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\title{Observations on IMO 2012}
\author{Geoff Smith}
\date{}
\begin{document}
\maketitle
\medskip

James Cranch has given a full report on events in Argentina,
so I will confine myself to matters disjoint from those
which he has already covered.

I take a taxi to Luton Airport, due north of London. I have
a lunchtime flight on easyJet to Madrid. I am travelling
independently of the British student party, and am trying to
be clever. From Madrid I have a ticket
on Aerol\'{\i}neas Argentinas, including a change of
airport in Buenos Aires, but taking me through to 
the IMO 2012 site of Mar del Plata.

The reason I am doing this, rather than taking a more sensible
London to Buenos Aires flight, is that I have been intimidated
by tales of systematic overbooking of domestic flights in
South and Central America, and hope that intercontinental 
passengers may get priority. It turns out that my overbooking
fears are entirely misplaced, and that a direct flight on BA to BA
would have been easier. 

The flight to Madrid is uneventful, and over the Bay of Biscay 
I observe the first sunshine I have seen since March. In Madrid,
in a prescient move, I change some euros into Argentinian pesos.
Argentina operates a Soviet style money exchange system. Once inside
the country, it is only possible to exchange hard currency for pesos
in that direction, and at a special rate which is officially 
determined, and has no relation to the peso exchange rate available
outside the country. Welcome to a Latin DDR. It takes me back
to the Friedrichstra\ss e crossing in Berlin during the Cold War.

The flight to Buenos Aires takes 12 hours, and I am seated
next to a snorer. I had better not discuss my darker thoughts.
On a happier note, Aerol\'{\i}neas Argentinas have issued me with
a voucher for a bus ticket. The plane lands later than it should 
have done, and I am torn between taking the official bus, or
jumping into a random cab, which according to Wikipedia is
the strategy favoured by those who wish to be robbed and murdered.

I decide to take the officially sanctioned
Manuel Tienda Lexi\' on bus, even though there is some risk that it will 
not quite make the connection in time. In fact the driver makes good time, 
save for a pause at the central bus station where there is a lot of 
fuss made about luggage. We arrive 10 minutes ahead of schedule,
and I rush to check-in my suitcase. That goes smoothly, and I am starting
to meet friendly IMO travellers. In particular, the IMOAB secretary
John Webb makes contact.

The short flight from Buenos Aires 
is scheduled to take just 40 minutes, and it seems that they 
operate a very friendly
policy of waiting for connecting flights, so we tarry for some time while 
several groups of IMO passengers board the plane from various parts of the
world. My anxiety about missing this connection was ill--founded.
There are only two flights scheduled per day to Mar del Plata, so it would
not be polite to leave people behind.

Mar del Plata airport terminal is comparable in scale to
Ia\c si, Romania, and to the old Kharkov, Ukraine effort 
(until it got redeveloped for Euro 2012) and the happily demolished 
Bath Bus Station of grotesque memory. The IMO organizers are
there and efficiently transport John and myself to the leaders' 
residence, a sports complex due south of the city. 

In his leader's report, James has alluded to the mysterious underfloor 
heating in the rooms. Mine had no off-switch, and since I prefer
life on the cold side, I have to get it turned off by special appeal
to the reception desk.

I have arrived in the middle of the question selection process, and
I am astonished to learn that (a) the jury has not implemented
my suggestion as to how to choose problems, as explained
in my 2011 \emph{Observations} and (b) the Problem Selection
Committee has decided not to put the problem I submitted for
IMO 2012 into the shortlist. I blame James Cranch for not
completely reorganizing jury problem selection protocols 
according to my whim. As for the responsibility for not using
my problem, it is less clear who is the guilty party. 

On reflection, I now realise that my ploy of constructing
culturally relevant problems might be viewed as a transparent
ruse to avoid judgements about mathematical good taste, and to
appeal instead to the jury's sentimental and craven desire 
to pander to the host nation. That was my plan anyway.

I tax Problem Selection Committee member K\' os G\'eza as to
why my question has been omitted. After he reassures me that
it was left out because it was a boring question of no intellectual
merit, I am happier. My fear had been that it was the 
intense eroticism of the question (which involves The Tango)
which had caused it to be set aside. 

Anyway, the problem is tailored to the Argentinian IMO,
and since we probably won't have another one for a generation,
here it is. The original version came in various forms, so that
you could tune the level of difficulty. This is the
straightforward version of what I think of as
\emph{not Problem 4}.

\begin{center}\large{The Tango}\end{center}

\medskip

Let $n$ be a positive integer.
A \emph{tango} is a dance on the number line.
Each dancer makes steps, each of which adds 1 to, or subtracts 
1 from, their current position.
Two people stand at 0 to begin. Each of them takes $n$ steps in sequence.
They keep in time with one another, so for all $i$, they both take their $i$-th
step at the same moment. They must each conform to the three rules of tango:

\begin{enumerate}
\item[(i)] Each dancer must finish at 0.
\item[(ii)] If, during the dance, they are in the same position, then they kiss
(this does not include the start or the finish).
\item[(iii)] The steps of each dancer are scored as follows. Each right step
scores the number of left steps that the dancer has already taken.
Each left step scores the number of right steps that the dancer 
has already taken. Each dancer must dance so that the sum of the scores
of his or her right steps is the sum of the scores of his or her left
steps. 
\end{enumerate}

\begin{enumerate}
\item[(a)] For which positive integers $n$ is a tango possible?
\item[(b)] Show that each tango must include a kiss.
\end{enumerate}

Last year, the usual Australian leader Angelo di Pasquale
did not attend the IMO because he was getting married. This
year, by way of compensation, Mrs Hellen di Pasquale accompanied
Angelo, and was able to witness what a hard time we give
him over breakfast. I personally felt a duty to lay it on thick.

