribute it as a joint effort by
Kevin Buzzard and Edward Crane.
The running five year IMO rank order at the back of the American
Booklet now has the UK at 14th, which is not so bad.
In 2007 we had a relatively low ranking of equal 28th. At IMO
2007 there was something of a shock when Russia pipped
China for first place. Congratulations to Nazar Agakhanov, his
organization and students. Note the continuing strong performance
of Italy. At the UKMT retreat the Italian Leader Roberto Dvornicich
made out that their excellent showing in 2006 was a lucky chance.
This was, of course, nonsense.
For all other details, I refer you to
\begin{center}
\tt{www.imo-official.org}
\end{center}
It would be particularly helpful if more senior readers could look
at information concerning past IMOs, and supply corrections and
missing data to the organizers of this website. There was a special
plea at IMO 2007 for this to happen.
\section*{Leader's Diary}
The extra cost of flying on to Australia is not that great,
so we decide to repeat our 2003 scheme of adjusting to
far eastern time by spending a few days in Queensland, Australia.
James Cook University has a campus in Cairns where we stayed
before, and had a happy time. Of course the fact that the team
of 2003 performed very well may have something to do with my
enthusiasm for this plan.
\noindent \textbf{July 11th}
We arrive at Sydney International at 5:30am. We have
a mid-afternoon internal flight to Queensland, so we take
our bags over to the domestic terminal and leave them in left luggage.
We catch a double decker train to the city in order to take a ferry
across Sydney harbour.
In the afternoon we take an internal flight to Cairns. We sit
towards the front of the plane, mixed up with a jovial bunch of
hearty Australian blokes who quickly drink the plane dry. The fun
starts when the exhausted and jetlagged Alison Zhu falls fast asleep,
and for a few moments uses one of the drinking
party as a snuggle rug. He is a very happy man, and looks rather
pleased with himself as his jealous colleagues celebrate his good
fortune. The banter disturbs Alison, and she straightens up in her
sleep. It turns out that this happy band of brothers are on their
way to Queensland to hunt feral pigs with spear guns. They invite us
to join them in the bush for some murderous fun. I am all for it,
but Ceri mutters pathetically about health and safety issues and
the absence
of any reference to wild boar in the risk assessment document,
and so I am pressed to decline. What can you expect from a
vegetarian?
We go on a snorkelling trip on the Great Barrier Reef. When
the time comes to enter the water, we are told to simply jump in from
the slowly moving boat. I go first, and enter the water
with a certain force for some reason. I was expecting this, and
my body (with, thanks to Descartes, mind in tow) plunged deep
beneath the surface. As team member Dominic happily pointed out,
this was presumably a life-changing experience for the
single celled organisms making up the reef. As photosynthesis
was suddenly switched off, it must have seemed like aquatic
Armageddon was imminent. The
four seahorses of the apocalypse swam by.
I had taken a deep breath on the way
down, and without too much discomfort I eventually kicked
to the surface to be greeted by a concerned professional diver.
Heaven knows what must have happened to the seismological arrays
which now monitor the Pacific. This was not the ideal start
to a snorkelling tour, but after grabbing a float to
boost my confidence,
I eventually forced my heart rate down and even started to enjoy the
experience. Praise be to Archimedes.
While all this going on, Joseph Myers has gone Ocean walking.
Being a man of doubtful buoyancy, he has taken the
\emph{Futurama} option. His head is inside a giant transparent
bubble, and teams of Australian lackeys pump life support down
plastic tubes while Joseph strolls beneath the waves. Presumably
oxygen, ice cream and combinatorics problems are pumped down
separate tubes by retainers.
We also went on a short trip to a local zoo, and Ceri busies herself
playing housemother to vast numbers of amiable kangaroos. The most
remarkable inmate is the crocodile keeper, who seems to think
that he deserves his corn for poking a croc with a stick while
delivering patter which only exceptionally strays from the
vacuous to the contradictory. Any celebrity
bestowed by Warhol should in this case be confined to 15
nanoseconds.
By way of contrast, a we are royally
entertained by an erudite bus driver who takes us to
Kuranda in the rainforest. His casual familiarity with
the social and economic history of Cairns, and working
knowledge of local natural history leaves me
impressed.
From Kuranda we make a prodigious journey
by cable car over the forest back to settled Australia.
