Putnam and Beyond 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 in one and several variables, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. 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. Key features of Putnam and Beyond * 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 every section according to the specific topic. * The exposition is driven by more than 1100 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. * Complete solutions to all problems are given at the end of the book. The source, author, and historical background are cited whenever possible. 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 self-study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to to expand their mathematical horizons.
From the reviews: "This work contains carefully selected problems in Algebra, Real Analysis, Geometry and Trigonometry, Number Theory and Combinatorics and Probability. ... The book is mainly intended to offer the principal skills and techniques for solving problems in elementary Mathematics. ... 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) "I enjoyed this book ... . Not just because of the collection of problems, but also because of their sheer scope and depth. This is a great collection which is extremely well-organized! ... 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) "A 935-problem and almost 800-page super-problem book with solutions, whose reading would certainly challenge, attract, and keep really busy any undergraduate student interested in acquiring various problem-solving techniques. ... the array of remarkable problem books has gained a new addition that could be really useful to undergraduate students. ... a book about excellence in mathematics, coming from a long cultural tradition whose history and experience can only help us deepen our understanding of how mathematics could be taught in a more attractive and inquisitive way." (Bogdan D. Suceava and Jack B. Gaumer, The Mathematical Intelligencer, Vol. 33 (2), 2011)