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. ⃝∞ 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 Harvard in the fall of 1931 with what I thought was some knowledge of the humanities,but no mathematics beyond quadratic equations. To address this deficiency, I enrolled in Math A, Analytic Geometry and Calculus. The texts were Analytic Geometry by Osgood and Graustein and Calculus by Osgood. These books were casual in their treatment of real numbers and limits but had challenging problems that assumed a good knowledge of physics. The best-prepared students in Math A were put in the section taught by the eminent Professor Julian Lowell Coolidge, and the rest were taught by junior instructors. I was put in the section taught by Sumner B. Myers. I skipped a good many of the classes (being occupied with certain extracurricu lar interests most of the year) but did well in the tests. At the end of the course in May, 1932, I was invited to be part of the team that would represent Harvard in the mathematics competition with West Point the following spring. Since cadets took two years of mathematics, the Harvard team was restricted to students who would be completing their second year of mathematics at the time of the test. Selection for this team led me to continue my study of mathematics for a second year. The Harvard Mathematics Department assigned Professor Marston Morse to coach the team, and we met with him about four times during the fall and winter. It was assumed that our Harvard intellects would easily carry the day, and our meetings with Morse were spent in general conversation rather than problem-solving.However, these sessions were very important to me, for I became impressed by Morse as a person and by the incomprehensibility of the mathematics he spoke of. I had many conversations with Morse in addition to the Putnam sessions. Morse was in a low mood then because his wife had recently left him to marry Professor William Fogg Osgood, a distinguished Harvard mathematician with a long white beard who was 28 years Morse's senior (this scandal forced Osgood to leave Harvard). However, my association with Morse was not enough to persuade me to pick mathematics as my major just yet; I was experimenting then with majors in the sciences and philosophy. One spring weekend the Harvard team traveled to West Point for the competition. There was a morning and afternoonsession, and the problems were rather cut and dried, technical integrations and the like, with little call for originality. The highlight of the weekend for me was a date Saturday night in New York City with a girl I had met the previous summer. Back at Harvard we found out, to our shame, that we had lost the competition to Army. I was told that I had done well on the exam (we never saw our exam books) and decided to be a math
xi xii THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION major. I never would have studied more than a year of mathematics, much less have become for a time a mathematician, were it not for my experience with the Putnam competition. The next year Morse left Harvard to go to the Institute for Advanced Study. He told me to continue my studies, to get a PhD in math at Harvard, and then to get in touch with him. I had no further contact with Morse until five years later when I defended my thesis at Harvard and sent him a telegram: "Have PhD in mathematics." His response was equally brief: "You are my assistant starting September l." It would seem that the reason I finally became a math major was that most Harvard mathematics professors were rather pompous know-it-alls and that I wanted to show them that any reasonably bright person could do mathematics. Unfortunately, I won the battle but lost the war.
