The William Lowell Putnam Mathematical Competition is the premier mathematical competition at the undergraduate level in North America. This volume, the third to cover the competition, contains the 192 problems from the years 1985-2000, as they appeared in the competition, with solutions and extensive commentary.

It is unlike the first two Putnam volumes and unlike virtually every other problem-based book, in that it places the problems in the context of important mathematical themes. The authors highlight connections to other problems, to the curriculum, and to more advanced topics. The best problems contain kernels of sophisticated ideas related to important current research, and yet the problems are accessible to undergraduates.

The heart of the book is in the solutions, which have been compiled through extensive research. The authors present the best solutions from *The American Mathematical Monthly*, *Mathematics Magazine*, past competitors, and many problem enthusiasts. Often the authors have simplified these solutions, or have developed new solutions of their own. Multiple solutions are common. In editing the solutions, the authors have kept a student audience in mind, explaining techniques that have relevance to more than the problem at hand, suggesting references for further reading, and mentioning related problems, some of which are unsolved.

In addition to the problems and solutions, the book contains:

- a hint to each problem, separate from the full solution;
- background information about the competition;
- a list of winning individuals and teams, with current information about the career paths of winners;
- a topic index;
- Putnam Trivia for the Nineties, by Joseph A. Gallian; and
- Some Thoughts on Writing for the Putnam, by Problems Committee member Bruce Reznick.

The authors of this volume are active research mathematicians who are renowned also for their expository skills. They were themselves winners of the Putnam Competition in most of the years covered by the volume: together they achieved the rank of Putnam Fellow eleven times.

This volume will appeal to students, teachers, and professors, or anyone interested in problem solving as an entrée to beautiful and powerful ideas.

### Table of Contents

Introduction

Problems

Hints

Solutions

Results

Putnam Trivia for the Nineties

Some Thoughts on Writing for the Putnam

Bibliography

Index

About the Authors

### About the Authors

**Kiran S. Kedlaya** is from Silver Spring, MD. He received an AB in mathematics and physics from Harvard (where he was a Putnam Fellow three times), a MA in mathematics from Princeton, and a PhD in mathematics from MIT, and currently holds a National Science Foundation postdoctoral fellowship at the University of California, Berkeley. Other affiliations have included the Clay Mathematics Institute and the Mathematical Sciences Research Institute. His research interests are in number theory and algebraic geometry.

He has been extensively involved with mathematics competitions and problem solving. He has taught at the Math Olympiad Summer Program, served on the USA Mathematical Olympiad committee and on the executive committee of the 2001 International Mathematical Olympiad (held in Washington, DC), served as a collaborating editor for *The American Mathematical Monthly* problems section, maintained problem information on the World Wide Web for the American Mathematics Competitions, and edited Olympiad compilations for the Mathematical Association of America.

**Bjorn Poonen** is from Boston. He received the AB degree *summa cum laude* in mathematics and physics from Harvard University, and the PhD degree in mathematics from the University of California at Berkeley, where he now holds the title of associate professor of mathematics. Other affiliations have included the Mathematical Sciences Research Institute, Princeton University, the Isaac Newton Institute, and the Université de Paris-Sud.

He is a Packard Fellow, a four-time Putnam Competition winner, and the author of over 50 articles. His main research interests lie in number theory and algebraic geometry, but he has published also in combinatorics and probability. Journals for which he serves on the editorial board include the *Journal of the American Mathematical Society* and the *Journal de Théorie des Nombres de Bordeaux*. He has helped to create the problems for the USA Mathematical Olympiad every year since 1989.

**Ravi Vakil** is from Toronto, Canada. He received his undergraduate degree at the University of Toronto, where he was a four-time Putnam Competition winner. After completing a PhD at Harvard, he taught at Princeton and MIT before moving to Stanford, where he is a tenure-track assistant professor. He is currently an American Mathematical Society Centennial Fellow and an Alfred P. Sloan Research Fellow. His field of research is algebraic geometry, with connections to nearby fields, including combinatorics, topology, number theory, and physics. He has long been interested in teaching mathematics through problem solving; he coached the Canadian team to the International Mathematical Olympiad from 1989 to 1996, and one of his other books is titled *A Mathematical Mosaic: Patterns and Problem Solving*.