We are in competition for seats with an army of teenaged Americans
who are on a Pacific tour as some kind of goodwill
ambassadors. We never quite
work out how they are selected, but apparently it is all to do
with being good.
This tourism leavens the diet of daily exams which form the
staple of our days.
Well before IMO 2007, the shortlist of questions for IMO 2006
leaked onto the internet. This disrupted our preparations, because
one of our students looked at the offending document. It seems perfectly
clear who facilitated this breach of IMO protocol and etiquette.
If this
happens again (future organizers please note), then surely the
credentials of the person or people involved should be withdrawn,
and they should not be permitted to attend IMOs. Growl.
\noindent \textbf{July 17th} We all fly to Sydney to join the Australian team.
In case you are not from Australia, think of Madrid.
However, Sydney is in the grip of a completely untypical
cold snap. We find that our accommodation at the University
of New South Wales is not really adapted to antarctic living,
and it is not at all funny. We arrive late in the evening
and very hungry. To my astonishment, pizza delivery seems
difficult. We are driven to walk the streets, and can only
find that burger chain open.
\noindent \textbf{July 18th}
Joseph and I are heading north towards the IMO today.
The Australian team and leaders are very hospitable, and we leave
amid cheery schemes to deal with the cold. Breaking up the furniture,
and tossing in a petrol soaked Tom Lovering
or Tim Hennock with a
magnesium flare is the obvious way forward.
Ceri and the Australian deputy Norm seem to have more sensible
plans. Time to go. We spend the night in Singapore.
\def\dong{{\leavevmode\setbox0=\hbox{d}d\llap{\raise.75\ht0\vbox{\hrule width0.67\wd0}}}{\leavevmode\setbox0=\hbox{\^o}\rlap{\raise.1\ht0\hbox{\kern.4\wd0\char'22}}\^o}ng}
\def\idong{{\leavevmode\setbox0=\hbox{d}d\kern0.2\wd0\llap{\raise.75\ht0\vbox{\hrule width0.67\wd0}}\kern-0.2\wd0}{\leavevmode\setbox0=\hbox{\^o}\rlap{\raise.1\ht0\hbox{\kern.4\wd0\char'22}}\^o}ng}
\noindent \textbf{July 19th} We head for Hanoi.
At Hanoi airport team leaders and Observers~A
are directed to a dedicated
queue. Immigration and customs formalities are dealt with at
speed, and very quickly Joseph and I find ourselves in the
arrivals hall. We are engulfed by charming IMO volunteers
who are determined to make us feel welcome. There is much
jovial banter from the Vietnamese, and for some reason
the words \emph{Father Christmas} keep being mentioned.
Maybe we should go as Lapland next year.
Local organizers help us to change money and purchase water.
Although I have a little wad of one dollar bills
(universal currency units) tucked away, it seems sensible
to purchase some local
currency. This is called the \emph{\idong}, and you get about
$2^{15}$ \dong{}s to the
pound. Thus a pound is 32 \emph{kilo\idong{}s} and in turn
this means that
a million \dong{}s is easy to work out and a \emph{mega\idong} is 32 pounds.
There is a TV camera and much shaking of hands. The arrival hall
has a mild form of air-conditioning which is good enough
to stop foreigners from wandering off.
Leaders are pouring in from all over the world, and my old
pal Jim Cruickshank of Ireland suddenly bursts on the scene.
Now the crack can begin. He and particularly his Observer~A
Mark will become the engines of many an adventure. I can see
that Mark is a man of destiny, and will soon thrust Jim aside
in the harsh world of Irish mathematics enrichment politics.
After a while we are shown to our bus.
This involves negotiating a
pedestrian crossing. First of all an IMO official jumps
into the road
in an attempt to stop the traffic. He merely diverts
the flow as vehicles
swerve around him. Then a policemen steps out in an impressive uniform
and raises his hand. This has an entirely similar effect. Eventually
we spot a gap and make a dash for the coach. This is air-conditioned,
comfortable and under-populated. By now we have discovered that
the jury site is at Halong Bay. Different people give different
opinions as to how long it will take, but clearly it will be several hours.
Nomatter, with air-conditioning and the promise of a comfort
stop en route,
this will be fine.
Apart from taxis, there are not many cars on the roads. There are
big buses and trucks and small motorcycles, but not much in between.
