\documentclass[a4paper]{article}
\usepackage{booktabs,enumerate,amsmath,amssymb,hyperref}
\usepackage[T1]{fontenc}

\title{53rd International Mathematical Olympiad}
\date{Mar del Plata, Argentina, 4th--16th July 2012}
\author{UK leader's report}

\newcommand{\bZ}{\mathbb{Z}}

\begin{document}
\maketitle

\section*{Introduction}

The \href{https://www.imo-official.org/default.aspx}{International
  Mathematical Olympiad} is the world's foremost mathematics
competition for school-age students. About a hundred countries
participate annually; each country can send a team of up to six
students.

This year, the 53rd IMO was held in Mar del Plata, Buenos Aires
province, Argentina. The UK has participated in every IMO since
1967. This year, the team were:
\begin{center}
\begin{tabular}{cll}
\toprule
UNK1 & James Aaronson      & \href{http://www.stpaulsschool.org.uk}{St Paul's School}, London \\
UNK2 & Sam Cappleman-Lynes & \href{http://www.shebbearcollege.co.uk/}{Shebbear College}, Devon \\
UNK3 & Andrew Carlotti     & \href{http://srms.kent.sch.uk/}{Sir Roger Manwood's School}, Kent \\
UNK4 & Daniel Hu           & \href{http://www.clsb.org.uk/}{City of London School for Boys} \\
UNK5 & Joshua Lam          & \href{http://www.theleys.net/}{The Leys School}, Cambridge \\
UNK6 & Matei Mandache      & \href{http://www.lesgrammar.org/}{Loughborough Grammar School} \\
\bottomrule
\end{tabular}
\end{center}

The first reserve was Adam Goucher
(\href{http://www.netherthorpe.derbyshire.sch.uk/}{Netherthorpe
  School}, Derbyshire), and the other reserves were Gabriel Gendler
(\href{http://www.qebarnet.co.uk/}{Queen Elizabeth's School}, Barnet,
London) and Katya Richards (\href{http://www.shsk.org.uk}{St Helen and
  St Katharine School}, Oxfordshire).

I (Dr James Cranch, \href{http://www.shef.ac.uk/}{University of
  Sheffield}) led the team; Jack Shotton
(\href{http://www3.imperial.ac.uk/}{Imperial College London}) was the
deputy leader.

Dr Geoff Smith (\href{http://www.bath.ac.uk/}{University of Bath})
attended as Observer A and in his capacity as member of the IMO
Advisory Board. Bev Detoeuf from the
\href{http://www.mathcomp.leeds.ac.uk/}{UKMT}'s office in Leeds
attended as Observer C, accompanying the team to provide pastoral
support.

\section*{Questions}

As ever, this year's IMO consisted of two exam papers, sat on
consecutive days. Each exam consists of three questions, and lasts
four and a half hours. Each question is worth seven points.

The questions this year were as follows:

\begin{enumerate}[\bf Problem 1.]

\item Given triangle $ABC$ the point $J$ is the centre of the excircle
  opposite the vertex $A$. This excircle is tangent to the side $BC$
  at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$,
  respectively. The lines $LM$ and $BJ$ meet at $F$ , and the lines
  $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of
  the lines $AF$ and $BC$, and let $T$ be the point of intersection of
  the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.

(The \emph{excircle} of $ABC$ opposite the vertex $A$ is the circle
  that is tangent to the line segment $BC$, to the ray $AB$ beyond
  $B$, and to the ray $AC$ beyond $C$.)

  \emph{(Proposed by Evangelos Psychas, Greece.)}

\item Let $n \geq 3$ be an integer, and let $a_2, a_3, \ldots, a_n$ be
  positive real numbers such that $a_2 a_3 \cdots a_n = 1$. Prove that
$$(1 + a_2)^2 (1 + a_3)^3 \cdots (1 + a_n)^n > n^n.$$

  \emph{(Proposed by Angelo di Pasquale, Australia.)}

\item The \emph{liar's guessing game} is a game played between two
  players $A$ and $B$. The rules of the game depend on two positive
  integers $k$ and $n$ which are known to both players.

At the start of the game $A$ chooses integers $x$ and $N$ with $1 \leq
x \leq N$ . Player $A$ keeps $x$ secret, and truthfully tells $N$ to
player $B$. Player $B$ now tries to obtain information about $x$ by
asking player $A$ questions as follows: each question consists of $B$
specifying an arbitrary set $S$ of positive integers (possibly one
specified in some previous question), and asking $A$ whether $x$
belongs to $S$. Player $B$ may ask as many such questions as he
wishes. After each question, player $A$ must immediately answer it
with yes or no, but is allowed to lie as many times as she wants; the
only restriction is that, among any $k + 1$ consecutive answers, at
least one answer must be truthful.

After $B$ has asked as many questions as he wants, he must specify a
set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then
$B$ wins; otherwise, he loses. Prove that:
\begin{enumerate}[1.]
\item If $n \geq 2^k$, then $B$ can guarantee a win.
\item For all sufficiently large $k$, there exists an integer $n \geq
  1.99^k$ such that $B$ cannot guarantee a win.
\end{enumerate}

  \emph{(Proposed by David Arthur, Canada.)}

\item Find all functions $f\colon\bZ\rightarrow\bZ$ such that, for all
  integers $a, b, c$ that satisfy $a + b + c = 0$, the following
  equality holds:
$$f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).$$

(Here $\bZ$ denotes the set of integers.)

