The 20th International Mathematical Olympiad
JOHN HERSEE
School Mathematics Project

Romania, where the first International Olympiad was held in 1959, was again the host country in 1978. Seventeen countries participated, each sending a team of eight pre-university students. The team from Great Britain included three members of the 1977 team and was placed third. Romania was first and the U.S.A. second.

Each team is accompanied by a leader and deputy leader. The leaders form the international jury which selects the questions for the competition and agrees on the final results. Participating countries are invited to submit questions for the competition and from these the jury selects three for each of the two four-hour papers. Of course, the questions have to be carefully checked for correctness, and their wording (in English) agreed before being translated into different languages for the competitors.

The two papers are taken on consecutive days and the scripts of each team are first assessed by the team leader and deputy. In addition, for each question the host country provides a panel of two or three coordinators. These coordinators review the assessments of the team leaders to ensure a uniform standard for all teams. It will be appreciated that, while mathematical symbols are generally unambiguous and widely understood, students write in their own language, so the coordination process takes some time. The methods used and the details of the working are carefully studied by the coordinators, for a correct answer is by no means all that is required. The solution must be carefully argued, with no lapses or omissions in the reasoning; any such flaws are penalised. In addition, elegance, originality, or a generalisation of the problem set can earn a special prize.

Cuba sent only four competitors.

This year only one competitor, an American, scored full marks. Richard Borcherds, of the British team, lost one point for the omission of part of a proof. However, he produced an extremely neat solution to Question 6 and for this was awarded a special prize. First, second and third prizes are awarded to all students who achieve appropriate standards; the British team received one first, two second and two third prizes this year.

After the papers have been taken there is a good deal of time during which the teams are entertained. Our hosts provided a programme of visits to museums and places of interest, and a stay of three days on the Black Sea coast. In the meantime, the work of marking the scripts and agreeing on the results was going ahead at Busteni in the Carpathians where the team leaders, deputies and coordinators stayed. For the last two days, everyone met again in Bucharest. There was time for more sightseeing, shopping and two specially arranged concerts, before the presentation of prizes and the final dinner brought our very enjoyable visit to a close.

The results of the different teams reveal interesting patterns of success in the various questions, reflecting differences in mathematics syllabuses and team selection and training methods. Almost all countries begin their selection process with a national mathematical competition, followed by further tests, but some follow up with an intensive training period for the team.

In Great Britain selection of the team begins with the National Mathematics Contest (NMC) which takes place in March. Any school pupil may enter for this competition which is a ‘multiple choice’ type of paper. Those who score high marks in the NMC are invited to enter for the British Mathematical Olympiad (BMO); the best entrants here take the Further International Selection Test (FIST) on which the team is selected. A problem-solving correspondence ‘course’ is the only form of training given to the team, but it is important to emphasize that participation in this course is not restricted to members of the team. (A stamped, addressed envelope sent to Dr D. Monk, University Department of Mathematics, James Clerk Maxwell Building, Mayfield Road, Edinburgh, will bring details of how to take part.) The limited training that the British Team receives makes our results over the past years all the more pleasing. We shall hope to do well in 1979 when the competition is held in London for the first time.

Details of how to enter can be obtained from the Mathematical Association, 259 London Road, Leicester.

Questions set in the 1978 Olympiad

1. m and n are natural numbers with . In their decimal representations, the last 3 digits of 1978^m are equal, respectively, to the last 3 digits of 1978ⁿ. Find m and n such that m + n has its least value.

2. P is a given point inside a given sphere and A, B, C are any three points on the sphere such that PA, PB and PC are mutually perpendicular. Let Q be the vertex diagonally opposite to P in the parallelepiped determined by PA, PB and PC. Find the locus of Q.

3. The set of all positive integers is the union of two disjoint subsets { f(1), f(2), ..., f(n), ... }, { g(1), g(2), ..., g(n), ... }, where f(1) <f(2) < ... < f(n) < ..., g(1) < g(2) < ... < g(n) < ..., and g(n) = f(f(n)) + 1 for all . Determine f(240).

4. In the triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of the triangle ABC and also to the sides AB, AC at P, Q, respectively. Prove that the midpoint of the segment PQ is the centre of the incircle of the triangle ABC.

5. Let {a_k} (k = 1, 2, 3, ..., n, ...) be a sequence of distinct positive integers. Prove that, for all natural numbers n,

   

6. An international society draws its members from six different countries. The list of members contains 1978 names, numbered 1, 2, ..., 1978. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

DETAILED RESULTS

Reproduced with permission from Mathematical Spectrum volume 11 (1978–9) pages 33–35
© 1979 Applied Probability Trust.

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