International Mathematical Olympiads 1959–1977

VOL ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 27 27 AMS / MAA IInternationalnternational MMathematicalathematical OOlympiadslympiads 11959–1977959–1977 Compiled and with Solutions by Samuel Greitzer 10.1090/nml/027 INTERNATIONAL MATHEMATICAL OLYMPIADS 1959-1977 NEW MATHEMATICAL LIBRARY PUBLISHED BY THR MATHEMATICAL ASSOCIATION OF &ERICA Editorial Committee Ivan Niven, Chairman (1978-80) Anneli Lax, Editor University of Oregon New York University W. G. Chinn (1977-79) City College of Sun Francisco Basil Gordon (1977-79) University of California,Lm Angeles M. M. Schiffer (1976-78) Stanford Uniwrsity The New Mathematical Library (NML) was begun in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In a decade the NML matured into a steadily growing series of some twenty titles of interest not only to the originally intended audience, but to college students and teachers at all levels. Previously published by Random House and L. W. Singer, the NML became a publication series of the Mathematical Association of America (MAA) in 1975. Under the auspices of the MAA the NML will continue to grow and will remain dedicated to its original and expanded purposes. INTERNATIONAL MATHEMATICAL OLYMPIADS 1959-1977 Compiled and with solutions @v Samuel L. Greitzer Rutgers University 27 THE MATHEMATICAL ASSOCIATION OF AMERICA Drawings by Buehler & McFadden ©1978 by The Mathematical Association of America, (Inc.) All rights reserved under International and Pan-American Copyright Conventions. Published in Washington, D.C. by The Mathematical Association of America Library of Congress Catalog Card Number: 78-54027 Print ISBN 978-0-88385-627-7 Electronic ISBN 978-0-88385-942-1 Manufactured in the United States of America To my wife Ethel In loving memory NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F. Beckenhach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinofl 5. The Contest Problem Book I Annual h.s. math. exams, 1950-1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by f. J. Davis 7. Uses of Infinity by Lao Zippin 8. Geometric Transformations I by 1. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. OIds 10. Replaced by NML-34 11. Hungarian Problem Books I and 11, Based on the Eotvos 12. I Competitions 1894-1905 and 1906-1928, translated by E. Rapaport 13. Episodes from the Early History of Mathematics by A. Aaboe 14. Groups and Their Graphs by E. Grossman and W. Magnus 15. The Mathematics of Choice b.y Ivan Niven 16. From Pythagoras to Einstein by K. 0. Friedrichs 17. The Contest Problem Book I1 Annual h.s. math. exams 1961-1965. Compiled and with solutions by Charles T. Salkind 18. First Concepts of Topology by W. G. Chinn and N. E. Steenrod 19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations I1 by I. M. Yaglom, translated by A. Shields 22. Elementary Cryptanalysis-A Mathematical Approach by A. Sinkov 23. Ingenuity in Mathematics by Ross Honsberger 24. Geometric Transformations 111 by I. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book I11 Annual h.s. math. exams. 1966-1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26. Mathematical Methods in Science by George f olya 27. International Mathematical Olympiads 1959-1977. Compiled and with solutions by S. L. Greitzer 28. The Mathematics of Games and Gambling by Edward W. fackel 29. The Contest Problem Book IV Annual h.s. math. exams. 1973-1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione, and N. Shell 30. The Role of Mathematics in Science by M. M. SchifSer and L. Bowden 3 1. International Mathematical Olympiads 1978-1 985 and forty supplementary problems. Compiled and with solutions by Murray S.Klanikin 32. Riddles of the Sphinx by Martin Gardner 33. U.S.A. Mathematical Olympiads 1972-1986. Compiled and with solutions by Murray S. Klamkin 34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson 35. Exploring Mathematics with Your Computer by Arthur Engel 36. Game Theory and Strategy by Philip Strafin 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger Other titles in preparation. Editors’ Note The lathematica Association of America is pdased to add the Interna- tional Mathematical Olympiad Contests, 1959-1977, to the distinguished problem collections published in the New Mathematical Library. The basic text was prepared by S. L. Greitzer. The educational impact of such problems in stimulating mathematical thinking of young students and its long range effects have been eloquently described both from the viewpoint of the participant and that of the mature mathematician in retrospect by Gabor Szego in his preface to the Hungarian Problem Books, volumes 11 and 12 of this NML series. Our aim in the present