  \emph{(Proposed by Liam Baker, South Africa.)}

\item Let $ABC$ be a triangle with $\angle BCA = 90^\circ$, and let
  $D$ be the foot of the altitude from $C$. Let $X$ be a point in the
  interior of the segment $CD$. Let $K$ be the point on the segment
  $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the
  segment $BX$ such that $AL = AC$. Let $M$ be the point of
  intersection of $AL$ and $BK$. Show that $MK = ML$.

  \emph{(Proposed by Josef Tkadlec, Czech Republic.)}

\item Find all positive integers $n$ for which there exist
  non-negative integers $a_1, a_2, \ldots, a_n$ such that
$$\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
  \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}}.$$

  \emph{(Proposed by Du\v{s}an {\DJ}uki\'{c}, Serbia.)}

\end{enumerate}


\section*{Results}

Here are the results of the UK team:
\begin{center}
\begin{tabular}{rcccccccl}
\toprule
name                &Q1 &Q2 &Q3 &Q4 &Q5 &Q6    & total & award \\
\midrule
James Aaronson      & 7 & 0 & 4 & 7 & 2 & 0    & 20 & Bronze medal\\
Sam Cappleman-Lynes & 7 & 0 & 0 & 7 & 0 & 0    & 14 & Bronze medal\\
Andrew Carlotti     & 7 & 7 & 5 & 6 & 0 & 4    & 29 & Gold medal\\
Daniel Hu           & 7 & 0 & 0 & 7 & 2 & 0    & 16 & Bronze medal\\
Joshua Lam          & 7 & 1 & 0 & 7 & 7 & 0    & 22 & Silver medal\\
Matei Mandache      & 1 & 4 & 0 & 7 & 0 & 2    & 14 & Bronze medal\\
\bottomrule
\end{tabular}
\end{center}

The medal boundaries were 14 for a bronze medal, 21 for a silver
medal, and 28 for a gold medal. Hence everyone on our team obtained a
medal, which is a fine achievement. In addition, we were joint first
in Western Europe (jointly with the Netherlands, who have had an
extraordinary year). However, we placed only fourth in the
Commonwealth of Nations: we were beaten by Canada, Singapore and
India, all of whom deserve congratulations.

For completeness, Table \ref{table:top-25} shows the scores and
ranking positions of the top thirty-three countries, and various
others whose performance has traditionally been of interest to us.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{rcc}
\toprule
country & total & rank\\
\midrule
Republic of Korea & 209 & 1\\
People's Republic of China & 195 & 2\\
United States of America & 194 & 3\\
Russian Federation & 177 & 4\\
Canada & 159 & 5\\
Thailand & 159 & 5\\
Singapore & 154 & 7\\
Islamic Republic of Iran & 151 & 8\\
Vietnam & 148 & 9\\
Romania & 144 & 10\\
India & 136 & 11\\
Democratic People's Republic of Korea & 128 & 12\\
Turkey & 128 & 12\\
Taiwan & 127 & 14\\
Serbia & 126 & 15\\
Peru & 125 & 16\\
Japan & 121 & 17\\
Poland & 119 & 18\\
Brazil & 116 & 19\\
Bulgaria & 116 & 19\\
Ukraine & 116 & 19\\
Netherlands & 115 & 22\\
\emph{United Kingdom} & 115 & 22\\
Belarus & 114 & 24\\
Croatia & 110 & 25\\
Greece & 107 & 26\\
Australia & 106 & 27\\
Hong Kong & 106 & 27\\
Saudi Arabia & 105 & 29\\
Republic of Moldova & 104 & 30\\
Germany & 102 & 31\\
Israel & 102 & 31\\
Mexico & 102 & 31\\
& & $\vdots$\\
Belgium & 93 & 38\\
France & 93 & 38\\
Hungary & 93 & 38\\
Italy & 93 & 38\\
& & $\vdots$\\
New Zealand & 75 & 53\\
& & $\vdots$\\
Ireland & 34 & 78\\
& & $\vdots$\\
\bottomrule
\end{tabular}
\end{center}
\caption{The top thirty-three and selected other countries in IMO 2012}
\label{table:top-25}
\end{table}

Clearly, this has been an excellent year for South Korea. Another
noteworthy international performance was by Teodor von Burg of Serbia:
he now moves to the top of the
\href{http://www.imo-official.org/hall.aspx}{IMO Hall of Fame} with a
bronze, a silver and four gold medals from the last six IMOs.

Moving on to discussion of our own team, Andrew is still on course to
be the most successful British IMO contestant of all time, ranked by
medals obtained: to do this he will need a gold medal at
\href{http://www.uan.edu.co/imo2013/en/}{IMO 2013} in Colombia, which
will be no easy task.

James will be disappointed that his uncharacteristically modest
performance has left him with a bronze medal, but this is due to his
extraordinarily high personal standards: we are as satisfied with him
as with the rest of our team.

Next year, Andrew, Daniel and Matei are all available to compete
again, as are reserves Gabriel and Katya. We greatly look forward to
their contributions.

