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\centerline{\fonttwobf
38TH INTERNATIONAL MATHEMATICAL OLYMPIAD}
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\centerline{\fonttwobf MAR DEL PLATA, ARGENTINA}
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\centerline{\fonttwobf{18-31 July 1997}}
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\centerline{\fonttwobf{Report by ADAM McBRIDE (UK Team Leader)}}
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\noindent
{\bf Introduction}

\noindent
This report chronicles events leading up to and during this year's International
Mathematical Olympiad (IMO).  The 38th IMO was notable for a variety of reasons.
For example, 82 countries took part, the highest number ever.  The organisation
was superb from start to finish.  Accommodation for the teams was good, with
lots of facilities in the hotel to let students from all over the world socialise
and get to know each other.  The hotel for the Team Leaders (plus Deputy Leaders
and Observers who moved there after the second paper) was palatial.  The weather
was extremely pleasant and the general ambience excellent.
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\noindent
On the mathematical front, it is probably fair to say that the problems chosen
for the competition did not play to the strengths of the UK team this year and
did not allow them to display fully their outstanding ability.  Nevertheless,
there was much to be proud of.  Throughout the months of training, the attitude
of the whole squad, including our reserve, was first-rate.  All of us were
under considerable pressure but the show stayed on the road.  It was a privilege
and a pleasure to work with the squad and to lead the team to Argentina.  I
congratulate all the students on their achievements.
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\noindent
Of course, the IMO represents just the apex of a very large pyramid.  During
the entire selection procedure, which is discussed in more detail below,
thousands of pupils took part, supported and encouraged by hundreds of teachers
throughout the country.  Thanks must go to all of them.  Not everyone can win
a place in an IMO team but, in the spirit of the original Olympic Games, taking
part is what matters.  I hope that in 1997/98 the number of participants will
continue to increase.  As I mentioned in last year's report, we need more
representation from schools in Scotland, Wales and Northern Ireland, while the
number of girls involved in the later stages was again extremely disappointing.
Let us see if this can be rectified next year.
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\noindent
{\bf Mathematical Competitions in the UK}
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\noindent
During the last year significant changes have been taking place in the organisation
of a number of the UK-wide mathematical competitions.  It seems appropriate to
describe some of these changes here.  (However, those of a tender disposition
may wish to proceed to the next section!)
\vskip 0.2truecm

\noindent
For many years the Mathematical Association has run the National Mathematics
Contest (NMC) aimed at senior pupils.  The NMC has acted as the gateway to
Rounds 1 and 2 of the British Mathematical Olympiad (commonly called BMO 1 and
BMO 2) and ultimately to the IMO.  During the last ten years, thanks to the
enthusiasm and hard work of Dr.Tony Gardiner (University of Birmingham), the
UK Junior Mathematical Challenge (UKJMC) and UK Intermediate Mathematical
Challenge (UKIMC) have been established to cater for students in earlier years
of secondary education.  Both the latter competitions have follow-up activities
for which pupils are selected on the basis of their performance in the first
stage.  These activities include the UK Junior Mathematical Olympiad (UKJMO),
the International Intermediate Invitational Mathematical Challenge (IIIMC) and
the European Kangaroo.
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\noindent
A new body called the United Kingdom Mathematics Trust (UKMT) has been set up
to co-ordinate the organisation of all the competitions mentioned above.  Most
of the day-to-day running of the competitions will be done by four subtrusts.
One subtrust will deal with the UKJMC and UKIMC, while a second subtrust
will look after the UKJMO, IIIMC and Kangaroo. At the senior level, the Senior
Challenges Subtrust (SCS) will be responsible for the UK Senior Mathematical
Challenge (UKSMC), which is the new name for the NMC.  Finally, the Senior
Olympiad Subtrust (SOS) will be responsible for everything from BMO 1 through
to the IMO.  To preserve some measure of continuity, the familiar title of
British Mathematical Olympiad Committee (BMOC) will be retained for use by
the SOS.
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\noindent
At this stage the reader could be forgiven for being totally bamboozled by
all the acronyms.  (The use of SMC is particularly confusing because it also
stands for Scottish Mathematical Council, a venerable body which has run its
own Mathematical Challenge for over 20 years.  The latter competition, which
complements the others because of its different format, remains under the sole
control of the Scottish Mathematical Council.)  Membership of the various subtrusts
is now almost complete and the new system should be fully operational in
1997/98.  For information, the Chairman of the UKMT is Dr. Peter Neumann (Queen's College,
Oxford), the Chairman of the SCS is Mr. Bill Richardson (Elgin Academy) and
the Chairman of the BMOC (SOS) is Prof. Jim Wiegold (University of Wales, Cardiff).
With a group of new enthusiastic recruits joining some of the seasoned campaigners,
the new structure should relieve the pressure on individuals and ensure the
long-term future of all the competitions.
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\noindent
Now we can get back to our main business.
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\noindent
{\bf Selecting the UK IMO Team}
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\noindent
As usual the process began in November 1996 with the UKSMC.  The name may
have changed from NMC but the format was the same.  The $\displaystyle 1{1\over 2}$-hour paper
contained 25 multiple-choice questions, the first 15 meant to be accessible
to most contestants and the remainder designed to stretch the field.  Around
35000 pupils took part.  Based largely on their performances in the UKSMC,
over 700 pupils entered BMO 1, a $\displaystyle 3{1\over 2}$-hour paper with
5 questions held in mid-January.  Thereafter, 100 pupils were invited to take
part in BMO 2, another $\displaystyle 3{1\over 2}$-hour paper with just 4
questions, held at the end of February.  From this group, exactly 20 were
selected for a residential Training Session at Trinity College, Cambridge in
April.  Selection for this Training Session was based on several criteria.  In
addition to the strongest contenders for this year's IMO team, some younger
students were blooded as an investment for the future.  The chosen 20 had 4
or 5 intensive 2-hour sessions each day dealing with Algebra, Combinatorics,
Functional Equations, Geometry, Inequalities and Number Theory.  In each session
the emphasis was on tackling problems, with the bare minimum of exposition
from the person leading the session.  Thanks are due to all who gave so willingly
of their time to prepare material and lead sessions.  We were specially
pleased to have with us Professor Derek Holton (University of Otago), a regular
Leader of the New Zealand IMO team and well known in this country through his
series of problem-solving booklets.  (During our stay in Cambridge, the weather
was glorious.  Walking through the Fellows' Garden each morning on the way to
breakfast was a delight, as was punting on the Cam.  Courtesy of Bill Richardson
we also had a trip to the Observatory to look at the Hale-Bopp comet.)
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\noindent
The climax of the Trinity Training Session was the Final Selection Test (FST).
There may be more relaxing ways of spending $\displaystyle 4{1\over 2}$ hours
on a Sunday morning under a cloudless sky but duty calls.  The mock IMO
paper contained just 3 questions.  When the students had gone, a group of 6
staff stayed on to mark FST, after which the IMO squad of 7 was selected.  The
7 students then embarked on a correspondence course.  They received sets of
8-10 problems every 10 days and were required to submit solutions in accordance
with strict deadlines.  Towards the end of May, our selection was finalised as
follows:
\vfill\eject

