Putnam and Beyond [Gelca]

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Putnam and Beyond
Authors: Răzvan Gelca & Titu Andreescu
Publication date: August 2007
Publisher: Springer

Putnam and Beyond is an advanced problem-solving textbook designed to bridge the gap between high-school competition mathematics, undergraduate mathematics, and the beginnings of mathematical research. The William Lowell Putnam Mathematical Competition provides the inspiration, but the book is considerably broader than a collection of Putnam problems. Gelca and Andreescu use challenging problems as a way of teaching mathematical ideas and techniques, emphasizing how to discover proofs and develop problem-solving strategies rather than simply memorize theorems. It begins with fundamental proof methods—including contradiction, induction, the pigeonhole principle, ordered sets, and invariants—and then develops increasingly sophisticated mathematics. 

The main mathematical areas are algebra, real analysis, geometry and trigonometry, number theory, combinatorics, and probability. Within these areas the reader encounters subjects such as polynomials, linear algebra, inequalities, sequences and series, calculus, differential equations, coordinate geometry, elementary number theory, and combinatorial arguments. A particularly valuable feature is the enormous solutions section: in the first edition, the main chapters occupy roughly the first 320 pages, while solutions begin on page 321 and extend through most of the remainder of the 798-page volume. This reflects the book's philosophy: the problems are the primary vehicle through which mathematics is learned. 

The book is best suited to strong undergraduate students, mathematics competition participants, teachers, and anyone wanting to become substantially better at solving unfamiliar mathematical problems. It is not an introductory textbook and can be demanding for readers without a solid foundation in calculus and undergraduate mathematics. For a mathematically mature reader, however, its strength lies precisely in that difficulty: problems encourage experimentation, recognition of hidden structures, and the combination of ideas from different areas. Springer describes it as suitable for self-study by undergraduate and graduate students and for readers wishing to prepare for more advanced mathematics. A substantially expanded second edition appeared in 2017, adding material on quadratic polynomials, functions in real analysis, plane curves, quadratic fields, graph theory and other subjects, with more than 1,300 problems. 

Key takeaways

• Problem solving is the central subject. Algebra, analysis and number theory provide the setting, but the deeper goal is learning how to think when the solution is not obvious.
  
• It bridges Olympiad and university mathematics. The book helps readers move from clever competition techniques toward rigorous undergraduate and eventually research-level mathematical thinking. 
  
• The extensive worked solutions are essential. They allow the reader to compare approaches and turn individual tricks into reusable mathematical techniques.
  
• Best used actively: attempt each problem seriously before reading its solution. Reading the solutions without struggling with the problems removes much of the book's educational value.
  
  

Overall: ★★★★★ — An excellent choice for a mathematically ambitious undergraduate or teacher who wants a large, challenging collection that develops genuine mathematical problem-solving ability rather than routine computational technique.

Putnam and Beyond — Springer
Putnam and Beyond — Goodreads