Miscellaneous Problems and Essays Ross Honsberger

AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 19 In Pólya’s Footsteps Miscellaneous Problems and Essays Ross Honsberger In P6lya's Footsteps Miscellaneous Problems and Essays The inspiration for mfulY of these problems cLune from the Olympiad Comer of Crux MathenulticorlUl1, now Crux Mathel1UlticorU111 }vith Mathenultical MaYhe111, published by the Canadian Mathematical Society with support from the University of Calgary, Memorial University of Newfoundland, and the lJniversity of Ottawa. Full attribution ell1 be found with each problem. @1997 by The Mathematical Association ofAmerica (lncorporatedj Library (?f Congress Catalog Card Number 97-70506 Complete Set ISBN 0-88385-300-0 Vol. 19 ISBN 0-88385-326-4 Printed in the United States ofAmerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 10.1090/dol/019 The Do/ciani Mathenlatical EX/Jositions NUMBER NINETEEN In P6lya's Footsteps Miscellaneous Problems and Essays Ross Honsberger University) o.f Waterloo Puhlished and Distrihuted hy THE yLt\THEMATICAL ASSOCL\TIO~ OF AMERICA THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications JAMES W. DANIEL, Chair Dolciani Mathematical Expositions Editorial Board BRUCE P. PALKA, Editor CHRISTINE W. AYOUB EDWARD J. BARBEAU IRL C. BIVENS The DOLCJANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association ofAlnerica was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initi ating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Comlnittee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an alnple supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. 1. Mathematical Gelns, Ross Honsberger 2. Mathematical Gelns 11, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (A.fter 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Prohlem, David Barnette 9. Mathematical Gems Ill, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geolnetry and Numher Theory, Victor Klee and Stan Wagon 12. Problems .lor Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Intefplay oj'the Continuous and the Discrete, Robert M. Young 14. The Wohascum County Prohlenl Book, George T. Gilbert, Mark Krusemeyer, and Loren C. Larson 15. Lion Hunting and Other Mathenlatical Pursuits: A Collection 0.( Mathematics, Verse, and Stories by Ralph P Boas, if:, edited by Gerald L. Alexanderson and Dale H. Mugler 16. Linear Algebra Prohlem Book, Paul R. Halmos 17. From ErdtJs to Kiev: Proble/ns 0.( Olympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go? ... and Other Intriguing Mathenlatical A{vster ies, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 19. In P6~ya 's Footsteps: Miscellaneous Problems and Essays, Ross Honsberger Preface Just as a recording of a Mozart concerto makes no pretense of teaching one to compose music~ these mathematical perfonnances are not motivated by a desire to teach mathematics; they are offered solely .for your enjoyment. There is no denying that a certain degree of concentration is required for the appreciation of their beautiful ideas~ but it is hoped that a leisurely pace and generous explanations \vill make them a pleasure to read. The technical demands are very modest; a high school graduate should be well equipped to handle many of the topics and a university undergraduate in mathematics ought to be perfectly comfortable throughout. I hope you will find something exciting in each of these topics-a sur prising result~ an intriguing approach~ a stroke of ingenuity-and that you will approach them as entertainment. You are certainly not required to at tempt these problems before going through the solutions~ but if you are able to give them a little thought first~ I'm sure you will find them all the more exciting. It is a pleasure to acknowledge the great debt this volume owes the un dergraduate problems journal Crux Mathematicarum, now Crux Mathemati carunl vvith Mathematical Mayhem, published by the Canadian Mathematical Society with support from the University of Calgary~ Memorial University of Newfoundland~ and the University of Ottawa. For interesting elementary problems~ this publication is in a class by itself. I came across the great major ity of the problems discussed in the present volume in the Olympiad Comer columns of Crux Mathenlaticorum. Sometimes a solution given in the present collection has been based on a solution published in this column~ but unless otherwise acknowledged~ the solutions