\varlist{5}
\varitem
{{\bf Team:}} Mansur Boase \ \ (St. Paul's School, London)
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\varitem
{}Michael Ching \ \ (Oundle School, Oundle nr. Peterborough)
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\varitem
{}Toby Gee \ \ (John of Gaunt School, Trowbridge, Wilts.)
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\varitem
{}Adrian Sanders \ \ (King's College School, Wimbledon)
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\varitem
{}Amit Shah \ \ (Haberdashers' Aske's School, Elstree)
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\varitem
{}Bennet Summers \ \ (St. Paul's School, London)
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\varitem
{{\bf Reserve:}} Colin Phipps \ \ (Bristol Grammar School)
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\varitem
{{\bf Team Leader:}} Adam McBride \ \ (University of Strathclyde, Glasgow)
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\varitem
{{\bf Deputy Leader:}} Philip Coggins \ \ (Bedford School)
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\varitem
{{\bf Observer:}} Michael Davies \ \ (Westminster School)
\endvarlist
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\noindent
Of the team, Michael and Toby went to Bombay in 1996 while Adrian and Bennet
were last year's reserves.  Sending an Observer to the IMO allows interested
parties to see what is involved in being either the Leader or the Deputy
Leader.  On this occasion Michael Davies shadowed Philip.
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\noindent
{\bf Final Preparations}
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\noindent
The correspondence course continued until the beginning of July, with occasional
interruptions because of A-levels and other exams.  During the period 2-6 July,
the whole squad gathered at Queen's College, Birmingham in conjunction with the
Summer School being run there by Tony Gardiner.  As well as tackling more hard
problems, including another mock IMO paper, the squad were able to get to know
each other better and to develop team spirit.  Cultural interludes were
provided by a walk along the canal to see a performance of ``The Importance
of Being Earnest'' and a ``Mathematical M$\acute {\rm e}$lange'' in which
the younger students at the Summer School displayed their musical, dramatic
and juggling abilities to a remarkably high standard.  The Summer School was
enormously successful and we are all most grateful to Tony Gardiner for the
huge amount of work he put in to get it organised.
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\noindent
The Birmingham Summer School provided a fitting conclusion to our preparations.
I am particularly grateful to Christopher Bradley for his major contribution
to the mathematical training, especially in geometry, and to Philip Coggins
for helping with the correspondence course, for meticulous care in arranging
flights and for dealing with all manner of domestic and medical matters relevant
to our trip.
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\noindent
As previously mentioned, the attitude of the squad was excellent throughout.
Special thanks to our reserve, Colin Phipps, who stayed fully involved right
up to departure and contributed to the team effort by making the others fight
every inch of the way.
\vfill\eject

\noindent
{\bf Timetable of the 38th IMO}
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\noindent
The Jury, comprising the Team Leaders of all competing countries, met for the
first time on 19 July and spent the next three days selecting the problems for
the two papers and approving the translation of the papers into all the required
languages (48 in all).  The Team, Deputy Leader and Observer arrived on 21 July
and the Opening Ceremony was held on 23 July.  The two examination papers took
place between 08.30 and 13.00 on 24 and 25 July.  Thereafter the contestants
could relax and go on excursions while the Leaders and Deputy Leaders embarked
on marking and co-ordination, lasting three days.  The Closing Ceremony, including
the presentation of medals, took place on 30 July.
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\noindent
{\bf The Problems}
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\noindent
All contestants sat two papers on consecutive days.  Each paper contained
three problems, each problem being worth 7 points.

