INTERNATIONAL MATHEMATICAL OLYMPIAD 1974

Table 1. The British Team

II = 2nd prize   III = 3rd prize

Table 2. The National Order.

*Teams with less than eight members

Tables reproduced from Science Teacher volume 18 number 1 (October 1974) page 4.

16th INTERNATIONAL MATHEMATICAL OLYMPIAD 1974

DR. DAVID MONK, Deputy Leader of the British Team

Eighteen countries, the largest number ever, took part in the 16th International Mathematical Olympiad in the German Democratic Republic in July. Newcomers were the United States of America and North Vietnam. Each country was represented by a team of eight school students, accompanied by a leader and deputy leader.

The two contest sessions, each of four hours duration, were held at the Pädagogische Hochschule “Dr. Theodor Neubauer”, a teachers’ training college at Erfurt in which the teams were accommodated. Erfurt, a city some 150 miles south-west of Berlin, is a regional capital with various industries, situated in pleasant countryside, and important as a horticultural centre.

Mr. Robert Lyness, the leader of the British team, took part in the preliminary discussions on the Olympiad jury, at which the six contest problems were selected. The team and deputy leader left London (Victoria) on 5 July. We reached Erfurt punctually, and were greeted by IMO representatives. A coach took us to the Hochschule, where a friendly reception and a much-needed meal awaited us. There followed a brief ceremony at which the British flag was raised alongside those of other competing nations. Then the deputy leader left by car to join the other deputies and leaders at the Interhotel “Elephant” in Weimar.

Weimar, a smaller town about 12 miles east of Erfurt, is rich in literary and musical associations. Goethe, Schiller and Liszt lived there at various times and their houses are carefully preserved as tourist attractions. Sightseeing interludes during the mathematical activities therefore presented no problems! There is a College of Architecture where the jury met, and working rooms were provided for the delegations.

The day after their arrival was spent by the teams with sightseeing in Erfurt and a youth concert. Meanwhile, the jury was settling fine points of wording of the questions and checking the typescripts of the papers. Strict precautions are always taken to preserve the confidentiality of the questions and to ensure scrupulous fairness, so the teams and their seniors did not mix at the opening ceremony. Immediately afterwards, the first session began, and the leaders had another duty—considering queries about the problems made in writing by contestants. While they worked their deputies enjoyed substantial “elevenses”!

Then, leaving the teams to their labours, the jury went on a coach tour of Erfurt. Martin Luther studied and preached in the city, and its sights include the Cathedral and the adjacent Severikirche. The leaders’ exclusion from the earlier refreshments may have proved an advantage at the next event on the jury programme—a sumptuous lunch as guests of the Erfurt Regional Council!

By the time the jury returned to Weimar the scripts from the day’s session were ready for marking. This, with the attendant “co-ordination” and jury meetings to resolve difficulties and assign prizes, was to occupy the leaders and deputies intermittently for the next three and a half days. In addition the leaders made another early journey to Erfurt to adjudicate on queries at the start of the second session.

The scripts of each team are marked initially by its leader and deputy leader. They are then scrutinised by co-ordinators. The German mathematicians who performed this formidable task worked in groups of three, each group dealing with just one question. Of all the aspects of the generally excellent organisation of the Olympiad we admired this most. The co-ordination was carried out with the greatest courtesy and good humour, adhering to a carefully planned timetable. Sometimes the co-ordinators were more generous than we expected, occasionally a little stricter, but there was never any difficulty in agreeing on a mark.

At the final reckoning, the British team was placed 9th in the national order. Although naturally we should have liked a higher place, it must be recognised that most of the other teams receive coaching and preparation to an extent which we have not so far been able to provide.

At its final meeting the jury agreed on the following inclusive mark ranges for prizes: 38–40: 1st; 30–37: 2nd; 23–29: 3rd. This accorded with the custom of giving prizes to about half the contestants. The rough guide-line is that the numbers of 1st, 2nd and 3rd prizes should be in the ratios 1 : 2 : 3; the actual numbers were 10, 24 and 37. As a result the British team took one 2nd and three 3rds; the individual scores are listed in the first table. Three contestants, from America, Hungary and Sweden, were awarded prizes for special elegance. Six gained the maximum possible mark of 40.

Special mention should be made of the performances by the two newly participating countries. The Americans, to their great delight, came second after Russia, beating the Hungarians and East Germans, always strong contenders, into 3rd and 4th place respectively; every member of the American team gained a prize. Only five Vietnamese competed but four were prizewinners, including one 1st. The Cuban team, too, was below strength, with seven members. The complete national order is given in the second table.

