The 21st International Mathematical Olympiad
COLIN GOLDSMITH
Marlborough College

Readers of Mathematical Spectrum will have seen reports of previous years’ contests, which are for teams of up to eight pre-university students. New and old readers will appreciate the tough nature of the Olympiad if they settle down and tackle the questions, allowing about an hour for each.

A record number of countries (23) were represented, including for the first time Brazil, Israel and Luxembourg. The results of the leading nations were as shown in the table.

	Competitor number								Total	Prizes
	1	2	3	4	5	6	7	8		1st	2nd	3rd	Special
U.S.S.R.	19	27	35	34	36	40	36	40	267	2	4	1
Romania	35	23	17	33	32	28	39	33	240	1	4	2
West Germany	21	14	35	33	39	30	34	29	235	1	5	1
United Kingdom	23	23	34	30	32	34	21	21	216		4	4
U.S.A.	17	9	27	17	39	24	32	34	199	1	2	2
Vietnam	32	29	33	40						1	3		1
East Germany	33	13	31	20	19	19	26	19	180		2	2
Czechoslovakia	16	20	24	17	23	40	12	26	178	1		4
Hungary	15	20	16	18	34	22	17	34	176		2	2
Yugoslavia	13	27	13	11	23	26	24	35	172		1	4

Since this year’s Olympiad was held for the first time in Britain, it is appropriate to describe some of the arrangements. The various countries had sent in 80 possible questions altogether. A short list of 26 was made by Dr T. Fletcher, Dr D. Monk and Mr R. Lyness, and these were discussed by the team leaders, with Dr Fletcher as chairman, three days before the competition. This process went exceptionally smoothly, and the final choice of six questions emerged without serious disagreement. Since all the really hard questions were voted against and question 3 was subsequently eased considerably when the detailed wording was considered, the papers were ultimately a little less testing than usual and seven students obtained full marks or only dropped one point.

The competitors were housed in Westfield College, London, and the domestic side, travel and entertainment were all administered by the School Mathematics Project under an organizing committee set up by the National Committee for Mathematical Contests. Special mention must go to Mr J. Hersee who was in charge of all these aspects. Group visits were made to Hampton Court, Greenwich (by boat), Windsor Castle and Stratford, and the interpreter-guides attached to each team took their students on further sightseeing and shopping trips to central London. A most lively concert was put on by the Royal Academy of Music, and an enjoyable reception was held by the Greater London Council on the terrace of County Hall, with a steel band playing. Two nights at the end were spent in Oxford.

All previous Olympiads have been government financed. This one was the responsibility of the School Mathematics Project. The Department of Education and Science made a substantial contribution, and generous amounts were given by the Royal Society, the British Council and various firms and schools. Dr B. Thwaites chaired the organizing committee and was primarily responsible for finance.

The Olympiad was officially opened by Sir James Hamilton, Permanent Secretary of the DES. The prizes were presented by the Duchess of Gloucester, and a message from the Duke of Edinburgh was read at the formal dinner at which the guests of honour were representatives of the sponsors.

We can congratulate ourselves on the relaxed and efficient staging of the 21st IMO, which was patently enjoyed by the teams and their leaders. The British team were admirable hosts, and it was gratifying that they were the only team to secure eight prizes.

The six questions were:

Let p and q be natural numbers such that

Prove that p is divisible by 1979.
A prism with pentagons A₁A₂A₃A₄A₅ and B₁B₂B₃B₄B₅ as top and bottom faces is given. Each side of the two pentagons and each of the line-segments A_iB_j, for all i, j = 1,... , 5, is coloured either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been coloured has two sides of a different colour. Show that all 10 sides of the top and bottom faces are the same colour.
Two circles in a plane intersect. Let A be one of the points of intersection. Starting simultaneously from A two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point P in the plane such that, at any time, the distances from P to the moving points are equal.
Given a plane $\pi$ , a point P in this plane and a point Q not in $\pi$ , find all points R in $\pi$ such that the ratio (QP + PR)/QR is a maximum.
Find all real numbers a for which there exist non-negative real numbers x₁, x₂, x₃, x₄, x₅ satisfying the relations

$\sum_{k=1}^5 (k x_k) = a, \sum_{k=1}^5 (k^3 x_k) = a^2, \sum_{k=1}^5 (k^5 x_k) = a^3.$
Let A and E be opposite vertices of a regular octagon. A frog starts jumping at vertex A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches vertex E, the frog stops and stays there. Let a_n be the number of distinct paths of exactly n jumps ending at E. Prove that

$a_{2n-1} = 0, a_2n = (1/\sqrt{2})(x^{n-1} - y^{n-1}), n = 1, 2, 3, ...;$

where $x = 2 + \sqrt{2}$ and $y = 2 - \sqrt{2}$ .

Note: A path of n jumps is a sequence of vertices (P₀, ..., P_n) such that
1. P₀ = A, P_n = E;
2. for every i, , P_i is distinct from E;
3. for every i, , P_i and P_i + 1 are adjacent.

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Online HTML version last updated: 5 January 2016

The 21st International Mathematical Olympiad COLIN GOLDSMITH Marlborough College

The 21st International Mathematical Olympiad
COLIN GOLDSMITH
Marlborough College