This book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies.

Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.

Table of Contents

1. Methods of Proof

2. Algebra

3. Real Analysis

4. Geometry and Trigonometry

5. Number Theory

6. Combinatorics and Probability

Reviews of the first edition:

The reviewer recommends this book to all students curious about the force of mathematics, especially those who are bored at school and ready for a challenge. Teachers would find this book to be a welcome resource, as will contest organizers.

―Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1122 (24), 2007

…This extraordinary book can be read for fun. However, it can also serve as a textbook for preparation for the Putnam … for an advanced problem-solving course, or even as an overview of undergraduate mathematics. … it could certainly serve as a great review for senior-level students.― Donald L. Vestal, MathDL, December, 2007

About the Author

Răzvan Gelca, Texas Tech University, works in Chern-Simons theory, a  field of mathematics that blends low dimensional topology, mathematical physics, geometry, and the theory of group representations. He is also involved in mathematics competitions such as the mathematical Olympiads and the W.L. Putnam Mathematical Competition. He is co-author of 2 published books (with Titu Andreescu), namely “Mathematical Olympiad Challenges” and the first edition of “Putnam and Beyond.” In 2015 Gelca and Andreescu will also publish a monograph on Pell’s Equations.

Titu Andreescu, University of Texas-Dallas, is highly involved with mathematics contests and olympiads. He was the Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and olympiad-style mathematics including the first edition of “Putnam and Beyond” (with Razvan Gelca), “Mathematical Olympiad Treasures” and “Mathematical Olympiad Challenges” (with Razvan Gelca). Additional Springer publications include “Mathematical Bridges”, “Complex Numbers from A to …Z”, “Number Theory” and a new monograph on Pell’s Equations to be published in 2015.