\noindent
On each day the time allowed was $\displaystyle 4{1\over 2}$ hours.

\noindent
The problems were proposed by the countries indicated.
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\centerline{\bf FIRST DAY}
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\item{1.} In the plane the points with integer co-ordinates are the vertices of
unit squares.  The squares are coloured alternately black and white (as on a
chessboard).

\item{}For any pair of positive integers $m$ and $n$, consider a right-angled
triangle whose vertices have integer co-ordinates and whose legs, of lengths $m$
and $n$, lie along edges of the squares.

\item{}Let $S_1$ be the total area of the black part of the triangle and $S_2$
be the total area of the white part.  Let
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$$
f(m, n) = \vert S_1 - S_2\vert \ .
$$

\list
\item{(a)} Calculate $\displaystyle f(m, n)$ for all positive integers $m$ and $n$ which
are either both even or both odd.

\item{(b)} Prove that $\displaystyle f(m, n) \leq {1\over 2} \max \{m, n\}$
for all $m$ and $n$.

\item{(c)} Show that there is no constant $C$ such that

$$
f(m, n) < C \ \ {\rm for \ all} \ \ m \ {\rm and} \ n \ .
$$
\hfill{(Belarus)}
\endlist
\vfill\eject

\item{2.} Angle $A$ is the smallest in the triangle $ABC.$

\item{}The points $B$ and $C$ divide the circumcircle of the triangle into
two arcs.  Let $U$ be an interior point of the arc between $B$ and $C$ which
does not contain $A$.

\item{}The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$
and $W$, respectively.  The lines $BV$ and $CW$ meet at $T.$

\item{}Show that
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$$
AU = TB + TC \ .
$$
\hfill{(United Kingdom)}
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\item{3.} Let $\displaystyle x_1, x_2,\ldots , x_n$ be real numbers satisfying
the conditions
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$$
\vert x_1 + x_2 +\ldots + x_n\vert = 1
$$

and\hskip 3truecm $\displaystyle \vert x_i\vert \leq (n+1)/2\qquad {\rm for}\qquad i = 1,2,\ldots, n.$

\item{}Show that there exists a permutation
$\displaystyle y_1, y_2,\ldots, y_n\quad {\rm of}\quad x_1,x_2,\ldots, x_n$
such that
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$$
\vert y_1 + 2y_2 +\ldots + ny_n\vert \leq (n+1)/2 \ .
$$

\hfill{(Russia)}
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\centerline{\bf SECOND DAY}

\item{4.} An $n \times n$ matrix whose entries come from the set
$\displaystyle S = \{1,2,\ldots , 2n-1\}$ is called a {\it silver} matrix if,
for each $i = 1,\ldots, n,$ the $i^{{\rm th}}$ row and $i^{{\rm th}}$ column together
contain all the elements of $S$.  Show that

\item\item{(a)} there is no silver matrix for $n = 1997;$
\item\item{(b)} silver matrices exist for infinitely many values of $n$.

\hfill{(Iran)}
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\item{5.} Find all pairs $(a,b)$ of positive integers $a$ and $b$ that satisfy
the equation
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$$
a^{b^2} = b^a \ .
$$
\hfill{(Czech Republic)}
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\item{6.} For each positive integer $n$, let $f(n)$ denote the number of ways
of representing $n$ as a sum of powers of 2 with non-negative integer exponents.

\item{}Representations which differ only in the order of their summands are
considered to be the same.  For instance, $f(4) = 4$ because the number 4 can
be represented in the following four ways:
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$$
4 \ ;\qquad 2 + 2 \ ;\qquad 2 + 1 + 1 \ ;\qquad 1 + 1 + 1 + 1 \ .
$$
\vfill\eject
\item{}Prove that, for any integer $n \geq 3,$
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$$
2^{n^{2}/4} < f(2^n) < 2^{n^{2}/2} \ .
$$
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\hfill{(Lithuania)}

{\it You are invited to send in solutions, enclosing an SAE please, to:}
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\centerline{\it Adam McBride, Department of Mathematics,}
\centerline{\it University of Strathclyde, Livingstone Tower,}
\centerline{\it 26 Richmond Street, GLASGOW  G1 1XH.}
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\noindent
{\bf Comments on the Problems}
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\noindent
Participating countries submitted over 150 problems, with geometry the most
popular area.  The organisers produced a short list of 26 problems for
consideration by the jury.  The United Kingdom submitted 6 problems, of which
3 were included in the short list.  One of these, composed by David Monk
(formerly University of Edinburgh) was chosen as the second problem on the
first day.  Another of David's problems was short-listed, as was one by
Christopher Bradley (Clifton College, Bristol).  Christopher and David must
be two of the most prolific and imaginative composers of problems in the world,
both having a keen interest in geometry.  It is fitting that we should congratulate
Christopher on being declared 1996 Problem Solver of the Year by the journal
{\it Crux Mathematicorum}.  We look forward to many more ingenious problems
from both Christopher and David in the future.

