This book presents the mathematical methods that students need when preparing for competitions, whether for the International Mathematical Olympiad (IMO) for high school students or the Putnam competition for undergraduate students. The book features six parts, each subdivided into several chapters. The six major sections address counting methods, number theory, inequalities and the theory of equations, metrical geometry, analysis, and number representations and logic. Each chapter provides detailed solutions to a series of problems as well as approximately 30 more problems; the text thus provides about 1,000 problems to be solved. In addition to its value to students preparing for competitions, this volume is a useful resource for those seeking a thorough and practical review of mathematical methods. For high school, undergraduate, and graduate students in mathematics