\magnification=\magstep1
\tolerance=10000
\lineskip=3mm
\lineskiplimit=3mm
\def \sizeof#1{{ \left \vert #1 \right \vert }}
\def \setof#1{{ \left \lbrace #1 \right \rbrace }}
\def \half{{${1 \over 2}$}}
\null\vskip1truecm
\centerline{{\bf 40TH INTERNATIONAL MATHEMATICAL OLYMPIAD}}
\bigskip
\centerline{{\bf BUCHAREST, ROMANIA}}
\bigskip
\centerline{{\bf 10-22 July 1999}}
\bigskip
\centerline{{\bf Report by Imre Leader (UK Team Leader)}}
\bigskip
\bigskip
This report is about the 40th International Mathematical Olympiad, which
was held in Romania in July 1999. The IMO is the pinnacle of excellence
in mathematics for
school
pupils the world over. Each year, around 80 countries each send a
team of 6 contestants
to the IMO. There they sit two
4\half-hour exams, each containing just three questions. Medals are
awarded for good performances. This year the host country was Romania, which
organised the first two IMOs (in 1959 and 1960), as well as the IMOs of 1969
and 1978 (the expert reader will wonder why the 1999 IMO is number 40, instead
of 41 -- it is because there was no IMO in 1980).
Let us start with some of the events leading up to the IMO. The selection of
the team started with the Senior Mathematics Challenge, a multiple-choice
paper sat by more than 40000 students, taken in November 1998. The SMC lasts
90
minutes, and consists of 25 questions, of which the first 15 are meant to be
widely accessible and the last 10 rather more testing. Based on their
performance in the SMC, around 1000
contestants proceed to the next round, the BMO1.
This is a far, far harder paper, containing
just 5 questions to
be done in 3\half~hours. Anyone who solves a BMO1 question
has reason to feel pleased with himself/herself!
The BMO1 is held in mid-January, and is followed by an
amusing weekend in which 20 or so
academics, teachers and ex-olympians gather together to mark the scripts.
After BMO1, about 100 pupils qualify for BMO2, which is a still harder exam,
consisting of 4 questions to be attempted in 3\half~hours. Based on BMO2,
20 pupils are
selected for the Trinity Training Session at Easter -- these 20 include
those
who we feel are realistic contenders for the team, and also some
younger pupils who we believe are good
prospects for the future.
The Trinity Training Session is an extremely intense and exciting experience
for everyone. It lasts four days. For the first three days, the students have
a variety of sessions, some taught to the whole group of 20 and some taught in
groups of 6 or 7. The emphasis is on the students trying problems: the actual
amount of `lecturing' is kept to a bare minimum. The final day is by contrast
rather different. The main event is the last of the selection exams, the
Final Selection Test. The FST is designed to resemble a real IMO paper: there
are just 3 questions, and the time allowed is 4\half~hours.
In the next day or so, a squad of 8 is selected. The choice is based on
performance in FST, BMO2 and BMO1, and also on how the students have
performed during the Training Session. The 8 are notified within a few days of
leaving Trinity, and they then embark upon the final and most gruelling part
of the selection. This is the dreaded Correspondence Course. Each week to 10
days, the students are sent a sheet of about 8 hard problems. They send in
their solutions, which are marked by the Leader and the Deputy Leader (both new
this year: I took over from Adam McBride as Leader, while Richard Atkins, Head
of Maths at Oundle, took over as Deputy from Philip Coggins). After about 5
rounds of this, the team of 6 is chosen, with the other 2 acting as reserves.
Of course, the two reserves contribute immeasurably to the success of the
team, as their presence during the training course has forced people to work
hard for their places in the team!
In the week before the IMO itself, the team gather at Birmingham, where the
Summer School for younger pupils is held. As well as participating in some of
the events of the Summer School, the team receive some final training and
preparation.
This year, the squad of 8 was as follows.
\noindent{\bf Team}:
\noindent Thomas Barnet-Lamb (Westminster School)
\noindent Rebecca Palmer (Clitheroe Royal Grammar School)
\noindent Marcus Roper (Northgate High School)
\noindent Oliver Thomas (Winchester School)
\noindent Oliver Wicker (Cockermouth School)
\noindent Jeremy Young (Nottingham High School)
\noindent {\bf Reserves}:
\noindent Stephen Brooks (Abingdon School)
\noindent Michael Spencer (Lawnswood High School)
Of these 8, Jeremy was the only `returner' from last year's IMO in Taiwan
(where he won a Bronze medal). Rebecca was one of last year's reserves.
Next came the IMO itself. The IMO, for the students, was to start in Bucharest
on July 13th, but the Team Leaders flew in three days early, to select the
questions that would be used. Each country has the right to submit some
questions (months in advance); the host country then narrow these down to a
short-list of about 25 questions, and it is from these that the six must be
chosen. For those three days, the Leaders were kept in a secret location, far
from Bucharest -- in fact, they were allowed no contact at all with the teams
or the Deputy Leaders until the last exam had finished, for obvious reasons!
This year, the Leaders met in Poiana Brasov, a ski resort high in the
Carpathians (Dracula territory).
Some countries send an `Observer' with the Leader or Deputy: this is usually
someone who will do the job in a later year, and is coming along to see how
things work. This year, I had Adam McBride as an Observer with me. Needless
to say, this was not for him to see how things worked, but to show me!
The Jury chose the questions, and supervised the various translations. This
year, 81 countries participated, which necessitated more than 50 languages.
The Jury
consists of all the 81 Leaders: as one can imagine, a committee of this size
is a
rather curious beast. Once the exam was ready, the Leaders
travelled to Bucharest, but were kept well away from where
the teams were staying.
Meanwhile, the team, led by Richard Atkins, had arrived in Bucharest. After
acclimatising, and a long Opening Ceremony, the actual exam dates were July
16th and 17th. There then followed a period of 48 hours of intense activity
by the Leaders and Deputies. The Deputies move to the Leaders' hotel, to help
with the marking of the exams. Each country marks its own students' scripts,
and then
goes to `coordination' for each question: this involves meeting with two
Romanian mathematicians and agreeing on marks. Finally, totals are worked
out, and the
cutoffs for medals established. The rough principle is that
the ratio of Gold to Silver to Bronze to no medal should be
very close to 1 to 2 to 3 to 6.
After a few days of sightseeing and socialising, there was a Closing Ceremony,
at which the medals were awarded. Everyone flew home the next day.
Now on to the papers. Each day has three problems, to be done in 4\half~hours,
with each question worth 7 points.
\bigskip
\centerline{{\bf FIRST DAY}}
\bigskip\noindent
Problem 1. Determine all finite sets $S$ of at least three points in the plane
which satisfy the following condition: for any two distinct points $A$ and $B$
in $S$, the perpendicular bisector of the line segment $AB$ is an axis of
symmetry for $S$.
\bigskip\noindent
Problem 2. Let $n$ be a fixed integer, with $n \geq 2$.
(a) Determine the least constant $C$ such that the inequality
$$\sum_{1 \leq i