\noindent
As regards Problem 4, there was much discussion about what to call a matrix
with the stated property.  The word ``silver'' was mentioned as a joke.
(Remember where the IMO was being held!)  It is fair to say that not all
countries were happy with ``silver'' but it was voted through.
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\noindent
{\bf How the UK Team Performed}
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\noindent
A total of 460 contestants from 82 countries took part.

\noindent
The UK team finished 16th out of 82 with 144 points (out of 252).

\noindent
Team members won

\centerline{1 Gold Medal, \ 2 Silver Medals and 2 Bronze Medals.}

\noindent
Individual scores were as follows:

$$
\vbox{
\halign{#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&
\quad#\hfil&\quad#\hfil&\quad#\hfil\cr
&{\bf Q1}& {\bf Q2}& {\bf Q3}& {\bf Q4}& {\bf Q5}& {\bf Q6}& {\bf Total}\cr
Mansur Boase & 0 & 7 & 6 & 7 & 3 & 5 & 28 & Silver\cr
Michael Ching& 4 & 7 & 0 & 7 & 7 & 0 & 25 & Silver\cr
Toby Gee& 0 & 7 & 1 & 5 & 4 & 1 & 18 & Bronze\cr
Adrian Sanders& 4 & 7 & 0 & 7 & 0 & 3 & 21 & Bronze\cr
Amit Shah& 4 & 0 & 0 & 4 & 6 & 0 & 14\cr
Bennet Summers& 7 & 7 & 7 & 7& 7& 3& 38 & Gold\cr}
}
$$

\noindent
Bennet's score put him 15th equal.  Amit was unlucky to miss a bronze medal
by just one mark.

\noindent
The papers contained quite a lot of things that could be classed as combinatorics.
Problem 1, which was meant to be a reasonably accessible question, was found
hard by the UK team (and many others).  Given the large number of geometry
problems proposed, it seems surprising that only one saw the light of day.
Another geometry problem on the second day might have suited us better, as
might a functional equation.
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\noindent
{\bf Overall Performance of All Contestants}
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\noindent
There was a conscious effort to make the papers somewhat easier than last year
(when scores of 28, 20 and 12 were the lower cut-offs for gold, silver and
bronze, respectively).  The outcome was satisfactory.  The
ranges of scores for the various medals were:

$$
\vbox{
\halign{#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil\cr
Gold & From 35 to 42 & (39 contestants)\cr
Silver & From 25 to 34& (70 contestants)\cr
Bronze & From 15 to 24& (122 contestants).\cr}
}
$$
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\noindent
Four students scored full marks.  Alas, 15 students scored 0.  (These students
came largely from Latin/South American countries competing for the first time.)
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\noindent
For the statistically minded, the individual problems produced the following
means ($\mu$) and standard deviations ($\sigma$):

$$
\vbox{
\halign{#\hfil&\quad#\hfil&\quad#\hfil&\quad#\hfil&\quad#
\hfil&\quad#\hfil&\quad#\hfil\cr
&{\bf Q1}& {\bf Q2}& {\bf Q3}& {\bf Q4}& {\bf Q5}& {\bf Q6}\cr
$\mu$& 2.48 & 3.89& 1.78& 3.74& 3.35& 0.82\cr
$\sigma$& 2.23 & 3.31 & 2.75 & 2.72& 2.97 & 1.72\cr}
}
$$

\noindent
Although the IMO is an individual competition and, officially, there is no
team competition, considerable interest still attaches to team totals.  For
the record, here are the top 16:
\vskip -0.5truecm
$$
\vbox{
\halign{#\hfil&\quad#\hfil&\quad#\hfill&\quad#\hfil\cr
223 China & 219 Hungary & 217 Iran\cr
202 Russia, USA & 195 Ukraine & 191 Bulgaria, Romania\cr
187 Australia & 183 Vietnam& 164 South Korea\cr
163 Japan & 161 Germany & 148 Taiwan\cr
146 India & 144 United Kingdom.\cr}
}
$$
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\noindent
Interested readers can make what they will of all these statistics.  The rest
of us will move on.
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\noindent
{\bf Organisation of the 38th IMO}
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\noindent
This IMO will be remembered as one of the best ever from the point of view
of organisation.  A team of professionals had been engaged to deal with all the
preliminary planning (e.g. submission of problems, names of contestants,
flight details, transport requirements).  These ladies were on hand throughout
the proceedings and cheerfully helped to sort out any difficulties that arose.
Transport was meticulously planned, buses turned up at the right time and
checks were made to make sure that nobody got left behind.  Staff were present
at both airports in Buenos Aires and at the airport in Mar del Plata to meet
flights and arrange transfers.  Each team was allocated a guide, a local
student who could speak the appropriate language and made sure, among other
things, that nobody slept in on the morning of the exams.

