MATHEMATICAL OLYMPIADS

Dr. Geoff Smith, University of Bath, on 27 September 2002

The International Mathematical Olympiads (IMO) are held every year and started in 1959 as an international competition between 6 Eastern European countries and grew from the tradition of problem solving competitions especially in Hungary. The UK joined in 1968, the US in the late 1970's and now in 2002, 84 countries took part when it was held in Glasgow in July. The country with the largest population that did not take part was Egypt, Africa in general is under-represented, and the only Middle Eastern state to take part was Kuwait, but Bahrain sent an observer.

The contest consists of two papers on consecutive days, each of 4½ hours containing three questions each. On each day, the questions are graded so that the easiest is first, and the second day is intended to be harder than the first. They are based on the standard curriculum of 1959 of geometry, algebra, number theory and combinatorics, which has the advantage of being tried and tested and not subject to manipulation by any member state, as the curriculum would have to be agreed by so many states. As an aside, the international baccalaureate is similarly regarded in many circles as being of a consistent standard from decade to decade.

Since 1980, each state sends a team of upto 6 students, who must be not over 20 years of age and not in tertiary education. Each question is marked out of 7 points; if a contestant did not do the question, the score is zero, but if substantial progress had been made, then 1 is awarded. If the question was answered, it would score 7 unless some reason can be found to reduce it, for example, by not making it clear that all cases have been considered. When the totals are known, the highest twelfth are awarded gold medals, the next sixth silver, and the next third bronze. Those who complete a question but do not otherwise qualify for a medal, get an honourable mention.

At the beginning of the competition, the jury, consisting of the team leaders, meet in isolation (in 2002 this was in Dunblane) to discuss the questions that some have brought along, and it is understood that these have not been shown to anyone else. They select those that they believe are a good test of ingenuity, and the standard is such that a fair number of entrants will score zero, even though they are among the top 500 students in the world. These discussions are held in English, and the English version of the question, together with translations into French, German, Spanish and Russian are masters. They are translated into the native language of the participants, who can choose to receive the paper in their native language and one master language. This attempts to remove possible differences in interpretation introduced by the translation. The answers are written in the native languages and marked by the team leaders initially, and then discussed to agree the marks for each answer.

In the UK about half a million pupils take part in various mathematical competitions, but since we do not wish to interrupt the national curriculum, about 100 are invited to attend training camps in both the normal and the olympic syllabus. These are then whittled down through camps in Budapest, Trinity College, Cambridge at Easter, Arundel school and Birmingham, to a team of 9, from which the final 6 a re selected in July. This year the BBC followed them around and will show a programme next year. In the event, we collected two silver and two bronze medals, which shows a slight improvement after a number of years of steady decline.

In 2002, the highest scoring team was from China, second was Russia, third was USA all of which have large populations. The star of the occasion was Bulgaria, which came in fourth. The UK was 27th. Some countries have dedicated schools, and Korea and Eastern Europe in particular consider these competitions prestigious and put a lot of effort into them. There is a strong correlation between winning a gold medal and enjoying a successful research career in mathematics.

Geoff Smith, who led the UK team, then presented the hardest question of 2002. This came from the Ukraine, and when the jury saw it, it was obvious that it had the necessary attributes for a good question. It is easy to describe, and had an ingenious method of solution. Then the leader of the Colombian team found an even better method, making it even more attractive. Suppose there are a number, n, of circles of unit radius in a plane, and with centres O1, O2, On, such that one cannot draw a straight line intersecting more than two circles. Then it is clear that these circles must, in some sense, be "spread out". So, considering the distances between all possible pairs of centres, OiOj, then the question asked to show that the sum of the reciprocals of these distances was not more than (n-1)p/4.

However, no solution was divulged, and it was left as an exercise for the audience to attempt in their own time (14 out of the 484 students answered it correctly).

There are also Olympiads in physics, biology, chemistry, astronomy and computing.

Andrew Pepperdine