A personal view is that the six questions were, on average, a shade easier than those of recent Olympiads. If so, this probably reflects the need to conform with the school syllabuses of an increasing number of countries. Question 1 was particularly simple. Some of our team, remarking on this, added the fair comment that two questions, numbers 1 and 4, had common features, involving systematic elimination of cases.

Sightseeing and social events form an essential part of an IMO. While the marking was in progress the teams had two full-day outings—to Eisenach and a nearby castle, and to a textile spinning mill—and a dance. After the last jury meeting came the first excursion for teams and jury—to the former Buchenwald concentration camp, situated just outside Weimar and now a national memorial and museum. Here, among other similar ceremonies, Mr. Lyness and the American leader, Professor S. L. Greitzer, laid wreaths on a memorial to British and Canadian airmen.

On the following day, the entire personnel of the Olympiad travelled by rail to (East) Berlin, the teams leaving Erfurt to the accompaniment of farewell fanfares and singing, the seniors joining the same train at Weimar. Again things were tightly organised. Barely an hour after reaching our new accommodation—a hotel for the jury, a university residence for the teams—we were off once more, this time for a cruise through the rivers and lakes of Berlin, with lunch on the boat. Sunday was devoted to an all-day visit to Potsdam. Here we saw the Sanssouci Palace and the Cecilienhof, the scene of the Potsdam Conference in July 1945. Some of the British team were rebuked by a stern policewoman for crossing tramlines without using the subway: fortunately their only brush with authority!

The Berlin Kongresshalle was the setting for the closing ceremony on Monday, 15 July. The prizewinners received their diplomas and there were several addresses, including one by the only girl among the winners, from Czechoslovakia. The orchestra of the Georg-Friedrich-Hündel-Oberschule (a special school for the musically talented) and a solo pianist provided musical interludes. These formalities were followed in the evening by a cheerful farewell dinner and entertainment. A day remained for local sightseeing, including the breathtaking view of the city from the television tower. Finally, on 17 July, the British contingent left by rail for the Hook of Holland, the boat to Harwich and home.

The whole trip was constantly interesting and enjoyable. The German organisers are to be congratulated on their arrangements and warmly thanked for their helpfulness and hospitality, which has been merely sketched in the foregoing paragraphs. Accepting the grave risk of unfairness to others by naming just two persons, mention must be made of Helmut Schreiber, our patient interpreter, and Professor Wolfgang Engel of Rostock, the efficient and genial Chairman of the Jury. As always, informal conversation between the various leaders and deputies was a most valuable feature. The same may be said for the teams, though in their case language difficulties were perhaps more of an obstacle. Gifts were exchanged, and both team members and seniors received from the organisers generous spending allowances for the purchase of refreshment and souvenirs.

The 16th International Mathematical Olympiad has indeed set a high standard for its successors to match! Both Bulgaria and Mongolia have issued invitations to organise the 17th Olympiad. Moreover “soundings” are being made on the possibility of Austria being the host nation in 1976; this would be the first IMO to be held in a non-Communist country.

IMO Questions: see page 9

Report reproduced from Science Teacher volume 18 number 1 (October 1974) page 5.

International Mathematical Olympiad 1974: Questions

Three Qs were taken in two sessions on succeeding days. Time allowed for each session was four hours. The country of origin of the Q and the number of marks for it appear in brackets, e.g. (USA: 5).

1. Three players A, B and C play the following game: On each of three cards an integer is written. These three numbers p, q, r satisfy 0 < p < q < r.

   These three cards are shuffled and dealt so that each player has a card. Each then receives the number of counters indicated by the card he holds. Then the cards are shuffled again; the counters remain with the players.

   This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round A has 20 counters in all, B has 10 and C has 9. At the last round B received r counters. Who received q counters on the first round? (USA: 5)

2. In the triangle ABC, prove that there is a point D on the side AB such that CD is the geometric mean of AD and DB if and only if

   

   (Finland: 6)

3. Prove that the number

   

   is not divisible by 5 for any natural number n.

   (Romania: 8)

4. Consider decompositions of an 8 × 8 chessboard into p non-overlapping rectangles subject to the following conditions:

   1. Each rectangle is to have as many white squares as black squares.
   2. If a_i is the number of white squares in the i–th rectangle, then a₁ < a₂ < ... < a_p.

   Find the maximum value of p for which such a decomposition is possible. For this value of p, determine all possible sequences, a₁, a₂, ..., a_p.

   (Bulgaria: 6)

5. Determine all possible values of

   

   when a, b, c, d are arbitrary positive numbers.

   (Netherlands: 7)

6. Let P be a non-constant polynomial with integer coefficients. If n(P) is the number of distinct integers k such that [P(k)]² = 1, prove that , where deg(P) denotes the degree of the polynomial P.

   (Sweden: 8)

Questions reproduced from Science Teacher volume 18 number 1 (October 1974) page 9.

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