It is quite remarkable what you can carry on a motorcycle (a
refrigerator for example). On a later occasion we see a young blade
moving four female companions round Hanoi on his motorbike---the
packing strategy was one in front and three behind, in case
you are wondering.
We drive away from Hanoi and through an agricultural area. People
labour beneath broad conical hats to tend rice paddies. We see
many brick factories actually built in a great river. From time
to time we pass through villages, each one of which
has a motorcycle franchise.
The jury site is in a very luxurious hotel in Halong Bay. This coastal
resort is north of Hanoi, towards the Chinese border. From the point of
view of natural science, it is a celebration of exotic geology.
The bay
is littered with so many hemispherical islands that there is no
visible
route to the open sea. I first read about Halong Bay on the long drive
from the airport. Someone has some astonishingly frank tourist
literature which makes disparaging remarks about the particular
tourist clientele which Halong Bay attracts. Actually I may be
letting my prejudices show again, and the remarks may not have
been intended as disparaging at all. The phrases in question concerned
karaoke bars and sleazy nightclubs. I know that several members of
the jury will flourish in such an environment.
It is always a concern to protect the IMO shortlist, and especially
the IMO paper, from unauthorized access. There are contrasting
attitudes
around the world. Last year the postmodern Slovenian organizers took the view
that given the state of modern communications, the only possible
defence of the IMO paper is the collective honesty of the jury
and co-ordinators. This year the Vietnamese organizers take a different
view. The hotel is surrounded by the army and no-one is allowed
out of the hotel. In order to give a relaxed and
friendly impression,
the soldiers are only carrying handguns. We have access to an
emasculated version of the internet which allows us to trawl for
information but not to send e-mail.
Personally I find being outside in Vietnam a difficult experience,
the air being hot and thick, and need little encouragement to stay
in air-con heaven. After several days of incarceration, following
a plea from the Leader of Trinidad and Tobago, a concession was
made and we were allowed out in groups of five, subject to making
no attempt to communicate with the outside world.
Against my better judgement,
I agree to join one of these tropical expeditions. Donning a Panama
hat, I ambled gently
to the city centre, and returned via a bar.
I was a little surprised to find other westerners wandering round the
town. Presumably they were taking karaoke breaks.
The hotel lifts are the weak point of the establishment. There
are three of them, but two sets of calling buttons. Two of the lifts
respond to one set of buttons, and the third lift is controlled
by the other set. Therefore if you want a lift, all but the
Quakers (of which we had an empty collection) selfishly press
both buttons, thereby ensuring that half the calls to which the
lifts respond are hoaxes. The net effect was that the lifts flocked.
There was the option of using the outside stairs, but you would only
do this if you were planning a shower in the immediate future.
To be serious, notwithstanding minor shortcomings, the hotel
is of fabulous standard, and the leaders and observers are
stunned by the hospitality of our Vietnamese hosts who are
doing their very best to make this a great IMO.
The quest to set the IMO paper soon began. We first receive the
shortlist without solutions, and next day the
extended booklet with solutions is issued.
I am afraid that it is
a fantasy to think of the jury a wise collective mind,
selecting a delicious and
exquisite paper aimed at discriminating between students at
all medal boundaries. There is definitely no controlling
mind of this unwieldy body, and it proceeds using a mixture
of good judgement, herd instinct and obsession with geometry.
Democracy is not an effective method to set a mathematics
exam, and it remains a mystery how the jury manages to do
quite a good job most years.
We seem to have a shortage
of good relatively easy questions. My heart sinks when
I see G1, ostensibly the easiest Geometry question. The jury has a
weakness for G1, irrespective of what year it happens to be.
I have noticed that when ordering Chinese food, number 76 is
often quite delicious, and am happy to order it
without enquiring as to what it is. The jury's attitude to G1 is similar.
Now, since it is a racing certainty that the jury will
select G1, it is the duty of the Problem Selection Committee
to ensure that it is a good question. This year G1 (proposed
by the Czech Republic) is not the
worst problem on the shortlist, but it is not absolutely ideal.
This is because although a little ingenuity is needed to
find a solution by synthetic geometry, one can instead simply
lower ones head, scrape ones feet against the ground, and
charge forward deploying either trigonometry or areal co-ordinates.