On the other hand, James, Sam, Josh and first reserve Adam are all off
to \href{http://www.trin.cam.ac.uk/}{Trinity College, Cambridge}: we
are sure that they will continue to excel there.

\section*{Leader's diary}

Many of the events this year were overshadowed by the untimely death
of Sam Cappleman-Lynes's mother, at the beginning of the IMO. While,
clearly, it would be inappropriate to include these events in my
narrative, the reader should understand the difficulties involved. We
were all impressed by Sam's determination in successfully
participating under such conditions.

\subsection*{Sunday 1st July}

London's Piccadilly line takes me to Heathrow Terminal 5, and I am
only ten minutes later than the appointed meeting time. There I meet
the team, our pastoral assistant Bev, and James's parents who have
agreed to take delivery of the uniform.

Jack Shotton arrives half an hour late, but we have left ourselves a
vast amount of spare time and he has hardly dented it.

We check in, and decide to eat in the Giraffe restaurant the other
side of security. The waitresses, from their vantage point a
respectful distance away, admire Sam's absent-minded
solving of Rubik's Cubes (and a range of similar beasts, one of
them large and dodecahedral).

Then we embark on the journey, allegedly the single longest-distance
flight offered by British Airways. I am kept company by Andrew and
Josh.

I glance out of the window from time to time, and ascertain that our
cruising altitude is roughly two metres, and that the mid-Atlantic
ocean looks a lot like a big piece of sheet aluminium.

Blankets and pillows are provided, and we make good use of them: not
much happens.

\subsection*{Monday 2nd July}

We arrive at Buenos Aires Ministro Pistarini airport. Baggage reclaim
is a very slow process, but that gives me enough time to disappear and
wash my face and brush my teeth. After doing that I feel almost like a
human being again.

We are then met by a minibus driver and taken to St George's College,
Quilmes, a traditional British international school south of Buenos
Aires.

We are met by their head of maths, Stephen Kay. He shows the students
to the boarding accommodation, then welcomes Jack and me warmly to his
flat, introduces his wife who gives us tea and freshly-baked Canadian
snacks, and allows us to shower and change. He explains how he has
rearranged the timetable in order to provide us with unfettered use of
a classroom. In gratitude, Jack and I immediately conspire to allow
his dog to escape, causing him to spend fifteen minutes rounding it up
again.

After lunch we try to find old IMO shortlist problems our students
have not done, and talk through them. The Argentineans eat late and,
at this school, have afternoon tea at 5pm.

We play frisbee (except for Bev, who does not indulge) in the evening,
but it becomes dark rather suddenly soon after 6pm. Soon afterwards
the Australians arrive. It is nice to see their leader Angelo di
Pasquale and deputy Ivan Guo again. This year, Angelo has brought his
wife Hellen: he missed the IMO last year in order to marry her;
clearly he has waited until this year in order to take her on
honeymoon.

\subsection*{Tuesday 3rd July}

We get up feeling refreshed, and at breakfast time we familiarise
ourselves with \emph{dulce de leche}, a milky caramel substance which
is omnipresent in sweet Argentine food.

The students sit the first practice paper, during which I give a short
masterclass on continued fractions to a handful of school
students. They speak English very nicely, and are a joy to teach.

After lunch we mark the paper. The British students have performed
nearly uniformly well: all have done the same two questions, and
nobody has done the hard third question. We debrief the students on
the paper.

Before dinner, I give the students what I naively imagine to be a
rousing pep talk, and present important logistical advice. Mostly this
consists of reminding them that Jack and I have to argue with
non-native English speakers about the strengths of their writings, and
encouraging them not to indulge in partying until the point where
Paper 2 is a recent memory.

In the evening there is a party for us staff at the headmaster's
residence, which renders my advice to the students ironic.

\subsection*{Wednesday 4th July}

After breakfast, I bid the students \emph{au revoir}, then embark on
the journey south. A taxi takes me from the school to Buenos Aires's
bus station.

At the station I see Arturas, the Lithuanian leader: we strike up
conversation, and it turns out we will be sharing a bus. We become
confused together about the buses: our bus is not listed on any of the
boards. With a few minutes to spare, we strike up conversation with a
nearby family, who explain that they are in the same position. One of
them goes to ask, and discovers that the bus is late, and that late
buses are not listed on the boards until they have arrived.

In the end it arrives. The bus turns out to be extremely comfortable:
my seat is, in essence, a bed. Some kind of \href
{http://www.rottentomatoes.com/m/a-thousand-words/} {ghastly Eddie
  Murphy movie} is played, but it is easy to ignore, particularly as
Eddie Murphy spends most of the movie in silence.

Arturas and I arrive in Mar del Plata, and are met by smiling IMO
staff. We are beckoned into a taxi, which drives us about 10km south
to a countryside hotel. There we meet many old friends.

I have a nice room, but it is extremely warm. I spend a while trying to
locate the source of the heat so that I can neutralise it. Eventually
I realise that there is underfloor heating over which I have no
control. So I simply open the window and resolve to avoid the floor.

