Back by popular demand, the MAA is pleased to reissue this outstanding collection of problems and solutions from the Putnam Competitions covering the years 1938-1964.

Problemists the world over, including all past and future Putnam Competitors, will revel in mastering the difficulties posed by this collection of problems from the first 25 William Lowell Putnam Competitions. Solutions to all 347 problems are given. In some cases multiple solutions are included, some which contestants could reasonably be expected find under examination conditions, and others which are more elegant or utilize more sophisticated techniques. Valuable references and historical comments on many of the problems are presented. The book concludes with four articles on the Putnam competition written by G. Birkhoff, L. E. Bush, L. J. Mordell, and L. M. Kelly which are reprinted from the American Mathematical Monthly.

There is great appeal for all; teachers, students, and all those who love good problems and see them as an entree to beautiful and powerful ideas.

### Table of Contents

Dedication

Preface

List of Problems

Solutions to the Problems in the Various Competitions

Appendices: From *The American Mathematical Monthly*

General Index

### Preface

The William Lowell Putnam Mathematical Competition, since its inception in 1938, has had a substantial impact on the field of mathematics in the United States and Canada. It rivals in this respect the classic Tripos in Cambridge and the influential Eötvös competition in Hungary. While there have been many different reasons for the remarkable expansion of mathematics during the past forty years, we believe that the challenge provided by the Putnam Competition has led many gifted college students into serious involvement with mathematics, and our profession is the richer for it.

There are some who feel that isolated problems, especially competition problems, present and emphasize an inappropriate view of mathematics. Yet progress in mathematics has often been made by separating problems from their contexts, and curiosity about isolated problems has frequently led to significant mathematical discoveries. Hermann Weyl wrote

Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us … nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor. (From the Preface to *The Classical Groups*, Princeton University Press, 1939, 2nd ed., 1946.)

Although this book is primarily about the problems, we have included four articles about the competition from *The American Mathematical Monthly*: Garrett Birkhoff’s account of the founding of the competition, L. E. Bush’s summary of the first twenty-four contests (which we have revised with Professor Bush’s permission to include the twenty-fifth), and an important exchange of views between L. J. Mordell and L. M. Kelly. We hope that these articles will give additional perspective on the role of the competition in American mathematics.

The first twenty-five competitions involved a total of three hundred forty-seven problems. They are collected in Part I essentially as they were presented to the contestants. Each problem is reprinted in Part II together with its solution. Sometimes we have given two or more solutions to a problem. In such cases we present first a solution which in our judgment could reasonably have been found by a contestant under examination conditions. Alternative solutions may involve more sophisticated approaches. We have also included references and historical backgrounds for some of the problems.

We found that it is not always easy to decide just what constitutes a solution of a problem. This is particularly true of the problems in geometry. For example, A. M. 3 of the Seventh Competition involves the concept of a polygonal line “crossing” a segment. Without a formal definition of this term a rigorous proof is impossible, yet a complete discussion could hardly be expected of a contestant pressed for time. Obviously, some compromise between rigor and intuition must be made, but it is by no means clear that a compromise appropriate for an examination booklet is equally appropriate for a book whose authors are under no time constraints.

The early competitions always included a mechanics problem, and in this area standards are even less clear. We can attest that an argument in mechanics can satisfy one mathematician and leave another unconvinced. It is not surprising that some of the problems involve subtleties not envisioned by their proposers. Problem P. M. 12 of the Third Competition, concerning the director sphere of a central quadric, is a case in point. Lacking any instruction to the contrary, the problem is presumably set in Euclidean space. In this setting a complete description of the locus is quite involved; in fact, although the problem itself is a classic, we believe that ours is the first complete solution to be published.

Inevitable, there have been some errors in the exams. These have varied from typographical errors of various degrees of seriousness to outright mistakes. Of the latter, the most interesting is problem A. M. 1 of the Seventeenth Competition, in which the desired conclusion is true locally, but not globally.

By and large we feel that the examinations have been well constructed. The continually increasing popularity of the competition is prima facie evidence that the examinations are generally perceived as fair and challenging. Each of us has served at one time or another on the examination committee, so we are well aware how hard it is to produce fresh and interesting problems year after year. Looking back on the considerable effort we have spent in compiling this book of solutions (but with no feeling that we as examiners did better), we would offer one piece of advice to future examination committees: Spend more time on the question, What constitutes an acceptable solution?

Originally, it was planned that this volume should appear shortly after the Twenty-Fifth Competition was held. We thank the officers of the Mathematical Association of America for their patient understanding of the difficulties we have encountered and we apologize to them and to the interested mathematical community for the long delay in completing this project.

It is hardly necessary to observe that we and everyone else interested in the Putnam Competition owe much to the vision of William Lowell Putnam and to the generosity of his widow, Elizabeth Lowell Putnam. We would like to record here our appreciation of the encouragement given us by George Putnam, grandson of the donor, who continues to give close personal attention to all aspects of the competition.

We have received advice and assistance from many friends in the mathematical world, but we must take special note of the invaluable contributions of Robert Brooks, Harley Flanders, Alan Grenadir, David Harbater, Fritz Herzog, David Jerison, James McKay, and L. E. Bush. We are particularly indebted to Basil Gordon, Murray Klamkin, and E. G. Strauss. Their criticisms, often trenchant, but always helpful, served to improve the solutions immeasurably. Our thanks are also due to E. F. Bechenbach, chairman of the Editorial Committee, and Raoul Hailpern, Editorial Director of the Mathematical Association of America.

Glendora Milligan typed the manuscript and James R. Holmes prepared the figures. We are especially grateful to Mrs. Milligan for her patience in handling the many, many revisions.

Summer 1978

A. M. Gleason

R. E. Greenwood

L. M. Kelly