s Herbert Robbin was the subject of an interview in the January 1984 College Mathematit:.,Journal. He was co-author with Richard Courant of What is Mathematics? and is Higgins Professor of Mathematical Statistics at Columbia University. WINNING TEAMS 137
APPENDIX
WINNING TEAMS
Twenty-sixthCompetition-1965 Harvard University, Cambridge, Massachusetts Massachusetts Institute of Technology, Cambridge, Massachusetts University of Toronto, Toronto, Ontario, Canada Princeton University, Princeton, New Jersey California Institute of Technology, Pasadena, California
Twenty-seventh Competition-1966 Harvard University, Cambridge, Massachusetts Massachusetts Institute of Technology, Cambridge, Massachusetts University of Chicago, Chicago, Illinois University of Michigan, Ann Arbor, Michigan Princeton University, Princeton, New Jersey
Twenty-eighth Competition-1967 Michigan State University, East Lansing, Michigan California Institute of Technology, Pasadena, California Harvard University, Cambridge, Massachusetts Massachusetts Institute of Technology, Cambridge, Massachusetts University of Michigan, Ann Arbor, Michigan
Twenty-ninthCompetition-1968 Massachusetts Institute of Technology, Cambridge, Massachusetts University of Waterloo, Waterloo, Ontario, Canada University of California at Los Angeles, Los Angeles, California Michigan State University, East Lansing, Michigan University of Kansas, Lawrence, Kansas
ThirtiethCompetition-1969 Massachusetts Institute of Technology, Cambridge, Massachusetts Rice University, Houston, Texas University of Chicago, Chicago, Illinois Harvard University, Cambridge, Massachusetts Yale University, New Haven, Connecticut
Thirty-first Competition-1970 University of Chicago, Chicago, Illinois Massachusetts Institute of Technology, Cambridge, Massachusetts University of Toronto, Toronto, Ontario, Canada Illinois Institute of Technology, Chicago, Illinois California Institute of Technology, Pasadena, California 138 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION Thirty-secondCompetition-1971 California Institute of Technology, Pasadena, California University of Chicago, Chicago, Illinois Harvard University, Cambridge, Massachusetts University of California, Davis, California Massachusetts Institute of Technology, Cambridge, Massachusetts
Thirty-third Competition-1972 California Institute of Technology, Pasadena, California Oberlin College, Oberlin, Ohio Harvard University, Cambridge, Massachusetts Swarthmore College, Swarthmore, Pennsylvania Massachusetts Institute of Technology, Cambridge, Massachusetts
Thirty-fourthCompetition-1973 California Institute of Technology, Pasadena, California University of British Columbia, Vancouver, British Columbia, Canada University of Chicago, Chicago, Illinois Harvard University, Cambridge, Massachusetts Princeton University, Princeton, New Jersey
Thirty-fifthCompetition-1974 University of Waterloo, Waterloo, Ontario, Canada University of Chicago, Chicago, Illinois California Institute of Technology, Pasadena, California Massachusetts Institute of Technology, Cambridge, Massachusetts University of British Columbia, Vancouver, British Columbia, Canada
Thirty-sixthCompetition-1975 California Institute of Technology, Pasadena, California University of Chicago, Chicago, Illinois Massachusetts Institute of Technology, Cambridge, Massachusetts Princeton University, Princeton, New Jersey Harvard University, Cambridge, Massachusetts