I pick up the shortlist. By tradition, a shortlist contains seven or
eight problems in each of four areas: algebra, combinatorics, geometry
and number theory. The organisers list them in order of perceived
difficulty. So C7 might be expected to be a very hard combinatorics
problem, and G2 might be a much easier geometry problem.

After some time enjoying the shortlist, I decide to go and be
sociable. I walk back to the hotel bar: representatives of Germany,
the Netherlands, and New Zealand are there with Matja\v{z}, the IMO's
technical support man. So I sit with them for a while.

\subsection*{Thursday 5th July}

After a brief welcome meeting, where a few logistical points are
discussed, we have a free day to consider the shortlist.

After getting hopelessly stuck on one of the combinatorics shortlist
problems, I notice there is a little shop in the hotel which sells
bric-a-brac including replacements for forgotten swimming shorts, and
I pick up a pair.

Upon going to the pool I discover that the sizes, which I had taken to
be classified by waist size in Imperial inches, in fact follow some
other system, and that I have bought a very large pair of shorts. No
wonder the girls in the shop were laughing at me. I deploy ingenuity
of a sort that I lacked when attempting the combinatorics problem, and
find a way of tying one part of the shorts to another part of the
shorts. I now simply have a fashionably baggy pair of shorts, and can
go swimming.

\subsection*{Friday 6th July}

The day begins with the first proper jury meeting. The first item of
business is the task of discarding questions which are already known,
or which are uncomfortably close to known questions.

One leader begins by standing up and suggesting that problem A2 bears
some resemblance to a problem on IMO 2000. The chair of the jury nods
sagely, and the problem is struck off. There is an uproar, where
various members of the jury explain that it would be normal to have a
formal motion, translate it into the other official languages of the
jury, and then vote on it. The chair of the jury indicates that he is
happy for us to do so, ``if we want to spend all day voting''. He
pushes it to a vote, and does not permit discussion, though we have
not even been shown the original question to which A2 is being
compared. Naturally, many people wish to abstain, but the chair of the
jury is uninterested in having abstentions counted. ``Why do you want
to play these games?'', he asks.

After this crash course, the chair of the jury rapidly comes up to
speed with standard procedure, and happily we are able to scrutinise
all further removals. This year, the IMO question gods will be pleased
with their sacrifice: a total of six questions are burnt.

\subsection*{Saturday 7th July}

While yesterday felt unproductive, it turns out that sleeping on
everything has led to us having a good understanding of the
problems. Today's debate is largely well-informed.

The jury start by picking two easy questions. There are two obvious
candidates, A1 and G1, and those are rapidly chosen.

Then we have a difficult fight over the two hard problems. There is a
number theory problem, N7, which has clearly won the jury's hearts,
but it is not obvious whether its partner will be geometry or
combinatorics. Eventually, after much talking, two geometry problems
are rejected (including G5 which is much-liked but slightly too easy
to be a hard problem) leaving the combinatorics question C6 standing.

During this process, Geoff walks in and is greeted by many old
friends.

After lunch, one leader suggests it might be possible to save
ourselves time: he proposes that we vote to select the
recently-rejected geometry problem G5 straight away. So we vote on
whether to vote on this. The results are 38 all, with seven
abstentions. The large number of abstentions leads me to think we
should have voted whether to vote on whether to vote this problem onto
the paper first.

The mechanism of selecting questions is traditional. First people
propose individual problems, until the jury tire of this. Then people
propose pairs of problems, until the jury tire of this. Then, finally,
there is a voting procedure, following a devil-take-the-hindmost
system over several rounds. The numerical details are, by tradition,
agreed on an \emph{ad hoc} basis every time, but it turns out
surprisingly often that we agree to eliminate $\lfloor\sqrt{n}\rfloor$
pairs of problems every time there are $n$ remaining.

The medium questions are eventually selected. They do indeed include
G5, and also the algebra question A3.

It is time for the English Language committee to meet: we have two
hours to reformulate the problems, taking into account any
inadequacies or ambiguities, and hopefully making it easier for
translation into the fifty-four other languages in which the papers
will be sat. It seems to have become traditional for the UK leader to
chair this committee (and I tend to invite the New Zealand leader
Chris Tuffley to be ``secretary'', which, among other things, involves
doing all the work). By the time we are done persuading the jury of
the merits of our work, and incorporating their several excellent
suggestions, I am quite tired. Accordingly I spend a relaxing evening
annoying the Francophones by sitting in on their translation meeting
with a beer.

In the evening some good news reaches us: the results of our annual
pre-IMO contest against the Australians, the Ashes of Mathematics. The
UK has won it by the comfortable margin of 72 points to 52,
corresponding to three more problems solved. Geoff is understandably
unhappy at having to break the news to Angelo.

\subsection*{Sunday 8th July}

Today is a big day. This is not least because it is the first time a
British man has been in a Wimbledon men's singles final since
1938. However, naturally enough, our man Andy Murray loses to the
rampaging Roger Federer. The Swiss leader Julian Kellerhals acts
magnanimously enough about the whole matter.

Informed partly by sleeping on it, and partly by other people's
problems in translation, Chris decides to further polish the
English. There are few complaints.

Now the English version is done, we are left with little to do for the
day, while many other language versions are produced.

