AMS / MAA PROBLEM BOOKS VOL 30 The William Lowell Putnam Mathematical Competition Problems and Solutions 1965–1984 Edited by Gerald L. Alexanderson Leonard F. Klosinski Loren C. Larson 10.1090/prb/030 The William Lowell Putnam Mathematical Competition Problems and Solutions 1965–1984 Originally published by The Mathematical Association of America, 1985. ISBN: 978-1-4704-4968-1 LCCN: 2003110418 Copyright © 1985, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2018 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 23 22 21 20 19 18 AMS/MAA PROBLEM BOOKS VOL 30 The William Lowell Putnam Mathematical Competition Problems and Solutions 1965–1984 Edited by Gerald L. Alexanderson Leonard F. Klosinski Loren C. Larson MAA PROBLEM BOOKS SERIES Problem Books is a series of the Mathematical Association of America consisting of collections of problems and solutions fromannual mathematical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematics; books on the art and practice of problem solving, etc. Committee on Publications Gerald Alexanderson, Chair Roger Nelsen Editor Irl Bivens Clayton Dodge Richard Gibbs George Gilbert Gerald Heuer Elgin Johnston Kiran Kedlaya Loren Larson Margaret Robinson Mark Saul A Friendly Mathematics Competition: 35 Years of Teamwork in Indiana, edited by Rick Gillman The Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and Loren C. Larson Mathematical Olympiads 1998-1999: Problems and Solutions From Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 1999-2000: Problems and Solutions From Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 2000-2001: Problems and Solutions From Around the World, edited by Titu Andreescu, Zuming Feng. and George Lee. Jr. The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964, A. M. Gleason, R. E. Greenwood. L. M. Kelly The William LowellPutnam Mathematical Competition Problems and Solutions: 1965-1984, Gerald L. Alexanderson, Leonard F. Klosinski, and Loren C. Larson The WilliamLowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Com­ mentary, Kiran S. Kedlaya, Bjorn Poonen. Ravi Vakil USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and Zuming Feng USA and InternationalMathematical Olympiads 2001, edited by Titu Andreescu and Zuming Feng DEDICATED TO THE PUTNAM CONTEST ANTS PREFACE Let us make clear from the start that we have not tried with this collection to imitate the scholarly and extensive treatment of the first twenty-five contests by Gleason, Greenwood, and Kelly (The William LowellPutnam Mathematical Competition/Problems and Solutions: 1938-1964. Washington: MAA, 1980). That splendid volume shows the years of work spent in following up on problems, compiling better solutions, and tracing effects of some of the problems in subse­ quent work. We have done none of that here. We have compiled material essentially already available in the American Mathematical Monthly and Mathematics Magazine, correcting in several cases solutions where errors had crept in. The present volume is mainly an attempt to put together in convenient form existing material. A volume comparable to the Gleason, Greenwood, Kelly book will have to wait for another time. We hope that in the meantime the present collection will benefit students interested in preparing for the Competition, faculty who wish to organize problem seminars, or any others just interested in problems. For information about the history of the Putnam Competition, we refer the reader to the excellent articles by Garrett Birkhoff and L. E. Bush in the earlier collection. These articles also appeared in the Monthly in 1965. We are happy to have in the present collection a further bit of information about the origins of the Competition, an essay on the first contest by Herbert Robbins as told to Alan Tucker. We have summarized lists of winning teams and individual participants; more extensive information on winners and teams appears in annual reports in the Monthly. Our work would have been much more difficult had we not had the reports of the Competition carefully prepared by formerdirectors of the Competition, James H. McKay (Oakland University) and Abraham P. Hillman (University of New Mexico). We wish especially to thank them for their many contributions over the years and specifically for their excellent reports. They are largely responsible for the presentation of solutions that have appeared in the Monthly during their directorships, though, of course, they had the benefit of having the solutions given them by members of the Questions Committees over those years. And, of course, had