Thirty-seventh Competition-1976 California Institute of Technology, Pasadena, California Washington University, St. Louis, Missouri Princeton University, Princeton, New Jersey * Case Western Reserve University, Cleveland, Ohio * Massachusetts Institute of Technology, Cambridge, Massachusetts
* Tied for fourth place. WINNING TEAMS 139 Thirty-eighthCompetition-1977 Washing ton University, St. Louis, Missouri University of California, Davis, California California Institute of Technology, Pasadena, California Princeton University, Princeton, New Jersey Massachusetts Institute of Technology, Cambridge, Massachusetts
1birty-ninthCompetition-1978 Case Western Reserve University, Cleveland, Ohio Washington University, St. Louis, Missouri University of Waterloo, Waterloo, Ontario, Canada Harvard University, Cambridge, Massachusetts California Institute of Technology, Pasadena, California
FortiethCompetition-1979 Massachusetts Institute of Technology, Cambridge, Massachusetts California Institute of Technology, Pasadena, California Princeton University, Princeton, New Jersey Stanford University, Stanford, California University of Waterloo, Waterloo, Ontario, Canada
Forty-firstCompetition-1980 Washington University, St. Louis, Missouri Harvard University, Cambridge, Massachusetts University of Maryland, College Park, Maryland University o1 Chicago, Chicago, Illinois University of California, Berkeley, California
Forty-secondCompetition-1981 Washington University, St. Louis, Missouri Princeton University, Princeton, New Jersey Harvard University, Cambridge, Massachusetts Stanford University, Stanford, California University of Maryland, College Park, Maryland
Forty-thirdCompetition-1982 Harvard University, Cambridge, Massachusetts University of Waterloo, Waterloo, Ontario, Canada California Institute of Technology, Pasadena, California Yale University, New Haven, Connecticut Princeton University, Princeton, New Jersey
Forty-fourthCompetition-1983 California Institute of Technology, Pasadena, California Washington University, St. Louis, Missouri University of Waterloo, Waterloo, Ontario, Canada 140 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION
Princeton University, Princeton, New Jersey University of Chicago, Chicago, Illinois
Forty-fifthCompetition-1984 *University of California, Davis, California *Washington University, St. Louis, Missouri Harvard University, Cambridge, Massachusetts Princeton University, Princeton, New Jersey Yale University, New Haven, Connecticut
* Tied for first place. WINNING INDIVIDUALS 141 WINNING INDIVIDUALS
Twenty-sixth Competition-1965 Andreas R. Blass, University of Detroit Robert Bowen, University of California, Berkeley Daniel Fendel, Harvard University Lon M. Rosen, University of Toronto Barry Simon, Harvard University
Twenty-seventh Competition-1966 Marshall W. Buck, Harvard University Theodore C. Chang, Massachusetts Institute of Technology Robert E. Maas, University of Santa Clara Richard C. Schroeppel, Massachusetts Institute of Technology Robert S. Winternitz, Massachusetts Institute of Technology
Twenty-eighthCompetition-1967 David R. Haynor, Harvard University Dennis A. Hejhal, University of Chicago Peter L. Montgomery, University of California, Berkeley Richard C. Shroeppel, Massachusetts Institute of Technology Don B. Zagier, Massachusetts Institute of Technology