\subsection*{Monday 9th July}

Today is Argentina's Independence day: the 196th anniversary of their
declaration of independence in 1816. The organisers have timed the IMO
so that the opening ceremony will fall on this auspicious day.

We are put on buses to a concert hall in Mar del Plata: we are met by
loud drumming coming from a crowd of carnival performers.

The opening ceremony is blissfully free of politicians: the only
people who speak are mathematicians, and the speeches are short. We
start by standing for the Argentine national anthem, which breaks into
rock music in places.

The IMO anthem, a gift to the world from the first Argentine IMO in
1997, is played by three accordionists and a singer.

The parade of nations occurs: every country marches across the stage
in turn. Our team look very smart in embroidered blazers. For the
first time, an IMO oath of fair play is sworn by all contestants.

Then two big cannons fire vast amounts of confetti into the hall. When
the air clears, we observe a man come onto stage equipped with two
small hammers on metre-long cords, one in each hand. When these are
whirled at speed, he becomes a tapdancing quadruped.

After being taken back to our base, the rest of the day is spent on
mark schemes. The coordinators have been hard at work devising
mark schemes, and they seem to be sensible (after a few corrections
suggested by the jury). My idea of a perfect mark scheme is one which
provides some structural insight into a problem. These mostly aren't
like that, but they do at least look like they might work.

One difficulty this year is that they have had to construct
mark schemes in such a way as to not reveal alternative solutions to
problems: this is a novel security measure devised by the advisory
board.

The other piece of business in this long evening's work is the
elections for the IMO advisory board. First is the election for
secretary, to replace John Webb of South Africa. There are two
candidates, Indra Haraksingh of Trinidad and Tobago, and Gregor
Dolinar of Slovenia; the latter wins. In a moment of high drama, Radu
Gologan of Romania ties with Rafael Sanchez of Venezuela for the post
of general member, with 45 votes each, and then concedes to him. While
this act of kindness was conspicuous, I am proud to be in a community
where elections are conducted in such a friendly manner.

\subsection*{Tuesday 10th July}

Today is the day of the first IMO paper. 

It is traditional for students to be allowed to ask questions in the
first half-hour of a paper, in case they do not understand the
problems. In recent years, the jury and the students have been far
apart during the paper, and questions have been relayed
electronically. This year, however, there is less than 20km separating
us, and instead of transporting electrons from the exam site to the
leaders' site, we will be transporting leaders from the leaders' site
to the exam site. This is a feature that invites nostalgia from the
more experienced leaders.

Our students ask no questions, which makes us happy. We had feared
that the complicated wording of Q3 would create huge problems, but
things turned out fine in the end.

After the end of the Q{\&}A session, we stroll downstairs and out of
the building, finding ourselves immediately amidst a crowd of deputy
leaders trying to board some buses. Since I am not supposed to
communicate with the deputies until after Paper 2, this may be
regarded as a bit of a security breach. Jack is nowhere to be
seen. The Canadian deputy waves at me from a bus, smiling
mysteriously, but then I realise with embarrassment that the Canadian
leader Jacob Tsimerman is right behind me. I see one pair, a leader
and a deputy, stroll down the street as the old friends that they are.

After we are safely put on separate buses, we are taken off to a
barbecue in a country estate. There is a quick tour of the vineyards
and fields, and then we are taken into a large tiled barn with several
log fires. There are \emph{empanadas} (traditional Argentine Cornish
pasties), sausages and black pudding, grilled chicken and grilled
steak.

We are kept entertained by some musicians, and later by some tango
dancers and a singer with a large collection of costumes made from a
small amount of fabric. Then some drummers come out, and the
aforementioned quadrupedal tapdancer. He uses his hammer-on-a-rope to
knock a cigarette out of the New Zealand leader's mouth. It's a filthy
habit, and I'm sure it dissuades him from adopting it.

After lots of food and several glasses of wine, and with the scripts
from day~1 looming later in the day, it seems sensible to join a group
who plan to walk back to our hotel. It turns out to be a very pleasant
walk of about 10km, across the coastal lowlands.

The scripts arrive, and I spend the rest of the evening assessing
them.

\subsection*{Wednesday 11th July}

Today is the second IMO paper. Today, again, our students make us
proud by not asking any daft questions (or indeed any questions at
all).

One question says that a certain named nearby student keeps making
loud noises with his hands, and asks if he can be made to shut up. We
decide, amidst some amusement, that the answer must be ``yes'', and
send orders to threaten him appropriately.

After this we move into a hotel across the road from the students. I
am reunited with Jack at once, who brings me news of our
students. Apparently, Sam's habitual Rubik's Cube solving led to him
being mobbed by admiring schoolgirls at St George's. Also, the team
members have developed a love-hate relationship with \emph{alfajores},
the Argentine snack obtained by inserting a layer of dulce de leche
between two sweet biscuits, and coating the whole thing in chocolate.

Afterwards I meet the students coming out of the exam. They feel
battered by the difficulty of the paper, but perk up after realising
most other countries feel similarly.

Jack has a hard day: not only must he get up to speed on the scripts
from Day 1, we must both get used to the scripts from Day 2. We spend
a long evening in, reading them.

\subsection*{Thursday 12th July}

Jack and I take an opportunity to visit the students' recreation room,
where they are based when nothing else is happening.