\noindent
Reference was made earlier to the standard of accommodation.  Although students
were three to a room, the quality of the rooms was more than adequate.
Various facilities, including a games room and a computer room, had been laid
on and there was ample opportunity for students to socialise.  At the Leaders'
5-star hotel, luxury was the name of the game.  Apart from rooms for formal
meetings, other rooms were available for working or relaxing, all with a constant
supply of tea, coffee, bottled water and small cakes.  A particular favourite
was a room right on the top with a panoramic view over the sea.  At both
hotels, the ambience was conducive to making the 38th IMO a very friendly
affair.
\vskip 0.2truecm

\noindent
All decisions concerning the actual competition were in the hands of the Jury
which comprised the 82 Team Leaders, each of whom had one vote.  We each had
a plastic wand with our own 3-letter code marked on it.  The UK code was UNK
so that, as usual, I was ``The Man from UNK''!  Votes came thick and fast and
majorities were based on the number of wands raised.  The business of the Jury
was conducted in English, with occasional translations into other languages when
required.  The first three days were taken up with the selection of the problems
and preparation of examination papers in nearly 50 languages.  This is a
complicated business at the best of times and firm control is needed to avoid
things turning into a shambles.  Our chairman during this period had no previous
experience of an IMO.  As a result, the selection of problems took a few
bizarre twists.  Sometimes we seemed to be voting for individual problems,
sometimes for pairs of problems and sometimes for triples to form an entire
paper.  Around noon on the second day, things were decidedly squiffy.  However,
by mid-afternoon, the selection process was complete.  Subsequent jury meetings
were chaired by Carlos Bosch, an IMO stalwart with a sure touch who ensured
that things went swimmingly.
\vskip 0.2truecm

\noindent
The other major task for Leaders and Deputy Leaders was the marking of scripts
and co-ordination of the marks.  We each marked our own team's scripts and
then had to justify our marks before a panel of co-ordinators.  It was up to
Philip Coggins, Michael Davies and myself to get as many marks as possible
for our students on the basis of the scripts which had been presented to us.
The three of us worked splendidly as a team and I should like to thank Philip
and Michael for all their hard work and for their ability to explain a
complicated solution in a highly convincing way.  After the exam, the organisers
had photocopied every sheet of paper submitted by every student.  This enabled
co-ordinators to read scripts in advance and get a feel for the various methods
of solution used by students.  The considerable time and cost involved in
photocopying every sheet was amply justified.  Co-ordination, which is always
an intense business, went smoothly.  Discussions were amicable and justice was
done.  The co-ordinators, who numbered well over 40 and came from all over the
world, deserve our thanks for all their labours.  As soon as a set of marks
had been agreed, they were entered into a computer.  Marks were posted as quickly
as possible on noticeboards in both hotels.  Lists were constantly updated, as
were various pie-charts and histograms.  Once again, the efficiency was commendable.
\vskip 0.3truecm

\noindent
{\bf Daily Diary}
\vskip 0.2truecm

\noindent
To try to give a flavour of how the IMO unfolded, I now offer a brief summary
of what happened day by day, as seen through my eyes.
\vskip 0.2truecm

\noindent
{\bf 17 July} \ \ Leave home at midday to catch flight to London, then on to
Madrid.  Meet Polish leader and Dutch contingent.  Settle down for long flight to
Buenos Aires.
\vskip 0.2truecm

\noindent
{\bf 18 July} \ \ Waken up to discover we have touched down at San Paolo in
Brazil.  After another $\displaystyle 2{1\over 2}$ hours reach Buenos Aires
where temperature is a bracing $-1^\circ$ C.  Transfer to domestic airport
by taxi.  The 45-minute drive gives me a glimpse of Buenos Aires.  Meet up with lots
of old friends for final flight to Mar del Plata, which is 400 km south of Buenos
Aires.  The 737 is full of IMO people.  Should so many mathematicians all travel
in the same plane?  Disembark at Mar del Plata on a glorious afternoon.  Taken
by minibus to hotel.  Door-to-door journey time of 30 hours.  This hotel is 5-star
OK.  It looks out over the Atlantic.  Across the road is a naval base which played a
major part in a certain conflict in the early 1980's.  Better not say too much
about that!  Have a very late lunch.  Pick up bag of goodies, 60 pesos and the
short-listed problems.  Try a couple of problems but soon give up.  Sleep for
12 hours.
\vskip 0.2truecm

\noindent
{\bf 19 July} \ \ Renew many more friendships at breakfast.  Tables groaning
with food.  Does anyone really eat lemon meringue pie at this time of day?
Apparently so.  First Jury meeting is brief.  After a few introductions, two
matters are debated with the chairman being overruled by the Jury both times.
Off to a flying start!  Adjourn to try more problems.  After lunch, stroll
to the end of the breakwater to get some fresh air.  Solutions to the problems
now available.  Try some more problems (without keeking at the solutions!).
At dinner we are serenaded by a harpist who plays a wide selection of music, even
parts of Verdi's {\it Rigoletto}!  After dinner, there is a demonstration of
how to do the tango.  Some preferred to get stuck into a book of crosswords
supplied by the Canadian leader.
\vskip 0.2truecm

