M a Crux t Published by the Canadian Mathematical Society. h e m http://crux.math.ca/ The Back Files a The CMS is pleased to offer free access to its back file of all t issues of Crux as a service for the greater mathematical community in Canada and beyond. i c Journal title history: ➢ The first 32 issues, from Vol. 1, No. 1 (March 1975) to o Vol. 4, No.2 (February 1978) were published under the name EUREKA. ➢ Issues from Vol. 4, No. 3 (March 1978) to Vol. 22, No. r 8 (December 1996) were published under the name Crux Mathematicorum. u ➢ Issues from Vol 23., No. 1 (February 1997) to Vol. 37, No. 8 (December 2011) were published under the m name Crux Mathematicorum with Mathematical Mayhem. ➢ Issues since Vol. 38, No. 1 (January 2012) are published under the name Crux Mathematicorum. * * it it it it it * & i?§ it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it %t %t it %t t* %t K if if %t %t if if if if if if if if if if if if if if if if if if if if if if CRUX MATHEMATICORI'M Vol. 11, No. 7 September 1985 ISSN 0705 - 0348 Sponsored by Carleton-Ottawa Mathematics Association Mathematique d'Ottawa-Carleton Pub lie par le College Algonquin* Ottawa Hie assistance of the publisher ®nd the support ©f the Canadian Mathematical Society9 the Carleton University Department ©f Mathematics and Statistics 9 the University ©f Ottawa Department of Mathematics, and the endorsement of the Ottawa Valley Education Liaison Council are gratefully acknowledged. ********************************************** ********************************************* CRUX MATHEHATICORtM is a problem-solving journal at the senior secondary and university under graduate levels for those who practise or teach mathematics « Its purpose is primarily educational, but it serves also those who read it for professional, cultural, or recreational reasons« It is published monthly (except July and August) . 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Return Postage Guaranteed, * * * CONTENTS The Olympiad Corner: 67 . - M.S, Klamkin 202 Problems - Probl&mes: 1061-1070 ...,.............••'••• 219 Solutions: 2909 879, 898, 926, 930f 939-9-47 . 222 Message from the Managing Editor ........••.....•• 234 201 - - 202 - THE OLYMPIAD CORNER: 67 M.S. KLAMKIN The Twenty-Sixth International Mathematical Olympiad (IMO) was held this year in Finland from June 29 to July 9. Teams from 38 countries took part in the compe­ tition. This was again a record number of participating countries, up from last year's record of 34 countries. The maximum team size for each country was 6 students, the same as for the last two years. If the number of participating countries con­ tinues to increase, the maximum team size will probably be reduced to 4 students (as occurred in Hungary in 1982). Having a smaller team size should make it easier for countries with relatively small populations to field better teams. Also, the expenses will be reduced and the logistics made easier. The total number of parti­ cipating students was a new record of 209, up from last year's record of 192. The countries participating for the first time were China, Iceland, Iran, and Turkey, The 1986, 1987, and 1988 IMO's are to be held in Poland, Cuba, and Australia, respectively. I fully expect that there will be a new record number of participating countries for the 1988 Australian IMO, The six problems of the competition were assigned equal weights of 7 points each (the same as in the last four lMOfs) for a maximum possible individual score of 42 (and a maximum possible team score of 252) . I believe that this year's compe­ tition was harder than last year's. This is evidenced by the facts that only 10 students had scores of at least 35 (last year there were 24) and only 2 had perfect scores (8 last year). For comparison purposes, seethe last four IMO reports in [1981: 