Twenty-ninthCompetition-1968 Don Coppersmith, Massachusetts Institute of Technology Gerald A. Edgar, University of California, Santa Barbara Gerald S. Gras, Massachusetts Institute of Technology Dean G. Huffman, Yale University Neal Koblitz, Harvard University
ThirtiethCompetition-1969 Alan R. Beale, Rice University Don Coppersmith, Massachusetts Institute of Technology Gerald A. Edgar, University of California, Santa Barbara Robert A. Oliver, University of Chicago Steven Winkler, Massachusetts Institute of Technology
Thirty-first Competition-1970 Jockum Aniansson, Yale University Don Coppersmith, Massachusetts Institute of Technology Jeffrey Lagarias, Massachusetts Institute of Technology Robert A. Oliver, University of Chicago Arthur Rubin, Purdue University Steven K. Winkler, Massachusetts Institute of Technology 142 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION Thirty-secondCompetition-1971 Don Coppersmith, Massachusetts Institute of Technology Robert Israel, University of Chicago Dale Peterson, Yale University Arthur Rubin, Purdue University David Shucker, Swarthmore College Michael Yoder, California Institute of Technology
Thirty-thirdCompetition-1972 Ira Gessel, Harvard University Dean Hickerson, University of California, Davis Arthur Rothstein, Reed College Arthur Rubin, California Institute of Technology David Vogan, University of Chicago Michael Yoder, California Institute of Technology
Thirty-fourthCompetition-1973 David J. Anick, Massachusetts Institute of Technology Peter G. de Buda, University of Toronto Matthew L. Ginsberg, Wesleyan University Arthur L. Rubin, California Institute of Technology Angelos J. Tsirimokos, Princeton University
Thirty-fifthCompetition-1974 Thomas G. Goodwillie, Harvard University Grant M. Roberts, University of Waterloo Karl C. Rubin, Princeton University James B. Saxe, Union College Philip N. Strenski, Armstrong State College
Thirty-sixthCompetition-1975 Franklin T. Adams, University of Chicago David J. Anick, Massachusetts Institute of Technology Ernest S. Davis, Massachusetts Institute of Technology Thomas G. Goodwillie, Harvard University Christopher L. Henley, California Institute of Technology
Thirty-seventhCompetition-1976 Philip I. Harrington, Washington University, St. Louis Christopher L. Henley, California Institute of Technology Paul M. Herdig, Case Western Reserve University Nathaniel S. Kuhn, Harvard University Steven T. Tschantz, University of California, Berkeley David J. Wright, Cornell University WINNING INDIVIDUALS 143
Thirty-eighthCompetition-1977 Russell D. Lyons, Case Western Reserve University Stephen W. Modzelewski, Harvard University Michael Roberts, Massachusetts Institute of Technology Adam L. Stephanides, University of Chicago Paul A. Vojta, University of Minnesota, Minneapolis
Thirty-ninthCompetition-1978 Randall L. Dougherty, University of California, Berkeley Mark R. Kleiman, Princeton University Russell D. Lyons, Case Western Reserve University Peter W. Shor, California Institute of Technology Steven T. Tschantz, University of California, Berkeley
FortiethCompetition-1979 Randall L. Dougherty, University of California, Berkeley Richard Mifflin, Rice University Mark G. Pleszkoch, University of Virginia Miller Puckette, Massachusetts Institute of Technology Charles H. Walter, Princeton University
Forty-first Competition-1980 Eric D. Carlson, Michigan State University Randall L. Dougherty, University of California, Berkeley Daniel J. Goldstein, University of Chicago Laurence E. Penn, Harvard University Michael Raship, Harvard University