Either way, a determined professional approach will quickly
yield a solution. Actually a solution by calculation turns out
to be safer, because one then need not allow for the singular
geometric configuration that arises in the isosceles case.
The jury selects a second question deemed to be easy from the
algebra list. This question is not hard, but it does involve the
casual manipulation of subscripts, and thereby plays to the
more experienced and well trained students. This A1 was
the brain child of Michael Albert, the New Zealand leader.
Next the jury moved to select the hard questions, the candidates for
Problems 3 and 6. Following a slight diversion where an originally
selected problem proves to be already known, they select one from
Russia, an excellent but unusual
combinatorics problem because all known solutions involve devising
an algorithm. The other choice is one which masquerades as
combinatorics, but the known solutions are actually algebraic.
The devastating effectiveness of algebra in this context wins
many friends on the jury. Later there will be chatter
on the internet that
the problem is unsuitable because it is trivialized by the
\emph{combinatorial Nullstellensatz}. This is a recurrent issue,
when a theorem of relatively advanced mathematics renders a
problem straightforward, should the jury throw out the question?
In this case there was no problem, because no-one alerted the jury
to the possibility of invoking this technique.
As far as I know, only a handful of students attempted to
invoke this advanced theorem in a solution. The Russian Problem 3
was manifestly combinatorial. Thus when Problem 6,
its paired hard question also looked combinatorial, I reasoned
that astute students would `smell a rat', and realise that
the solution to Problem 6 would not be combinatorial at all.
Problem 6 is a Dutch proposal.
Finally the jury moves to selecting the medium questions.
It plumps for G4, the creation of the Luxembourg Leader
Charles Leytem (he of the dynamic $z$-co-ordinate) and
the \emph{lemma of N6}. This reminds me of the way that, many
years ago, there was a pop singer called
\emph{the artist formerly known as Prince}.
Now we have \emph{the problem formerly known as N6}. The jury
likes the hard problem N6 so much that it selects the
key lemma from its solution, turns that into the problem,
and reclassifies it as medium so that it can be selected.
In fact N6 is a UK proposal, but being either saintly, or
at least well aware of my own fallibility, I refuse to look
at the UK submissions so that I can speak freely in the jury.
Indeed, I spoke in favour of N5.
Sometimes there are dark mutterings when students from country $X$
do exceptionally well in response to a problem posed by country $X$. Eyebrows
are bent, lips are pursed, and leaders exchange meaningful
glances. No doubt the UK will fall prey
to similar suspicions. Our six students will go on
to rake in a total of 7 marks between them on this problem.
The jury sends one of the proposed marking schemes back for
revision, but the jury chair seems surprised that we will
not vote to accept it anyway. There is a minor cultural
clash here, but happily it is easily resolved.
Some of the jurors are getting restive, and resentment
at being locked up in the hotel is starting to build up.
Unsurprisingly, some of the leaders banging on about their
human rights come from countries which habitually lock up
people in industrial quantities. Eventually Indra, Leader
of Trinidad and Tobago, makes a health and safety based case
that we be allowed out for exercise. Our hosts readily agree
to \emph{exeats} for large group as mentioned earlier.
\noindent \textbf{July 24th} It is time for the opening ceremony.
It is hard to be sure who
is in charge here, since Vietnam has parallel government and
communist party structures. We have ministers at all
ceremonies, the PM at the opening jamboree and the president
will turn up at the medalfest. We are escorted to the opening ceremony
by cops with \emph{blues and twos} switched on. This cuts down
the journey time, and unimaginative leaders such as myself seek
front seats to enjoy the chaos as oncoming traffic hurls itself
into ditches as we drive on random sides of the road and
use traffic lights
only for illumination. We have an excellent ceremony, and
our team's hats make quite a splash.
\noindent \textbf{July 26th} The second exam finishes today, and we start to
sweat over the scripts. Ceri joins us at the leaders' hotel, leaving
Pam in charge of looking after the team. Results are mixed.
Jack Shotton has done very well. He has solved 1, 2, 4 and 5 completely,
and has part marks on both 3 and 6. It is quite clear that
he will get a gold medal. The other students have mostly
underperformed slightly, and looking back I can see that
we should have gone pig hunting. The other five students
are about one category down from what one might have
expected. They get three bronzes and two honourable mentions,
where we hoped for at least three silvers and a couple of bronzes.