collection is not only to help the high school student satisfy his curiosity by presenting solutions with tools familiar to him, but also to instruct him in the use of more sophisticated methods and different modes of attack by including explanatory material and alternate solutions. For problem solvers each problem is a challenging entity to be conquered; for theory spinners, each problem is the proof of their pudding. It is the fruitful synthesis of these seemingly antithetical forces that we have tried to achieve. We are extremely grateful to Samuel L. Greitzer, the ingenious problem solver and devoted coach who helped lead the U. S. Olympiad team to victory in 1977, for having compiled the bulk of the solutions; some of them are based on the contestants’ papers. We also acknowledge gratefully the many alternate solutions and elaborations contributed by Peter Ungar. The editors of the present collection have occasionally departed some- what from the wording of the problems originally presented to the English-speaking contestants. This was done in the interest of clarity and smooth style; since translations from one language into another are seldom completely faithful, we felt that such small departures were permissible. We close this foreword by quoting G. Szegij’s concluding observation from his preface to NML volumes 11 and 12: “We should not forget that the solution of any worthwhile problem very rarely comes to us easily and without hard work; it is rather the result of intellectual effort of days or weeks or months. Why should the vii viii INTERNATIONAL MATHEMATICAL OLYMPIADS young mind be willing to make this supreme effort? The explanation is probably the instinctive preference for certain values, that is, the attitude which rates intellectual effort and spiritual achievement higher than material advantage. Such a valuation can be only the result of a long cultural development of environment and public spirit which is difficult to accelerate by governmental aid or even by more intensive training in mathematics. The most effective means may consist of transmitting to the young mind the beauty of intellectual work and the feeling of satisfaction following a great and successful mental effort. The hope is justified that the present book might aid exactly in this respect and that it represents a good step in the right direction.” Basil Gordon William G. Chinn Ivan Niven Max Schiffer Anneli Lax December, 1977 Contents This volume is a collection of all the problems in the International Mathematical Olympiads (IMO) from the First (1959) through the Nine- teenth (1977) together with their solutions. To explain how the problems are selected and the contests administered, I give a bit of the historical background. Various countries have conducted national mathematical contests for a long time. ‘fie Hungarian Eotvos Competition (begun in 1894, see NML vols. 11 and 12) is a famous example. In 1959 Rumania invited Hungary, Bulgaria, Poland, Czechoslovakia, the German Democratic Republic (G.D.R.) and the W.S.S.R. to participate in the First I.M.O. After a slow start the number of participating nations grew. Finland joined in 1965, Great Britain, France and Italy in 1967; and since then the number of participating nations grew rapidly, reaching twenty-one by 1977. The U.S.A. first participated in the IMO in 1974. Travel to this competition in Erfurt, G.D.R., was made possible by a generous grant by the Spencer Foundation. Also, the National Science Foundation funded a three-week training session at Rutgers University for the American team prior to its departure. Preparatory work started in 1972 when the Subcom- mittee on the U.S.A. Mathematical Olympiad of the Mathematical Association of America’s Committee on High School Contests organized the first U.S.A. Mathematical Olympiad (USAMO). This contest examination was written by Murray Klamkin of the University of Alberta, Alberta, Canada and administered by this writer to the top 100 scorers (out of 300,000) on the Annual High School Mathematics Examination. In 1975, the training session was again held at Rutgers University and funded by NSF. The 1975 IMO was held in Burgas, Bulgaria, and travel was made possible by grants from Johnson and Johnson Foundation, Minnesota Mining and Manufacturing Corporation, the Spencer Founda- tion, Standard Oil of California, and Xerox Corporation. x INTERNATIONAL MATHEMATICAL OLYMPIADS The 1976 training session was held at the U.S. Naval Academy and the 1977 Training Session was held at the U.S. Military Academy. These training sessions were funded by grants from the Army Research Office and the Office of Naval Research.

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