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.
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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. Suceavă and Jack B. Gaumer, The Mathematical Intelligencer, Vol. 33 (2), 2011)
From the Back Cover
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 expand their mathematical horizons.
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15 of 15 people found the following review helpful
5.0 out of 5 stars
An excellent high-level compilation of problems and problem-solving strategies
August 8, 2009
By nilkn
Format:Paperback
There are many books on problem solving. The majority are aimed at junior high and high school students preparing for either the International Mathematical Olympiad or national IMO selection tests (the American Mathematics Contests in the United States, such as the USAMO). A select few, such as Problem Solving through Problems by Loren Larson, are aimed at university students preparing for university competitions. This book, as the title suggests, belongs to this latter category, and it has a particular emphasis on the sorts of problems that occur on the Putnam exams.
This book is generally written at a higher level than most other problem solving books. Many problem solving books place a great emphasis on geometry. Just as the Putnam exam generally replaces synthetic geometry with analysis and abstract and linear algebra (although there are exceptions), so this book replaces the traditional focus on geometry with a focus on analysis and algebra. That said, there is an entire chapter on geometry, but it does not discuss synthetic geometry, instead focusing on vectors, the geometry of the complex plane, analytic geometry, and some special topics that are especially relevant to college mathematics (integrals in geometry, some higher-level results such as the fact that all conics are rational curves, and a brief but still substantive survey of trigonometric substitutions).
Putnam and Beyond discusses many areas of college mathematics that are likely to appear on the Putnam exam but would never appear on the IMO, such as abstract algebra, linear algebra, and real analysis (with a very tiny bit of complex analysis).
That said, this book still does overlap a bit with many other problem solving books. It opens with a chapter on general problem solving strategies, but I feel that these sections are written with students who have encountered the basic methods before. For example, most introductions to induction demonstrate it by summing some series, but the authors here show that if finitely many lines divide the plane into regions, the regions can be colored with two colors in such a way that no two neighboring regions receive the same color. Another example they offer is a particularly difficult inequality from a past Putnam exam. So in a way the opening chapter is appropriate more as a *second* introduction to problem solving techniques than as a first introduction. This leads to my next point.
The book's exposition is generally written at a high level, and I'd say that to fully appreciate it would impose somewhat high prerequisites, including a good amount of mathematical maturity and a good knowledge of basic college mathematics up through first courses in algebra and analysis. For example, a problem in the very first section of the very first chapter on argument by contradiction requires one to be familiar with the density of rationals in the reals.
To anyone interested in beautiful proofs or in competition math, I would heartily recommend this book along with Problem Solving through Problems by Larson. I think Putnam and Beyond is written at a slightly higher level than Larson's book and many of the problems here are more difficult than those in Larson, but together both books provide a very thorough and strong review of undergraduate mathematics through problem solving.
Finally, full solutions to every single problem (and by "full" I mean complete proofs written out in detail, often with accompanying figures) are in the back of the book (in fact, a little more than half of the pages are devoted to these solutions).
This book is generally written at a higher level than most other problem solving books. Many problem solving books place a great emphasis on geometry. Just as the Putnam exam generally replaces synthetic geometry with analysis and abstract and linear algebra (although there are exceptions), so this book replaces the traditional focus on geometry with a focus on analysis and algebra. That said, there is an entire chapter on geometry, but it does not discuss synthetic geometry, instead focusing on vectors, the geometry of the complex plane, analytic geometry, and some special topics that are especially relevant to college mathematics (integrals in geometry, some higher-level results such as the fact that all conics are rational curves, and a brief but still substantive survey of trigonometric substitutions).
Putnam and Beyond discusses many areas of college mathematics that are likely to appear on the Putnam exam but would never appear on the IMO, such as abstract algebra, linear algebra, and real analysis (with a very tiny bit of complex analysis).
That said, this book still does overlap a bit with many other problem solving books. It opens with a chapter on general problem solving strategies, but I feel that these sections are written with students who have encountered the basic methods before. For example, most introductions to induction demonstrate it by summing some series, but the authors here show that if finitely many lines divide the plane into regions, the regions can be colored with two colors in such a way that no two neighboring regions receive the same color. Another example they offer is a particularly difficult inequality from a past Putnam exam. So in a way the opening chapter is appropriate more as a *second* introduction to problem solving techniques than as a first introduction. This leads to my next point.
The book's exposition is generally written at a high level, and I'd say that to fully appreciate it would impose somewhat high prerequisites, including a good amount of mathematical maturity and a good knowledge of basic college mathematics up through first courses in algebra and analysis. For example, a problem in the very first section of the very first chapter on argument by contradiction requires one to be familiar with the density of rationals in the reals.
To anyone interested in beautiful proofs or in competition math, I would heartily recommend this book along with Problem Solving through Problems by Larson. I think Putnam and Beyond is written at a slightly higher level than Larson's book and many of the problems here are more difficult than those in Larson, but together both books provide a very thorough and strong review of undergraduate mathematics through problem solving.
Finally, full solutions to every single problem (and by "full" I mean complete proofs written out in detail, often with accompanying figures) are in the back of the book (in fact, a little more than half of the pages are devoted to these solutions).
4 of 5 people found the following review helpful
Format:Paperback
This book consists of a very useful collection of Putnam-like math problems. Putnam and Beyond is organized for self-study by undergraduate and graduate students who wish to try a lot of competitive math problems.It is also useful for teachers who are preparing their bright students for IMO type (or higher) math competitions. However the book assumes a level of mathematical maturity and prior mathematical knowledge that not many college students possess. Another very useful book for math competitions is The IMO Compendium.
1 of 1 people found the following review helpful
Format:Paperback
An excellent book! I found some nice classical and some non-standard methods for solving problems.
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