Shake the kaleidoscope. Visas and traveller’s cheques; the anxious meeting at Cromwell Road (so these are our companions for the next twelve days); the Danube from 29,000 ft; Bulgarians trotting into Sofia airport lounge by the OUT door; the pound-stokinki exchange rate (look again! stokinki?) 2.72 to the pound, Euler’s e; the heat; the friendly interpreter-companion; the traumatic isolation of team from jury at Burgus; the free pocket-money issue.
The final wording of the lucky six questions, chosen from dozens; prams and citizens promenading the Burgus boulevards in the cool of the evening; the open churches and a wedding; eating in a PECTOPAHT (transliteration from Cyrillic left as an exercise for the reader); never a dog.
The Opening Ceremony; flags of the nations; the heat; delay from lights failure; the heat; the start of the first four-hour session; tension in the jury room as questions on the questions are brought up to be answered or shrugged off; the heat; ice-cream, Coca-Cola, coffee rounds; only an hour to go.
See blue Black Sea (black Blue Danube we never see close); the inscrutable Mongolians, the smiling Vietnamese (they have something to smile about); sunflowers by the square mile (sorry: hectare); beaches, waves, sand, Golden Sands, Sunny Beach, tower hotels. The second session.
Marking, coordinating, discussing prizes; the jury’s turn on the rack; three days of that; but beaches again for the teams; and again the beaches.
The two-day cavalcade to Sofia; the competitors’ buses, police-escorted, holding the middle of the road; headlights glaring at noon; into the verge with oncoming traffic. The Balkan range and age-old passes, the invader’s way; yoghourt; the Turks and 1875; the welcomes in the towns; bouquets of carnations, roses; children with gifts; platforms, speeches (marguerites in the smaller towns); attar of roses, Thracian tomb. Turnovo, the Ancient Capital, with cyclopean gatehouse and Baldwin’s Tower; (breathless? weren’t we all?); fruit juices versus Coca-Cola; Roman inscriptions (a far cry from Hadrian’s Wall, the bounds of Empire, but Babel now); the Russian learner speaking his Russian to real live Russians with dust on their shoes.
Sofia; The Holy Wisdom and many-domed, golden-domed Alexander Nevsky; 1875 and the Turks; pink and peach stucco; Party HQ; cool mosque, the last of its tribe; Lenin’s frown; ancient churches; frescoes; bookshops; a dog (on a lead). The Closing Ceremony (already?); the prizes, hand-tooled leather folders; photographs; hands clasped; the partings of friends, photographs; the final dinner, 200 seated in a mountain-top restaurant; a double-bill farewell speech with American speaking Russian and Russian English (the night of space-craft rendezvous); dancing, more speeches; more wine (beer for the boys).
Sofia airport again; Vienna, only the airport (too far to the city, but a pastry, surely, and a coffee? At over £1?); IMO’s to be here next year, perhaps they’ll be free then? Heathrow; the bonds slip away; the crowds suck us in; ‘See you in October’, ‘Come and visit me’, ‘We’ll look at my slides’.
What was the mathematics like? The what? Oh, the mathematics; there were puzzles, tricks, in-jokes, chess; the official part was like this:
Let xi, yi (i = 1 to n) be real numbers such that
Prove that, if z1, z2, ..., zn is any permutation of y1, y2, ..., yn, then
Let a1, a2, ..., ar .. be any infinite sequence of strictly positive integers such that ar < ar + 1 for . Prove that infinitely many an can be written in the form
an = xai + yaj
with x, y strictly positive, and .
On the sides of an arbitrary triangle ABC, triangles ABR, BCP, CAQ are constructed externally with
Prove that and QR = RP.
When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A, B are written in decimal notation.)
Determine, with proof, whether one can find 1975 points on the circumference of a circle of unit radius such that the distance (along the chord) between any two of them is a rational number.
Find all polynomials P in two variables with the following properties:
P(tx, ty) = tnP(x, y)
(that is P is homogeneous of degree n), and
P(a + b, c) + P(b + c, a) + P(c + a, b) = 0,
Countries of origin:
1. Czechoslovakia, 2. Great Britain, 3. Netherlands, 4. USSR, 5. USSR, 6. Great Britain.
Our scores (out of 40) were 40, 40, 36, 32, 25, 24, 23, 19; we got 2 first prizes (only 8 competitors scored 40), 2 second prizes, 3 third prizes. As a team (team? the Olympic spirit?) we were 5th, 19 points behind the Hungarian winners, and the 7th team trailed us by 47 points.
Enter the NMC*, O reader; get chosen for the BMO; try for the XVIIIth IMO in Austria, 1976.
* National Mathematical Contest.
British Mathematical Olympiad.
Reproduced with permission from Mathematical Spectrum volume 8
(1975–6) pages 37–39
© 1976 Applied Probability Trust.
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