The 19th International Mathematical Olympiad
City of London School

At Belgrade, in July 1977, 156 pre-university students from 21 countries competed in the 19th International Mathematical Olympiad. Each delegation submits problems in advance, which are considered by all the leaders just before the competition, when a final selection of six is agreed. On two successive days the competitors tackle three problems in four hours. Their scripts are assessed first by the team’s leader and his deputy, who then discuss the solutions of each problem with two Yugoslav coordinators whose task is to maintain a common standard for that one problem. The few cases where coordinators and leaders disagree on the value of a solution are considered by the whole jury, a wearying process when conducted in five languages on a hot afternoon.

The total scores (maximum 320) and prizes for each team of eight were as shown in Table 1.

This year’s British team had only one overlap with last year’s, and included two representatives from Scotland and one from Northern Ireland. The team’s position has been bettered only once in the eleven years in which we have taken part (second in 1976, though then we were 36 points behind the winners, compared with 12 points this year). One individual deserves particular mention: John Rickard (City of London School), a team member for the third time, scored full marks and received two special prizes for generalisations of problems. This unique achievement was rightly acclaimed at the closing ceremony.

Of course there is much more to an international gathering like this than the hard grind of the competition. Although the teams and leaders came together only for the last two days, when assessment had been completed, both groups were well entertained by our generous and efficient Yugoslav hosts, with excursions, receptions, concerts, a simultaneous chess display (where Richard Borcherds of King Edward’s School, Birmingham, was one of only three to beat the Yugoslav grand master), and time for making friends and learning about other countries.

The degree of national support which the teams receive varies widely. The selection of teams always seems to start with regional or national competitions, such as our National Mathematics Contest and British Mathematical Olympiad, though often these competitions are much more part of the normal curriculum than they are here. We were told that the selection of this year’s teams had involved some three million pupils. Some countries (e.g. USSR and Yugoslavia) have special mathematical schools catering for those who are successful. Others select a pool of ‘possibles’ who are given training either by correspondence or at regular weekend or holiday sessions, and from whom the team is chosen. Of all the full teams the British, who met for the first time at Heathrow, seem to have the least preparation. This makes their good results all the more praiseworthy, but it also prompts one to wonder whether with a modest extra effort we could not win.


3=Great Britain1901332
7West Germany165114
8East Germany1632111

In addition, teams of fewer than eight from Cuba, Belgium, Italy and Algeria came 18th to 21st respectively. Greece and Vietnam had to withdraw at the last moment.


These were the problems for 1977.

  1. (6 points) Equilateral triangles ABK, BCL, CDM, DAN are constructed inside the square ABCD. Prove that the midpoints of the four segments KL, LM, MN, NK and the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN are the twelve vertices of a regular dodecagon.

  2. (6 points) In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

  3. (7 points) Let n be a given integer > 2, and let Vn be the set of integers 1 + kn, where k = 1, 2, .... A number m \in V_n is called indecomposable in Vn if there do not exist numbers p, q \in V_n such that pq = m. Prove that there exists a number r \in V_n that can be expressed as the product of elements indecomposable in Vn in more than one way. (Expressions which differ only in the order of the elements of Vn will be considered the same.)

  4. (6 points) a, b, A, B are given constant real numbers, and

    f(\theta) = 1 - a cos(\theta) - b sin(\theta) - A cos(2\theta) - B sin(2\theta).

    Prove that, if f(\theta) >= 0 for all real \theta, then

    a^2 + b^2 <= 2 and A^2 + B^2 <= 1.

  5. (7 points) Let a and b be positive integers. When a2 + b2 is divided by a + b the quotient is q and the remainder is r. Find all pairs (a, b), given that q2 + r = 1977.

  6. (8 points) Let f(n) be a function defined on the set of all positive integers and taking on all its values in the same set. Prove that, if f(n + 1) > f(f(n)) for each positive integer n, then f(n) = n for each n.

Reproduced with permission from Mathematical Spectrum volume 10 (1977–8) pages 37–39
© 1978 Applied Probability Trust.

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