I enter, and find that Sam is engaged in a chess tournament. A few
yards away, Andrew Carlotti has joined an international effort to make
progress on a 24000-piece jigsaw puzzle the size of a small
carpet. Daniel is learning circus skills, and can juggle hoops for
short periods of time. The others are all milling around.

The room is excellent: besides these activities there are all manner
of board and table games, table football, pool, video games,
ping-pong, people teaching tango dancing and acrobatics, and at some
hours, karaoke.

Rumours reach us of some unfortunate hijinks the night
before. Apparently a certain team had a raucous party in their rooms,
with various sorts of contraband present, and some damage to
property. Options with that team were mulled over, but in the end the
organisers felt they had no choice but to rusticate them, sending them
to a hotel the other side of town.

A stop-and-search policy is instituted, and Jack and I have our bags
inspected every time we cross the road to enter the students' hotel.

After lunch, we begin coordination, the process of agreeing scores
with committees of locals. This starts with problem 6. We were asking
for three points for Andrew and two for Matei.

The coordinators say that Andrew's work might even be enough to
receive four points, but that they wish to think about it some more in
comparison with other scripts. We're in no hurry, so this seems an
entirely adult approach.

In the evening, Jack and I feel well in control of all the scripts, so
we pop out to a pub in town, in the company of Mark from Ireland and
Marteinn from Iceland. The first appropriate venue we reach is in fact
the local Irish pub. The atmosphere and clientele are pleasant and
there is a local stout and a brown ale on sale.

Perhaps the only significant area where the pub is seriously deficient
is that of the local drunk. This pub's drunk, a man with the
implausible name of Dariak, besieges us, tells us repeatedly that he
is both an artist and is well-connected, and refuses to believe that
Jack is not German. His outgoings on business cards must be
substantial, for it seems giving us one each is probably not
enough. Eventually two nearby girls, Vicky and Camila, take pity on us
and magic him away; we are grateful for the change of company.

\subsection*{Friday 13th July}

Today the students will be making friends with dolphins at the local
aquarium, while Jack and I do the bulk of the coordination.

It starts with good news on Problem 6: the coordinators have decided
that Andrew's script (and all similar scripts) deserve the extra mark
mentioned earlier.

However, Andrew loses a mark on Problem 4. This is entirely
self-inflicted: he has a daft strategy for checking one of his
solutions, and tells the reader he can't be bothered to carry it
through to a conclusion.

We also have a fairly painless time on Problem 3, and pick up 5 for
Carlotti and 4 for Aaronson, both of whom manage all of the first part
and make inroads into the second.

Problem 2 is rather less straightforward. Five scripts are
uncontroversial, but the clash is over Matei's. We had planned to ask
for two points, but the coordinators seem intent upon giving it
six. Matei has proved the equality for all $n\geq 11$ (and we have
checked that the argument works equally well for $n=10$). However, the
given bound becomes pathetically weak as $n$ becomes large, and, as
Jack and I have spent a long evening verifying, the method can't
easily be adapted for the cases $4\leq n\leq 9$. So, in our opinion,
Matei has no programme for solving the full problem.

Jack and I ask permission to go talk in a corner. We are agreed that
this mark is inappropriate, and that Matei would prefer to receive an
honourable mention (the standard award for a student with a full
solution to some problem, but who receives no medal) than a decidedly
dishonourable bronze.

The coordinators offer four, since it's the arithmetic mean of what we
are asking and what they are asking. We explain that that is no
good. Indeed, our absolute demands are that the student gets a mark
which is both fair and in line with those given to other students;
this would achieve neither.

When we return this afternoon, having decided to deliver an ultimatum,
things are made uncomfortable for us: the organisers have decided to
impose no hierarchy on the coordinators this year. Ordinarily there
would be a chief coordinator responsible for all coordination, and
problem captains, one per problem. The former post is in fact
specified by the organisers' own contest regulations. As a result, we
have no option but to reason with all coordinators simultaneously, and
we are aware we have no recourse beyond that other than to the full
jury.

With twelve coordinators sitting in a semicircle around us, we ask
that they either reconsider the mark for Matei's script and for all
scripts with similar work (we later discover there are five of them),
or else we will be forced to bring the matter to the jury. They
concede, and go to work on remarking. Later, we meet them again. They
make some sensible points in favour of Matei's script -- that it is
closer to a solution than he is aware -- and Matei ends up with a much
more sensible mark of four.

Problem 1 is hard work, but of a less extraordinary sort. Five of the
scripts are uncontroversial solutions. Jack has found a mark in
Matei's script: in one place there is a mysterious formula. Jack has
determined that it is a slightly incorrect statement of Menelaus's
theorem, as applied to a certain triangle involving a novel
constructed point. Jack has also found a full solution which starts
with (a corrected version of) the same formula. The coordinator takes
ten minutes to reproduce these formulae herself, and then agrees that
this is worth a mark. This pushes Matei one step closer to an honest
bronze.

\subsection*{Saturday 14th July}

Coordination finishes for us rapidly with Problem 5. Josh has done it,
and there are points to be found for James and Daniel. The
coordinators congratulate us: apparently Josh's script is the only
script they have seen which succeeds by synthetic means.