220], ["1982: 223], [1983: 205], and [1984: 249]. First, Second, and Third Prizes were awarded to students with scores in the respective intervals [34,42]9 [22,33]9 and [15,21]. Congratulations to the 14 First- Prize winners: Name Country 1 Score Geza Kos Hungary 42 Daniel Taturu Rumania 42 Gabor Megyesi Hungary 38 Nikolai I. Chaudarov Bulgaria 37 Philippe Alphonse Belgium 36 Olga Leonteva U.S.S.R. 36 Andrew Hassell Australia 35 Vasil B. Daskalov Bulgaria 35 Waldemar Horwat U.S.A. 35 Nguyen T. Dung Vietnam 35 Hagen V. Eitzen West Germany 34 Radu Negulescu Rumania 34 Gelca Razvan Rumania 34 Jeremy K&hn U.S.A. | 34 As the IMO competition is an individual event, the results are announced offi­ cially only for individual team members. However, team standings are usually com­ piled unofficially by adding up the scores of individual team members. The team results are given in the following table (where the team size is given if less than 6). Congratulations to the winning team, that of Rumania, the country that origi­ nated the IMO in 1959. - 203 - Score I Prizes Total Rank Country I (max 252) 1st 2nd 3rd Prizes -1 Rumania J 201 " " 3 _ 6 "^ [T2 U.S.A* 180 2 4 _ 6 3 Hungary 168 2 2 2 6 14 Bulgaria 165 2 3 - 5 5 ! Vietnam 144 1 3 1 5 6 Ueb eSsRe 140 1 2 2 5 7 West Germany 139 1 1 4 6 8 East Germany 136 - 3 3 6 9 France 125 - 2 3 5 10 Great Britain 121 - 2 3 5 11 1 Australia 117 1 1 2 4 12,13 | Canada 105 - 1 4 5 12,13 Czechoslovakia 105 . 3 1 4 I 14 Poland 101 w 1 4 5 15 i Brazil 83 - - 2 2 16 1 Israel 81 i - 1 - 1 17 Austria 77 | _ _ 3 3 18 Cuba 74 - - 2 2 19 The Netherlands 72 - • 1 1 \ 20 j Greece 69 _ 1 1 2 j 21 Yugoslavia 68 - _ 2 2 22 I Sweden 65 - . 1 i ! 23 1 Mongolia 62 - 1 - 1 24,25 Belgium 60 1 ^ 1 2 24,25 Morocco 60 - - 2 2 26,27 ! Colombia 54 | _. - 2 2 26,27 Turkey 54 - - 2 2 28 Tunisia 46 w _ 2 2 | Team of 4 29 Algeria 36 - _ _ _ 30 Norway 34 - - - - 31 Iran 28 • 1 - 1 Team of 1 32,33 China 27 - 1 1 2 Team of 2 32,33 , Cyprus 27 "" - 1 1 34,35 Finland 25 " - - "" 34,35 Spain 25 - « - Team of 4 36 Italy 20 1 " _ - - Team of 5 37 Iceland 13 1 _~ - . "" Team of 2 J 38 j Kuwait 1 7 1 "~ - - 1 - 1 Team of 5 >ers. sec>res * prizes, and 1eader s of th e Ca nadia n an d U.S.A. teams were as follows s Martin Piotte 18 (3rd prize) Ron Scoins, University of Waterloo Eric Veach 20 (3rd prize) Tom Griffiths, University of Western Frank DfIppolito 16 (3rd prize) Ontario Minh Vo 26 (2nd prize) Moses Klein 15 (3rd prize) Giuseppe Russo 6 Bjorn Poonen 31 (2nd prize) Cecil Rousseau, Memphis State Univer­ David Moews 27 (2nd prize) sity Jeremy Kahn 34 (1st prize) Gre^g Patruno, Columbia University David Grabiner 29 (2nd prize) Joseph Keane 24 C2nd prize) Waldemar Horwat 35 (1st prize) - 204 - I was glad to note the improved Canadian performance compared to last year's, when the Canadian team ranked 20th with a score of 83 and only one prize. It is of interest to note that Jeremy Kahn (U.S.A.) has so far participated in three IMO's and won 2nd prizes in the first two. Since he has not yet graduated, he may even participate in his fourth IMO next year (he will be 16 years old in October of this year). The problems of this year's competition are given below. Solutions to these problems, along with those of the 1985 U.S.A. Mathematical Olympiad, will appear in a booklet, Olympiads for 1985? obtainable later this year for a small charge from Dr. W.E. Mientka, Executive Director M.A.A. Committee on H.S. Contests 917 Oldfather Hall University of Nebraska Lincoln, Nebraska 68588 26TH INTERNATIONAL MATHEMATICAL OLYMPIAD First days- July 4, 1985. Times A\ hours 1, A circle has centre on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB, (Great Britain) 2B Let n and k be given relatively prime natural numbers, o < k < n. Each number in the set Af = {1, 2, ,,., n-1} ts coloured either blue or white.