\noindent
{\bf 20 July} \ \ Selecting problems starts in earnest.  From the short list,
5 are chucked out because they have appeared in some competition or other.  More
time still needed to try problems ourselves.  Adjourn until lunch.  Write
postcards.  Walk along the beach in bright sunshine.  I really like this place!
After lunch there is a complex series of votes.  What exactly is going on here?
By dinner we have chosen the 3 problems for the first day.  Are we proceeding
in a sensible way?  Philip, Michael and the team will be on their way now.
\vskip 0.2truecm

\noindent
{\bf 21 July} \ \ A long day!  Morning spent trying to finalise paper.  Procedural
wrangles aplenty.  At 11.00 we seem to be going OK, but by 12.00 things are
distinctly pear-shaped.  Feelings running high.  Mercifully, lunch intervenes.
After lunch things get more and more bizarre.  Eventually we have our 6 problems.
Not the ones I'd have chosen individually (e.g. not enough geometry).  However,
could be worse.  Work starts on preparing master versions of problems in English.
Small group of English-speaking Leaders go off to do the business.  What {\it will}
we call the matrices in Q4?  We must be able to improve on ``coveralls''.  Someone
suggests ``silver'' as a joke.  Some Leaders think this is facetious but it gets voted
through!  (A message arrives.  The team have missed their connecting flight from
Buenos Aires to Mar del Plata because of late arrival from London.  Looks like a
night in B.A. or a 5-hour train journey starting at 23.30.  I reply that Philip
should decide on the better option as he is the man on the ground.)  English
version approved.  Next come versions in French, German, Russian and Spanish.
Fun and games in Q1.  We say ``black and white'', the French say ``blanc et noir'',
not ``noir et blanc''!  After several similar arguments, next four versions are
approved around 21.30.  Time to relax.  Hope the team are OK.  Just as well
tomorrow is a free day.
\vskip 0.2truecm

\noindent
{\bf 22 July} \ \ Hooray!  The team have arrived OK.  Apparently they managed
to get onto a later flight.  First meeting not until 11.30.  A cosmopolitan
group of Leaders (Canada, Colombia, New Zealand, Switzerland, UK) have an
invigorating 2-hour walk along the beach.  Cold to start with but soon the sun
comes out.  There are lots of breakwaters.  On every one, many local people
are fishing.  Back to business, with approval given for over 40 more translations.
After lunch there is a joint meeting of the IMO Advisory Board and the Jury.
The IMOAB is responsible for planning ahead, finding future venues, discussing
rules and regulations, etc.  There is a wrangle over elections to the IMOAB.
Then we have an open forum.  At the behest of BMOC I put the view that excessive
training for IMO may harm the mathematical development of students and I gain
support from a number of countries.  Venues (some provisional) up to 2006 are
announced.  Next year we go to Taiwan.  At dinner we get the first edition of
``La Helice'', a daily IMO newsletter with contributions from students.
\vskip 0.2truecm

\noindent
{\bf 23 July} \ \ Bus to Opening Ceremony.  Leaders are segregated in the
gallery, with Deputy Leaders and Teams down below.  (We know the contents of
the two papers and mustn't communicate with our students.)  The lads are
looking smart.  Philip throws up my team sweat-shirt, designed by Bennet.  Things
get going a little late.  After an introduction, we see a specially prepared
film of highlights of Argentina.  The football goes down particularly well.
Then come speeches including one from the delightful Minister for Culture and
Education (representing President Carlos Menem).  Next a surprise.  We have
a new IMO song which receives its premiere.  It is to be sung at all future
IMOs, apparently.  This year's version is in Spanish, contains such wondrous
lines as ``Sumamos, multiplicamos y llegamos a un total'' and is entitled
``Razonar es nuestro estilo'' (Reasoning is our style).  This little ditty
provoked a certain amount of merriment but it isn't quite as bad as it seemed
at first.  Translations (including some irreverent ones by certain teams) are
now available in many languages.  Proceedings end in a shower of balloons.  I
give the lads the thumbs-up as they disappear.  Off we go for a barbecued lunch
in the country.  A bus trip of less than 5 minutes (surely a record for IMO
bus trips?)  gets us to a Minizoo where a water-pig takes exception to a
peacock invading its enclosure.  Back to base for dinner and early bed.
\vskip 0.2truecm