Most of them have thrown away easy marks, and missed solutions
within their grasp. It is hard to know why this has happened.
I blame Pam, Ceri, Joseph and global warming.
Actually, it could have been a lot worse. Ceri did a fantastic
job of working out that a lot of incoherent gibberish associated
with two of our solutions to Problem 1 is actually correct mathematics
written down by orang-utans on acid. She writes out what these two
candidates were trying to say, and manages to get the Vietnamese
co-ordinators to agree that these solutions are indeed correct.
Joseph and I are stunned. She has done nothing dishonest, and
her case is sound, but without her insight into what passes for
the student mind, many marks would have been lost.
Jack's script for the hard combinatorics Problem 3 claims a solution
but has a fatal flaw in the last line and might score zero.
The marking scheme for this problem offends
Observer Joseph. The only known solution involved making
very special initial moves, for which part marks are
available. Joseph reckons that this is unfair to Jack's script because
he speculates that there are solutions which involve
completely different initial moves. He disappears for several
hours and returns with a new solution which is compatible
with various initial moves. This is dynamite, and he circulates
it the evening before co-ordination to stir up the marking scheme.
Jack gets 2/7.
I lead during the co-ordination of Problem 4. Here our students
have all solved the problem, but a stern
co-ordinator might want to take a mark off one script where
the student has been sloppy in two different ways. This
is a particular concern because the author is Takaki Oshima,
and this is his only chance of marks. If he gets 7/7 he will
get an honourable mention. Otherwise he will get nothing.
The co-ordinators are indeed programmed to subtract marks for
listed imperfections, but it turns out that, somewhat characteristically,
Takaki's flakiness is so bizarre as not to be in the marking scheme.
Happily he scrapes a 7/7.
Our Vietnamese hosts have done a wonderful job. There have
been some difficulties on the way, and some leaders are
annoyed by particular incidents. No IMO is perfect
(though admittedly some are less perfect than others). There
was a suggestion that there was some small inconsistency
in the marking of one problem. This seems to have been true, but I can
remember an IMO when the co-ordination was a real disaster:
idleness, stupidity and petulance seemed to be the co-ordinators
motto. This was certainly not the case in Vietnam, where
diligent and intelligent co-ordinators were all trying their
best, and were mostly doing an excellent job.
We moved hotels to Hanoi after co-ordination. The new site is
next to a large lake in the centre of town. Happily BBC World Service
TV is available at this new hotel, so there is a relatively
cerebral alternative to the pap put out by CNN. At the closing
ceremony I am interviewed by \emph{Voice of Vietnam}, a national
radio station. The IMO has been front page news in the papers
throughout the competition, and has had much TV coverage.
I meet the team's excellent local guide, a young woman called
Quy. Her English is excellent, and according to Pam
and the team she has been a first class guide.
\noindent \textbf{July 30th}
The closing ceremony is surprisingly relaxed, and the president
does not seem to be concerned with formalities. We dash for a table
at the banquet, and have a marvellous time. Happily I have failed
to hold on to the \emph{microphone d'or}, the prize for the
most garullous juror. The Romanian leader Radu Gologan has made
one more speech than I have, so he holds the shiny plastic this year.
All the students mix up in a deplorable festival of goodwill.
Later we go out to celebrate Jack Shotton's gold medal and 18th birthday
with ice cream, a party sponsored by Mr and Mrs Shotton.
There were some unusual features to the entertainment
at the ceremonies, including
a song by a Vietnamese game show host. However, I leave you
with the words of a song which featured at the opening ceremony.
It was sung by a girl band, all jump suits and lip gloss,
who wriggled and writhed through the
following number (thanks to the Leader of Norway,
seven times contestant D\'avid Kunszenti-Kov\'acs, for the text).
\begin{verbatim}
A new horizon is waiting for us
The broad horizon is inviting us over
The horizon for science is right in front of us
Awakening our passions, challenging our thirsts. (x2)
Together, let's realize our dreams about creativity
Together, let's indulge in the joy of science
With our hearts and brains, victory is within our reach
Shoulder to shoulder, we build our nation.
Together, let's realize our dreams about creativity
Together, let's indulge in the joy of science
With our hearts and brains, victory is within our reach
Fly to the future, young men and women.
\end{verbatim}
\end{document}