This leaves us with 115 points. We are pleased with this: it is clear
already that some strong countries have been damaged by this very
difficult pair of papers, and we are heartened that the UK has
weathered the storm.

I decide to seat myself in the hotel lobby, and gossip with all who
pass. I find this just as good as sitting in front of the scores
screens around the corner.

There are considerable rumblings about Problem 4: many people
apparently feeling that the coordinators have interpreted the
markscheme in an unacceptably strict fashion.

The Canadian staff invite me to join them in the swimming pool. I
spend an hour and a half with them, and with Mark from Ireland. In the
hot tub, meanwhile, one leader well known for such things is harassing
a female deputy leader: it is all rather sad.

Jack and I also spend some time in the students' hotel. It is Sam's
birthday, and we congratulate him.

We also are in time to see James Aaronson win the IMO
\emph{Ta-Te-Ti-To} tournament (this game is four-in-a-row, played by
two players on a $4\times 4\times 4$ cubical board). John Papantoniou
from Australia has reached the final of the chess contest, but we are
not around to see him beaten by a Lithuanian student.

Then I return to the hotel lobby. Mea Bombardelli, the Croatian
leader, has signed off with 110 points in total, which this year is
enough to beat Germany. She is intrigued that her team always seem to
get about 110, no matter how hard the papers are.

\subsection*{Sunday 15th July}

Bev comes over to the leaders' hotel for a cup of coffee, in advance
of the final jury meeting. After Jack and I have done with breakfast
we head down. There is a lengthy delay as the organisers work out how
to project things on a screen.

One leader is good enough to open the proceedings by telling the other
assembled leaders how coordination works: clearly he feels that the
point of it was lost on some of them.

It turns out that there are three scripts left unresolved, all on
Problem 4, and it is the job of the jury to settle them. We vote first
on two Estonian scripts, and in the end we uphold the suggestions of
the coordinators.

Then Zuming Feng, from the USA, gets to discuss his unresolved
script. He suggests that the organisation of the coordinators is the
worst he has seen in fifteen years or so: the coordinators wish to
penalise his student partly for verifying a symmetric two-variable
homogeneous fourth-degree polynomial identity by eye, rather than by
multiplying out both sides. This is not the International
Multiplication Olympiad, he thunders.

The coordinators want to give it five. The script looks very similar
to our student Andrew's six, so I am comfortable with a mark of six
for it. This is what happens, eventually, but there are procedural
misgivings all around.

A leader from Eastern Europe proposes that we change the definition of
``honourable mention'' this year, to award honourable mentions for
students with fives and sixes in Problem 4. I find the suggestion
amusing but unconstructive.

Then there is a short break, and we go into the more normal business
of the final jury meeting. There is a report from the chief
invigilator, and we approve the scores and medal boundaries. Two new
IMO hosts are announced: following the already-known hosts for the
next three years (2013 in Colombia, 2014 in South Africa and 2015 in
Thailand) will be IMO 2016 in Hong Kong and IMO 2017 in Brazil.

We take the team out for lunch at a restaurant down the road. There is
steak and pasta, all very pleasant. The students tell us that Andrew
spent the evening before partying until 6am and then spent the time
between then and breakfast solving last year's shortlist problem C8
(the ``napkins'' problem).

I give the students their scripts back, and award the \emph{golden
  pen} for most painful script which nevertheless receives a full
score to Andrew, for his work on Problem 2. On balance, this year has
been more characterised by the team setting about the problems like so
many angry ferrets, and obtaining large numbers of marks by bravely
and repeatedly trying to do the problems, but sadly we have no prize
for that. I also award the \emph{Sceptre of UNK}, my IMO jury voting
stick, to Matei Mandache. It will endow him with demigodlike powers
for the next year.

We hurry from lunch to the closing ceremony. There is more of the
national anthem, more of the IMO anthem, and the medals are
awarded. Jeck Lim of Singapore, who achieved the unique perfect score,
is ushered on stage alone to receive his gold medal.

After this we relax for a while, and drink a bottle of ros\'e
champagne that Josh Lam's father has donated. The students discuss
their principal source of intrigue: a female student from an Asian
country phoned their rooms overnight, asking if she could speak to
UNK4 (Daniel). However, events during the daytime suggest that she
miscalculated and that it was UNK5 (Josh) that has attracted her
interest. None of the team have any idea what to do about this.

Then there is the closing party, taking place in a large hall in the
students' hotel. Our team are ushered to a table, and we join the
Pakistani leader Barbu Berceanu, an expatriate Romanian. Barbu appears
to enjoy the chance to talk to our Romanian speaker Matei.

The food arrives eventually, and then entertainment is laid on from
the other end. First up is the annual award of the \emph{golden
  microphone}, to the leader who has made the most jury speeches. This
year there has been a clear winner: the jury has been pleased to
listen most often to the pronunciations of Lev Radzivilovsky of
Israel. This gives way rapidly to the finals of the \emph{IMO's Got
  Talent} competition. Highlights include some pleasant singing by a
Romanian student, and a bizarre routine by two Pakistani students, one
of whom plays the piano while performing contortionistic moves, while
the other spins a plate on a stick.

I retire early: the next two days will be hard work.