\noindent
{\bf 24 July} \ \ Up at 6.30.  Off to Hotel 13 de Julio for first paper.  Turns
out the teams are staying there, not 25 km away as expected.  Jury gets installed
before students go into exam room.  Students can ask questions during the first
30 minutes.  Questions have to be written down and are brought by runner to the
jury room.  Queries are dealt with in order of receipt.  The relevant Leader
has to explain to the Jury what the query says and how he/she intends to reply.
The reply has to be approved by the Jury before being written down and taken
back to the student by runner.  Today we get 56 queries, mainly about the
meaning of max$\{m, n\}$.  Also ``perpendicular bisector'' seems to present
problems in Spanish.  Off we go 70 kms into the pampas to visit the Juan Fangio
Museum at Balcarce.  Fangio was world motor racing champion five times in the
1950's.  Examine cars he drove (Ferrari, Maserati, Mercedes-Benz) as well as
other old cars, including a Model T Ford.  Next stop is a barbecued lunch at
an estancia (ranch).  Entertainment consists of traditional music and dancing.
Another glorious afternoon.  Back to base, expecting to collect scripts from
morning exam.  Won't be ready until at least 21.30 because every page is being
photocopied.  By 23.00 there are still no scripts.  Some Leaders are annoyed.
I go to bed.
\vskip 0.2truecm

\noindent
{\bf 25 July} \ \ Another early kick-off.  Today we get 44 queries, mainly about
the wording of Q4 and the meaning of $a ^{(b^2)}.$  After question time, back
to base to collect scripts.  Quick glance suggests that some of the lads have
struggled on the first day.  Bennet in good shape.  Walk down to meet the team
as they come out of second paper (a brisk 50-minute stroll).  Their view of
Q1-Q3 agrees with mine.  Philip and Michael return from a trip to the aquarium
and minizoo.  Meet Carla, the team guide, who seems to be getting on fine with
the team.  We all have lunch together.  First time we have been able to speak
since leaving the UK!  Bennet's hair is now red, white and blue, while Toby
has painted his finger-nails purple, presumably to express solidarity with
Bennet.  Philip and Michael move to my hotel.  After dinner, get stuck into
scripts for Q1-Q3.  Around 21.30 Q4-Q6 arrive.  Co-ordination schedule appears.
Reasonably early bed.
\vskip 0.2truecm

\noindent
{\bf 26 July} \ \ Spend morning polishing up Q1-Q3 ready for the afternoon.
Co-ordination very fair.  Get one or two more marks than expected.  Philip and
Michael in good form.  Work on Q4-Q6 before dinner.  Scores going up on boards.
Bennet's three 7's are pleasing on the eye.  After dinner, the crossword buffs
are reinforced by the Canadian and Irish Deputy Leaders.  We have to unravel
something to do with The Spice Girls if we want to crack The Listener Crossword,
or so it seems.
\vskip 0.2truecm

\noindent
{\bf 27 July} \ \ Spend morning polishing up Q4-Q6 ready for the afternoon.
One or two unfortunate errors become apparent and some of the lads lose marks
that would have helped their cause.  Once again Michael and Philip produce
order out of chaos.  After an adjournment to sort out Toby's construction in
Q4, we end up with as many marks as we could reasonably expect.  Meanwhile,
I supervise co-ordination of Argentina's scripts for David Monk's Q2, in accordance
with IMO rules for the host country.  Results coming in thick and fast but still
too early to make confident predictions.  At dinner we all get a free cassette
of ``La Cancion de la Olimpiada''.
\vskip 0.2truecm

\noindent
{\bf 28 July} \ \ Michael, Philip and I stroll down to the students' hotel to return
scripts.  The lads know how things stand from scoreboards displaying the marks
of all contestants.  We lunch together.  Then it is the lads' turn to experience
the delights of the minizoo.  The three of us stroll back leisurely.  We sample
alfajores, a local delicacy in the form of biscuits with toffee in the middle
and either chocolate or meringue on the outside.  We also try bocaditos, chocolate
cones with toffee inside.  They certainly like their toffee in Argentina!  The
beach is heaving with people.  It is the mid-winter break when people come down
to Mar del Plata from Buenos Aires to stay in their chalets.  Although it is
mid-winter the temperature is 21$^\circ$ C and the sun is shining.  We see a
variety of entertainment, including several games akin to boule or p$\acute{\rm e}$tanque
(a form of bowls).  We return to base 7 hours after leaving.  A most enjoyable
excursion giving us a chance to soak up the atmosphere.  Back at base, scores
are virtually complete.  We are 16th.  After dinner it's party time, courtesy
of the Colombians.  We learn the salsa, which is a mere bagatelle for those who
can do a Dashing White Sergeant or Strip the Willow.  As an interlude, Marta
Joltac, one of our superb organisers, entertains us with some Spanish love-songs.
Reinforcements arrive and the party goes on until 4 a.m.  I chicken out around
1 a.m.
\vskip 0.2truecm

\noindent
{\bf 29 July} \ \ Open the curtains and can't see a thing for fog.  Final Jury
meeting attended by Deputy Leaders and Observers.  After a few arguments, marks
are approved and cut-offs for medals agreed.  Vote of thanks to chairman,
co-ordinators, etc. closes proceedings.  Go to a large sports centre to take
part in all sorts of physical activities.  All in the line of duty, I help the
lads to win at tug-of-war and lose at 8-a-side football.  (I have still a bruise
on my shin to show for my efforts.)  A giant conga leads everyone back to the
buses.  It's the last night when we'll all be together and we spend it relaxing
in various ways.
\vskip 0.2truecm