Forty-second Competition-1981 David W. Ash, University of Waterloo Scott R. Fluhrer, Case Western Reserve University Michael J. Larsen, Harvard University Robin A. Pemantle, University of California, Berkeley Adam Stephanides, University of Chicago
Forty-thirdCompetition-1982 David W. Ash, University of Waterloo Eric D. Carlson, Michigan State University Noam D. Elkies, Columbia University Brian R. Hunt, University of Maryland, College Park Edward A. Shpiz, Washington University, St. Louis
Forty-fourth Competition-1983 David W. Ash, University of Waterloo Eric D. Carlson, Michigan State University 144 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION
Noam D. Elkies, Columbia University Michael J. Larsen, Harvard University Gregg N. Patruno, Princeton University
Forty-fifthCompetition-1984 Noam D. Elkies, Columbia University Benji N. Fisher, Harvard University Daniel W. Johnson, Rose-Hulman Institute of Technology Michael Reid, Harvard University Richard A. Stong, Washington University, St. Louis INDEX 145
INDEX OF PROBLEMS
Abstract algebra Differential calculus -binary operations 1971 B-1; 1972 A-2; -differentiation 1967 A-1 1978 A-4 -maxima & minima 1970 A-2; 1973 B-6; -extension fields 1968 B-3; 1980 A-4 1981 B-2 -finite fields 1979 B-3 -partial derivatives 1967 8-6 -group theory 1968 B-2; 1969 B-2; 1972 -rates 1970 B-4: 1972 8-2 B-3; 1975 B-1; 1976 B-2: 1977 B-6 Differential equations 1975 A-5; 1983 8-3 -mappings 1966 A-5 -higher order 1966 B-6 Area 1978 B-1; 1979 B-5; 1981 B-6; 1984 -homogeneous linear 1979 8-4 A-4; 1984 B-6 -nonhomogeneous linear 1979 B-4 Arithmetic geometric mean inequality 1968 -systems 1969 A-5; 1971 B-5; 1973 A-5; A-6: 1975 B-6; 1978 A-3 1982 A-4 Arithmetic progressions 1972 A-1; 1978 A-1; Diophantine equations 1971 A-5: 1978 8-4; 1979 A-3 1979 A-1 Balanced triples 1977 B-3 Dissections 1982 B-1 Bernoulli polynomials 1981 B-1; 1982 A-2 Equations Binary operations 1971 B-1; 1972 A-2; 1978 -functional 1971 8-2; 1979 A-2 A-4; 1984 B-3 -systems 1967 A-6; 1977 A-2; 1980 A-5 Binary representation 1981 B-5: 1973 B-1; Euler's formula for polyhedra 1969 A-3 1984 B-5 Extension fields 1968 8-3; 1980 A-4 Binomial coefficients 1965 B-4; 1967 8-5; Factorials 1984 A-6; 1984 B-1 1971 A-4; 1972 A-1; 1974 A-4; 1974 B-6; Fermat's theorem 1983 A-3 1977 A-5; 1983 A-4 Fields, characteristic 1979 B-3 Calculus ( See Differential calculus, Integral Finite fields 1979 8-3 calculus, Real analysis) Functions 1977 A-3 Cauchy-Schwarz inequality 1977 8-5; 1978 -continuous 1966 A-5; 1982 8-5 A-6; 1979 B-6; 1982 B-6 -convex 1980 B-5 Centroids 1982 A-1: 1984 B-4 -linear 1980 B-2 Chessboard 1981 A-2 Games 1971 A-5 Clocks 1983 A-2 Gamma function 1984 A-5 Coloring problems 1979 A-4 Geometry ( See Plane geometry, Plane ana Combinatorial identities 1965 A-2; 1974 A-4 lytic geometry, Solid geometry) Combinatorics 1965 A-5; 1965 B-5; 1967 Graphs 1965 A-4 A-6; 1973 A-6: 1974 A-1; 1980 B-4 -coloring 1979 A-4 -inequalities 1978 A-6 -Hamiltonian circuits 1968 A-3 -optimization 1974 B-1 -incidence matrices 1965 A-4 Complex numbers 1967 B-1; 1973 8-2; 1979 Greatest integer function 1973 A-3; 1979 B-6 A-5: 1983 A-5; 1983 8-2; 1983 8-4 -algebra 1975 A-4 Group theory 1968 8-2; 1969 B-2; 1972 B-3; -graphing 1975 A-2 1975 B-1; 1976 8-2; 1977 8-6 Complex variables 1972 B-6 Hamiltonian circuits 1968 A-3 Continuous functions 1966 A-5; 1978 B-5; Heron's formula 1982 B-6 1982 B-5 Higher plane curves Convex sets 1967 A-5; 1969 B-4; 1979 B-5 -cycloids, etc. 1971 8-5 Decimal representation 1984 A-6 Hyperbolic functions 1979 B-1; 1980 8-1 Determinants 1969 A-2; 1978 A-2; 1984 A-3 Incidence matrices 1965 A-4 Difference equations 1980 B-3 Inequalities Differences 1971 A-6; 1974 A-6; 1976 B-5: -arithmetic mean-geometric mean 1968 1983 A-4 A-6; 1975 B-6 146 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION
-Cauchy-Schwan 1966 B-3; 1977 B-5: -matrices 1968 B-5; 1969 B-6; 1981 B-4 1979 B-6 -permutation matrices 1967 A-2 -functional considerations 1967 A-1; 1967 -symmetric matrices 1967 A-2 B-6; 1973 B-4; 1973 B-6; 1974 B-5; Linear functions 1980 B-2 1978 A-5; 1980 A-6 Linear programming 1980 B-2 -geometric (triangle, polygon, etc.) 1966 Locker room problem 1967 B-4 A-2; 1966 B-1; 1971 A-3; 1973 A-1: Mappings 1966 A-5 1982 B-6 -binary operations 1978 A-4 -miscellaneous 1968 A-2; 1978 B-6; 1979 Mathematical induction 1978 B-3; 1978 B-6 A-6; 1980 A-4; 1982 A-5 Matrices 1967 A-2; 1968 B-5; 1969 B-6; -series 1966 B-3; 1974 B-5; 1980 B-1 1981 B-4; 1984 A-3 Infinite products 1969 B-3; 1970 B-1; 1977 Maxima 1983 A-2 B-1 Maximum & minimum problems 1975 A-3; Infinite sequences 1965 A-3; 1966 A-1; 1966 1978 B-5; 1979 A-1; 1981 B-2; 1984 A-4; A-3; 1969 A-6: 1969 B-3; 1969 B-5; 1970 1984 B-2 A-4; 1972 A-3; 1978 B-3; 1979 A-3; 1980 Multivariable calculus 1975 A-3 B-3; 1982 B-5; 1983 A-5; 1983 B-4; 1983 Normal lines 1979 B-1 B-5 Number theory 1966 A-4; 1974 A-3; 1976 Infinite series 1967 A-2; 1967 B-5; 1969 A-4; A-3 1970 A-1; 1972 B-1; 1972 B-6; 1973 A-2; -arithmetic 1975 A-1 1975 B-5; 1976 B-1; 1977 A-4; 1978 B-2; -congruences 1968 B-5; 1969 B-1; 1973 1979 A-6: 1980 B-1; 1981 B-5; 1982 A-6; B-1; 1977 A-5; 1983 A-3 1984 A-2 -Diophantine equations 1971 A-5: 1978 -convergence and divergence 1966 B-3; B-4; 1979 A-1 1969 B-5 -divisibility 1966 B-2; 1966 B-4; 1971 Integral calculus 1980 A-5 B-1; 1971 B-6; 1972 A-5; 1973 B-3: -definite integrals 1965 B-1; 1967 B-3; 1981 A-1; 1981 B-3; 1982 B-4; 1983 1968 A-1; 1969 A-4; 1970 B-1; 1970 A-1; 1983 A-3 B-2: 1970 B-4; 1972 A-6; 1973 B-4; -Euler phi-function 1968 B-3; 1972 A-5 1976 B-1; 1979 B-2; 1980 A-3; 1980 -Fermat's theorem 1983 A-3 A-6; 1982 A-3; 1983 B-5 -irrational numbers 1974 B-3; 1977 B-3: -double integrals 1981 A-3; 1981 B-6; 1980 A-4 1982 B-2; 1983 A-6 -least common multiple 1980 A-2 -improper integrals 1968 B-4; 1976 A-5; -multiplicative functions 1967 B-4 1982 A-3 -Pythagorean triples 1965 B-3 -triple integrals 1984 A-5 -relatively prime integers 1974 A-1 Integral equations 1967 A-4; 1980 A-5 -representation of integers 1970 A-3: 1973 Irrational numbers 1974 B-3; 1977 B-3; 1980 B-1; 1981 B-5; 1983 B-2 A-4 -sums of divisors 1969 B-1; 1976 B-6 Isoperimetric problems 1972 A-4 Partial derivatives 1967 B-6 Lattice points 1981 A-6 Permutation matrices 1967 A-2 Law of cosines 1972 B-5; 1983 A-2 Permutations 1982 A-6 L'Hopital's rule 1979 B-2; 1983 A-6 Pick's theorem 1971 A-3 Limits 1965 B-4; 1966 A-3; 1966 A-6; 1967 Pigeonhole principle 1971 A-1: 1978 A-1; B-3; 1969 B-5; 1970 A-4; 1972 A-3; 1974 1980 A-4 B-2: 1976 B-1; 1978 B-3; 1979 B-2; 1981 Plane analytic geometry 1965 A-6; 1972 A-4; A-1; 1981 A-3; 1981 B-1; 1982 B-3; 1983 1977 A-1 B-5 -ellipses 1976 B-4 Linear algebra -higher plane curves 1971 B-5 -determinants 1969 A-2; 1978 A-2 -hyperbolas 1967 B-2 -incidence matrices 1965 A-4 -parabolas 1974 A-5; 1980 A-1 INDEX 147