here are based on my own work (with one exception) and in all cases the responsibility for shortcomings in presen tation is mine alone. The exception is the essay The Infinite Checkerboard~ vii viii In P61ya's Footsteps which was written probably twenty years ago, and with my deepest apologies~ I cannot recall where I encountered the topic. The sections may be read in any order. No attempt has been made to bring together all the problems on a specific subject; in fact, this has been avoided in the hope of encouraging a feeling of spontaneity throughout the work. I would like to extend my wannest thanks to chainnan Bruce Palka ofthe Dolciani Subcommittee and to the members Christine Ayoub and Irl Bivens for their generous reception of this manuscript and their perceptive revie\vs. Finally, it has again been my great good fortune to have had Beverly Ruedi and Elaine Pedreira see this work through publication; their unfailing geniality and excellence in all phases of this trying process are deeply appreciated. CONTENTS Preface vii Four Engaging Problems 1 A Problem from the 1991 Asian Pacific Olympiad 6 Four Problems from the First Round of the 1988 Spanish Olympiad 9 Problem K797 from Kvant 14 An Unused Problem from the 1990 International Olympiad 16 A Problem from the 1990 Nordic Olympiad 19 Three Problems from the 1991 AIME 21 An Elementary Inequality 26 Six Geometry Problems 29 Two Problems from the 1989 Swedish Olympiad 36 Two Problems from the 1989 Austrian-Polish Mathematics Competition .42 Two Problems from the 1990 Australian Olympiad 45 Problem 1367 from Crux Mathematicorum 48 Three Problems from Japan 51 Two Problems from the 1990 Canadian Olympiad 59 A Problem from the 1989 U. S. A. Olympiad 65 A Problem on Seating Rearrangements 67 Three Problems from the 1980 and 1981 Chinese New Year's Contests .73 A Problem in Arithmetic 78 A Checkerboard Problem 81 Two Problems from the 1990 Asian Pacific Olympiad 85 Four Problems from the 1989 AIME 89 Five Unused Problems from the 1989 International Olympiad 98 Four Geometry Problems 111 Five Problems from the 1980 All-Union Russian Olympiad 121 The Fundamental Theorem of 3-Bar Motion 129 ix x In Polya's Footsteps Three Problems from the 1989 Austrian Olympiad 132 Three Problems from the Tournament of the Towns Competitions 138 Problem 1506 from Crux Mathematicorum 143 Three Unused Problems from the 1987 International Olympiad 145 Two Problems from the 1981 Leningrad High School Olympiad 151 Four Problems from the Pi Mu Epsilon Journal-Fall 1992 156 An Elegant Solution to Morsel 26 165 Two Euclidean Problems from The Netherlands 168 Two Problems from the 1989 Singapore Mathematical Society Interschool Competitions 174 Problem M1046 from Kvant (1987) 177 Two Theorems on Convex Figures 179 The Infinite Checkerboard 184 Two Problems from the 1986 Swedish Mathematical Competition 188 A Brilliant 1-1 Correspondence 192 The Steiner-Lehmus Problem Revisited 194 Two Problems from the 1987 Bulgarian Olympiad 197 A Problem from the 1987 Hungarian National Olympiad 202 A Problem from the 1987 Canadian Olympiad 206 Problem 1123 from Crux Mathematicorum 208 A Problem from the 1987 AIME 211 A Generalization of Old Morsel 3 213 Two Problems from the 1991 Canadian Olympiad 216 An Old Chestnut 221 A Combinatorial Discontinuity 223 A Surprising Theorem of Kummer 229 A Combinatorial Problem in Solid Geometry 234 Two Problems from the 1989 Indian Olympiad 239 A Gem from Combinatorics 243 Two Problems from the 1989 Asian Pacific Olympiad 247 A Selection of Joseph Liouville's Amazing Identities Concerning the Arithmetic Functions a(n), T( n), ¢(n), Jl( n), A(n) 251 A Problem from the 1988 Austrian-Polish Mathematics Competition 267 An Excursion into the Complex Plane 271 Two Problems from the 1990 International Olympiad 280 Exercises 289 Solutions to the Exercises 293 Praise God from Whom all blessings flow. Exercises These exercises are generally much easier than most of the problems we have been considering and they are included just for your amusement. They are questions, some slightly revised, that occurred on various Grade 13 Problems Papers ofOntario, a defunct examination which used to be taken by candidates writing for university scholarships involving mathematics. 1. (1960). If the lengths of the sides of a triangle are in the ratios 3 : 7 : 8, show that its angles are in arithmetic progression.

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