\noindent
{\bf 30 July} \ \ Early breakfast then off to the Closing Ceremony which starts
at 09.00 (a time chosen, presumably, to let teams fly out later in the day).
Medals dished out.  I manage to catch the lads on camera.  Bennet has reinforced
his red, white and blue hair-do specially for the occasion.  Extra loud cheers
for medallists from Argentina.  Then we see a video of highlights of the IMO,
prepared to a high standard during the night.  The lads are seen in pensive
mood after one of the papers and Philip is spotted having his lunch.  Representatives
from Taiwan extend their welcome for next year and proceedings end with farewells
from our hosts, accompanied by much singing and waving of flags.  Time for
team photos, both formal and in sweat-shirts.  Back to the team hotel for
lunch.  The first teams are starting to leave.  Everyone gets a copy of the
``Golden Book'' containing team pictures and statistics of the 38th IMO, again
produced very rapidly.  The team go off to play frisbee with the Americans.  Philip,
Michael and I stroll back to base.  No sooner have we got inside the hotel than
a short sharp thunderstorm breaks, the first real rain we have seen in Argentina.
At dinner we get free wine and another tango display.  People start dancing
again.  We go to pack but at midnight the party is still in full swing.
\vskip 0.2truecm

\noindent
{\bf 31 July} \ \ The long trek home begins.  Leave hotel at 09.40 with fond
farewells from our delightful hosts.  Having reached the main airport in Buenos
Aires, we have 7 hours to kill.  Apart from eating, we play a curious game which is
a mixture of blind man's buff and Cluedo, or so it seemed to me.  Take-off is
delayed by 100 minutes for ``fuelling''.  (Could they not have done it earlier?).
Eventually get going at 23.40.
\vskip 0.2truecm

\noindent
{\bf 1 August} \ \ Dawn breaks over the Atlantic.  The lads are asleep.  I'm
not because I find it virtually impossible on a plane.  Breakfast arrives, time
goes by and we land late at Madrid.  On the tarmac the temperature is 40$^\circ$ C.
A bus takes us to the sanctuary of the air-conditioned terminal.  We're off again
pretty sharpish and reach Heathrow at 18.15.  Alas, no luggage.  There wasn't
enough time to transfer it at Madrid.  Philip sorts things out, we fill in forms,
clear customs and emerge to be greeted by parents.  One more flight for me and
I get home at 22.20.  Journey time (allowing for 4 hours difference in local times)
32 hours 40 mins.  The odyssey is almost complete, but not quite.
\vskip 0.2truecm

\noindent
{\bf 2 August} \ \ At 19.40 my missing luggage is delivered.  Now I can start
writing this report!
\vskip 0.2truecm

\noindent
{\bf Concluding Remarks} \ \ IMO97 will go down as one of the best IMOs ever,
thanks to our Argentinian hosts.  As soon as we arrived, we received a warm
welcome which lasted right through to departure.  As previously remarked, the
organisation was superb, with some nice little touches which epitomised the
meticulous attention to detail.  The whole atmosphere was delightful, with old
friendships renewed and many new friendships made.  To everyone involved in
making our stay so enjoyable we extend our heartfelt thanks.
\vskip 0.1truecm

\noindent
Nearer home, I should like to thank
\vfill\eject

\varlist{2}
\varitem{$\bullet$} all the pupils who took part at any stage

\varitem
{$\bullet$} all the teachers who encouraged the pupils and supported our
endeavours

\varitem
{$\bullet$} Peter Neumann for his skilful chairmanship of BMOC (and now UKMT)

\varitem
{$\bullet$} the Problems Group for creating so many interesting problems

\varitem
{$\bullet$} Alan West and Brian Wilson, organisers of BMO 1 and BMO 2 respectively

\varitem
{$\bullet$} all those involved with the marking of BMO 1, especially Brian
 Wilson and Christine
Farmer

\varitem
{$\bullet$} all those who contributed to the Trinity Training Session

\varitem
{$\bullet$} all our sponsors, especially Trinity College, Cambridge for hosting
the April training session and the Royal Society for hosting the September
celebration

\varitem
{$\bullet$} DfEE for a grant covering travel to and from Argentina

\varitem
{$\bullet$} Tim Cross for producing the 1997 BMO booklet, 175 copies of which
were taken as gifts for Leaders, Deputy Leaders and our Argentinian hosts

\varitem
{$\bullet$} Philip Coggins for assistance with the correspondence course and,
along with Michael Davies, for help of all sorts before and during the visit
to Argentina.
\endvarlist
\vskip 0.2truecm

\noindent
That leaves us with the 7 members of the squad.  The majority have been in the
system for several years but are now leaving us to start the next stage of
their mathematical education at university.  Their enthusiasm over the years
has been remarkable and it has been a privilege and a pleasure to work closely
with them.  They have been excellent ambassadors.  Let us salute them for their
achievements this year and in previous years and let us wish them all the best
for the future.


\bye