Plane geometry -measure theory 1972 A-6 -circles 1965 B-6 -minimizing an integral 1978 A-3 -impossible constructions 1968 B--3 -Taylor's remainder theorem 1974 B-5 -lattices 1971 A-3 Recurrence relations 1966 A-3; 1967 A-2; -minimum problem 1976 A-1 1969 A-6; 1971 B-6; 1973 B-5; 1975 B-5; -octagons 1978 B--1 1979 A-3; 1980 B-3; 1980 B-6; 1982 B-5; -pentagons 1984 A-4 1983 B--2; 1983 B-4 -polygons 1966 B--1; 1966 B--5; 1967 B--1; Reflection 1981 A-4 1969 A-3; 1978 B--1; 1981 B-6; 1984 Representation of integers 1983 B-2 B--6 Rolle's theorem 1973 A-4; 1980 A-5; 1981 -quadrilaterals 1970 B--6; 1972 B-5 A-5 -triangles 1965 A-1; 1965 B-3; 1966 A-2; Sequences ( See Infinite sequences) 1971 A-3; 1973 A-1; 1981 A-6; 1982 Series (See Infinite series) B--1; 1982 B-6 Sets 1968 A-3; 1975 B-1; 1980 B-4 Plane geometry and mechanics 1974 A-2 Solid analytic geometry 1970 A-5; 1971 B-4 Points Solid geometry -collinear 1 979 A-4 -covering problems 1975 B-2 Polygons 1984 B--1 -cubes 1983 B--1 Polyhedra 1980 B-2 -lattices 1971 A-1 Polynomials 1967 A-3; 1968 A-5; 1968 A-6; -polyhedra 1980 B-2 1970 B-2; 1971 A-2; 1971 A-4; 1972 B-4; -quadrilaterals 1977 B-2 1973 B--5; 1974 A-6; 1975 A-4; 1976 B--5; -spheres 1968 A-4; 1983 B-1 1978 B-3; 1978 B-5; 1979 A-5; 1980 A-1; -triangles 1975 A-6 1981 A-5; 1983 B-6 Symmetric functions -irreducible 1979 B--3 -polynomials in several variables 1975 B--3 -two variables 1969 A-1; 1970 B--2; 1976 Tangents 1980 A-1 A-2 Theory of equations Probability 1968 B-1; 1970 A-6; 1976 B--3; -roots 1968 A-6; 1976 A-4; 1977 A-1 1982 B-3 Topology Progressions 1972 A-1; 1978 A-1; 1979 A-3 -closed curve in the plane 1977 B-4 Quickest descent problem 1974 A-2 -connectedness 1975 B-4 Rational numbers 1973 B--2; 1978 B-2; 1980 Triangle inequality 1979 A-4 B-6; 1981 B-5 Triangular numbers 1975 A-1 Real analysis 1970 B-3; 1971 A-6; 1972 B-2; Trigonometric functions 1967 A-1 1976 A-6; 1978 A-5; 1983 A-5; 1984 B-4 Trigonometry 1974 B-3 -compact sets 1968 B--6 -law of cosines 1983 A-2 -continuity 1970 B-5; 1972 A-3; 1973 Vectors 1968 A-4; 1975 A-6; 1983 A-2 A-4; 1974 B-4; 1977 A-6; 1979 A-2 Volumes 1983 B-1; 1984A-1 -limits 1974 B-2 Wallis product 1969 B-3; 1983 B-5 AMS / MAA PROBLEM BOOKS
The William Lowell Putnam Mathematical Competition Problems and Solutions 1965 –1984
Gerald L. Alexanderson, Leonard F. Klosinski, Loren C. Larson Editors
Since 1928, the Putnam Competition has been providing a challenge to gifted college mathematics students. This book, the second of the Putnam Competition volumes, contains problems with their solutions for the years 1965–1984. Additional solutions are presented for many of the problems. Included is an essay on recollections of the first Putnam Exam by Herbert Robbins, as well as appendices listing the winning teams and students from 1965 through 1984. This volume offers the problem solver an enticing sample of challenging problems and their solutions.
In 1980, the MAA published the first William Lowell Putnam Mathematical Competition book, covering the contests from 1938 to 1964. In 2002 the third of the Putnam problem books appeared, covering the years 1985 through 2000. All three books belong on the bookshelves of students, teachers, and all interested in problem solving.
PRB/30
For additional information and updates on this book, visit www.ams.org/bookpages/prb-30