the members of the Questions Committee not provided the questions (and in many cases solutions) there would have been no Competition. We therefore wish to thank the members of the Questions Committee: H. S. M. Coxeter (University of Toronto), AdrianoM. Garsia (California Institute of Technology), Robert E. Greenwood (University of Texas, Austin), Nicholas D. Kazarinoff (University of Michigan, Ann Arbor), Leo Moser (University of Alberta), Albert Wilansky (Lehigh University), Warren S. Loud (University of Minnesota, Minneapolis), Murray S. Klamkin (Ford Scientific Laboratories), Nathan S. Mendelsohn (University of Manitoba), Donald J. Newman (Yeshiva University), J. Ian Richards (University of Minnesota, Minneapolis), Gulbank D. Chakerian (University of California, Davis), Joseph D. E. Konhauser (Macalester College), Richard J. Bumby (Rutgers University, New Brunswick), Lawrence A. Zalcman (University of Maryland, College Park), Edward J. Barbeau, Jr. (University of Toronto), Kenneth B. Stolarsky (University of Illinois, Urbana-Champaign), Joel H. Spencer (State University of New York, Stony Brook), William J. Firey (Oregon State University), Douglas A. Hensley (Texas A & M University), vii viii THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION Melvin Hochster (University of Michigan, Ann Arbor), Bruce A. Reznick (University of Illinois, Urbana-Champaign), and Richard P. Stanley (Massachusetts Institute of Technology). We would further like to thank Alan Tucker, Chairman of the Publications Committee of the MAA, A. B. Willcox, Executive Director, and Beverly Joy Ruedi of the Editorial Office of the MAA. Gerald L. Alexanderson Leonard F. Klosinski Loren C. Larson March, 1985 CONTENTS PAGE DEDICATION . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. V PREFACE ................................................. , .. .. vii RECOLLECTIONS OF THE FIRST PUTNAM EXAMINATION BY HERBERT ROBBINS . .. xi LIST OF PROBLEMS . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3 SOLUTIONS TO THE PROBLEMS IN THE VARIOUS COMPETITIONS Twenty-sixth . .. .. .. .. .. .. .. .. .. .. .. .. 47 Twenty-seventh .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51 Twenty-eighth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 Twenty-ninth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 59 Thirtieth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 62 Thirty-first .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 66 Thirty-second .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 70 Thirty-third .. .. .. .. .. .. .. .. .. .. .. .. .. 76 Thirty-fourth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 Thirty-fifth .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 87 Thirty-sixth . .. .. .. .. .. .. .. .. .. .. .. .. 91 Thirty-seventh . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 95 Thirty-eighth .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 99 Thirty-ninth .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 103 Fortieth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 109 Forty-first . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 113 Forty-second . .. .. .. .. .. .. .. .. .. .. .. .. .. 117 Forty-third . .. .. .. .. .. .. .. .. .. .. .. .. .. 122 Forty-fourth .. .. .. .. .. .. .. .. .. .. .. .. .. .. 127 Forty-fifth . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 132 APPENDICES Winning Teams . .. .. .. .. .. .. .. .. .. .. .. .. 137 Winning Individuals .. .. .. .. .. .. .. .. .. .. .. .. .. 141 INDEX OF PROBLEMS ......................................... , . .. 145 RECOLLECTIONSOF THEFIRST PUTNAMEXAMINATION HERBERTROBBINS as told to A Ian Tucker The well-known story of the origin of the initial 1933 Putnam contest in mathematics is as follows (I believe this story to be mostly true). During half-time of the 1931 Harvard-Army football game, President A. Lawrence Lowell said to the Commandant of the U.S. Military Academy that while Army was showing that it could trounce Harvard in football, Harvard would just as easily win any contest of a more academic nature. TheCommandant took President Lowell up on his challenge and it was decided to have a mathematics contest between the two schools. I would guess that the field of mathematics was chosen because it is a subject that was studied at both West Point and Harvard (all cadets, then as now, took two years of math) and because a relative of President Lowell, George Putnam, was an amateur mathematician who was involved in arrangements for the contest and got it named after his relative William Lowell Putnam. I came to