\subsection*{Monday 16th July}

The day begins with a five-hour bus and half-hour taxi ride back to
Buenos Aires's main airport. The film this time is a
\href{http://www.imdb.com/title/tt1568911/}{lengthy melodrama} by
Steven Spielberg about a horse. By the time the film is halfway
through, I am simultaneously yearning for the
\href{http://www.imdb.com/title/tt0067023/}{early days of Spielberg},
and also hoping that the next character to appear will be a
glue-boiler or a Kazakh butcher or someone else with a similarly
robust attitude towards horses.

We are, because of scheduling issues, left waiting for several hours
in Buenos Aires airport. One eagerly-anticipated source of excitement
was waiting for Bev to check Daniel Hu in, as for some unknown reason
the website didn't let her do so last night. However, she has her
boxing gloves on and so this takes just fifteen minutes.

While she is remonstrating with the staff of Iberia, we are mobbed by
several crowds of Argentine teenage girls, who are apparently all
driven wild by British accents. Presumably, since chiselled Latin men
with designer stubble and impressive dance moves are two-a-penny
around here, what women around here really hunger for is clumsy
ill-dressed pasty types from the UK.

We decide to hide from the adoring hordes in an airport caf\'e. There
we order a large amount of pizza to tide us over. James, who doesn't
like food with ingredients in, gets a small pizza \emph{sin queso}.

We mostly engage in playing liar's poker, but Bev reports that at the
other end of the airport is a big display about
\href{http://en.wikipedia.org/wiki/Falklands_War}{the Falklands
  war}. I go to have a look. Every banner of photos is entitled, ``las
Malvinas son Argentinas y sus recursos naturales tambi\'en'' (``the
Falklands are Argentinean and so are their natural resources''). There
are lots of pictures of Argentine soldiers in successful moments, and
infographics about sunken British warships. I was startled by this as
an introduction to the country for visiting foreigners. However, I am
pleased to realise that our dealings with individual Argentine people
over the course of this trip have in no way reflected the unease
between our governments.

The queues for security and emigration passport control are lengthy;
the whole process takes a while, and I refresh myself with a fizzy
pomelo juice drink afterwards.

Eventually we get on a plane. Sam and Andrew, next to me, fall asleep
immediately, which makes for a very quiet flight.

\subsection*{Tuesday 17th July}

We all wake up, and calculate that Sam has spent seventeen of the last
twenty-six hours sleeping. Before too long we arrive in Madrid Barajas
airport, only about 50 minutes late. The landing is not exactly
smooth; many people experience reverse peristalsis on the way in.

The delay has caused us to miss our connecting flight; the staff
juggle us all on the next flight to Heathrow, and give us one-third as
many meal vouchers as there are people in the party.

To get to the next flight, we have to go to the very other end of the
airport. Madrid is very, very hot. I become stressed, and Jack beats
some sense into me.

There is some confusion and delay in working out where in the
departure hall the restaurant \emph{ARS} is, since this is the unique
restaurant at which the meal vouchers are redeemable. Eventually we
determine that the ARS is to be found at the rear end. They sell us
sandwiches, crisps and drinks to go and we catch our flight home. Soon
afterwards we arrive at Heathrow, tired and smelly, to be met by four
friendly parents.

\section*{Acknowledgements}

It is no easy matter to send six young people seven thousand miles, or
to prepare them for competition at the highest level against young
people from other countries, or to support them before and during this
process.

Many people helped in many different ways. Here are some whom I would
like to thank explicitly:
\begin{itemize}
\item the generous and helpful staff and students of St George's
  College, Quilmes, for hosting us in the runup to the IMO (and
  particularly Stephen Kay, Natalia Kovsch and Derek Pringle);
\item our team's fantastic guide Javier Corti, who looked after them
  so well;
\item many IMO leaders, for their friendliness and kindliness
  (particularly Gregor Dolinar, Roberto Dvornicich, Zuming Feng, Mark
  Flanagan, Hans-Dietrich Gronau and Julian Kellerhals);
\item John Webb, for some kind help at a difficult time;
\item the organisers and staff of IMO 2012 for their excellent efforts
  in creating an IMO that students greatly enjoyed, especially
  Patricia Fauring, who seemed omnipresent at times;
\item all the UKMT's volunteers, who gave freely of their time to make
  our team the problem-solving force that they are;
\item all the students' teachers, who ensured that the mathematical
  base level of the team was so high;
\item each of our families, who support each of us unconditionally;
\item Rachel Greenhalgh and Joseph Myers, for their careful and caring
  support of the team from back home;
\item Bev Detoeuf, for her great logistical and pastoral work;
\item Geoff Smith, for his constant excellent advice and support;
\item Jack Shotton, for being an academically formidable and pastorally
  inspiring deputy leader;
\item every dedicated olympiad student in the UK who failed to make
  the IMO team this year, for creating an atmosphere of strong
  mathematical achievement; of the hundreds of these I should single
  out Adam Goucher, Gabriel Gendler and Katya Richards, our wonderful
  reserves;
\item the team themselves, for doing all they could and doing it
  pleasantly. They have been excellent representatives of the country.
\end{itemize}

\emph{James Cranch, 19th July 2012}

\end{document}