There was quite a lot of IMOAB business to do. The Ethics
Committee thinks that there should be a public affirmation
of our commitment to honesty at the IMO, but time is at a premium.
Various people help to construct an item to insert
into the opening ceremony at short notice, for which
many thanks. Next year we will have time to construct
something a little more polished, possibly involving
human sacrifice (I have some suggestions).

We heard strong
presentations from Hong Kong SAR and Brazil concerning
their bids to host future IMOs, and we later recommended
to the jury that both generous and kind offers be accepted.
The forthcoming schedule looks very attractive, but with so
many IMO countries coming from Europe, I do wonder if some of
the smaller ones will be able to afford so many long journeys.
Of course, this is the usual problem for Asian and South American
countries.
 
The IMOAB also had to construct the manual for the new \emph{Ethics Committee}. 
Henceforth, the Ethics Committee will have all sorts of powers
and responsibilities, including looking for marking anomalies
in past scripts. From now on, if KLI7, the leader of 
The Klingon Empire, asserts in co-ordination
that \emph{tugh qoH nachDaj je chevlu'ta'} means
``and so, by the Combinatorial Nullstellensatz, we are done'',
and later the Ethics Committee discovers that it actually says
``soon a fool and his head will be separated'', then, in the
words of Irving Berlin, \emph{There may be trouble ahead \ldots}
Students of the Klingon Empire, please remind your leader of this
new and troubling circumstance.

I was, as usual, delighted to see so many old friends in the jury.
However, I had a most pleasant surprise when so many students
greeted me with such enthusiasm. 
This seems to be connected to \emph{The Windmill Problem}, IMO 2011/2. 
When walking through the student recreation room, or passing
through an airport going to or from the IMO, groups of superficially
friendly students would rush forward and ask to take photographs.
I pretended
not to know that they were actually looking for material 
to decorate their 
dartboards. 

Towards the end of the IMO, Charles Leytem called 
a meeting for people interested in EGMO 2, the second European
Girls' Mathematical Olympiad which will be held in Luxembourg
next April. The inaugural competition was held last April
in Murray Edwards College, Cambridge, in the UK. The EGMO room
was completely full, and it seems that the number of 
European nations participating will be sharply up from the 
16 countries which attended the Cambridge event.  
Charles has cleverly found some funding which will defray some
accommodation and travel costs.

This year there were some major European nations which did
not participate. Next year, it seems that there will be many fewer
such countries. There is good reason to hope that that EGMO 3
will be held in Turkey, but we await final confirmation of this.
Applications to host EGMO $n$ for $n \geq 4$ should be directed
to the Provisional EGMO Advisory Board. The chair
is Birgit van Dalen of the Netherlands, on
{\tt bevandalen@gmail.com}

The journey home was another adventure. First we stood
in the cold outside the students' hotel for an hour, until
the news came through that Mar del Plata air-traffic control
was taking industrial action. Why does an
airport that takes two planes per day need air-traffic control?
The people who have to make a sharp airport change in Buenos Aires
are put on a bus to the capital. I have a leisurely 
change, so I elect to gamble on there being a flight at some point.
People like me go by bus to Mar del Plata airport 
to do some serious waiting. There I run into the leaders
of the highly successful Saudi IMO team who have travelled to
the airport independently. One of them,
Abdulaziz Al-Harthi, has to make a very rapid change in Buenos Aires,
and his embassy has laid on a car for him. He offers me a lift,
and I ponder the relative attractions of this new offer, and
the worthy Manuel Tienda Lexi\' on bus, and make a rather easy choice.
Thank you Abdulaziz. The local flight takes off at lunchtime, and 
we cross Buenos Aires at pace.

I still have plenty of time left, and this time 
I am properly prepared for the journey north:
shorts and lots of water, and of course piano wire in case
they put me next to another snorer. In fact all is well, and I
have a very pleasant flight back to Europe.

Now, some shameless advertizing: my \emph{twitter} address is 
{\tt{@GeoffBath}}. My UKMT book ``A Mathematical Olympiad Primer'' is now
greatly extended, and includes an extra four years worth of problems.
It is available at http://www.mathcomp.leeds.ac.uk/publications/, and
since virtually all the 14--16 year olds of your country will no 
doubt wish to read it, note the UKMT bulk discount offer.

\end{document}
