Winning Solutions

Problem-solving competitions for mathematically talented sec ondary school students have burgeoned in recent years. The number of countries taking part in the International Mathematical Olympiad (IMO) has increased dramatically. In the United States, potential IMO team members are identified through the USA Mathematical Olympiad (USAMO), and most other participating countries use a similar selection procedure. Thus the number of such competitions has grown, and this growth has been accompanied by increased public interest in the accomplishments of mathematically talented young people. There is a significant gap between what most high school math ematics programs teach and what is expected of an IMO participant. This book is part of an effort to bridge that gap. It is written for students who have shown talent in mathematics but lack the back ground and experience necessary to solve olympiad-level problems. We try to provide some of that background and experience by point out useful theorems and techniques and by providing a suitable ing collection of examples and exercises. This book covers only a fraction of the topics normally rep resented in competitions such as the USAMO and IMO. Another volume would be necessary to cover geometry, and there are other v VI Preface special topics that need to be studied as part of preparation for olympiad-level competitions. At the end of the book we provide a list of resources for further study.

Texte du rabat

Problem-solving competitions for mathematically talented sec­ ondary school students have burgeoned in recent years. The number of countries taking part in the International Mathematical Olympiad (IMO) has increased dramatically. In the United States, potential IMO team members are identified through the USA Mathematical Olympiad (USAMO), and most other participating countries use a similar selection procedure. Thus the number of such competitions has grown, and this growth has been accompanied by increased public interest in the accomplishments of mathematically talented young people. There is a significant gap between what most high school math­ ematics programs teach and what is expected of an IMO participant. This book is part of an effort to bridge that gap. It is written for students who have shown talent in mathematics but lack the back­ ground and experience necessary to solve olympiad-level problems. We try to provide some of that background and experience by point­ out useful theorems and techniques and by providing a suitable ing collection of examples and exercises. This book covers only a fraction of the topics normally rep­ resented in competitions such as the USAMO and IMO. Another volume would be necessary to cover geometry, and there are other v VI Preface special topics that need to be studied as part of preparation for olympiad-level competitions. At the end of the book we provide a list of resources for further study.

Résumé
This book provides the mathematical tools and problem-solving experience needed to successfully compete in high-level problem solving competitions.

Contenu
1 Numbers.- 1.1 The Natural Numbers.- 1.2 Mathematical Induction.- 1.3 Congruence.- 1.4 Rational and Irrational Numbers.- 1.5 Complex Numbers.- 1.6 Progressions and Sums.- 1.7 Diophantine Equations.- 1.8 Quadratic Reciprocity.- 2 Algebra.- 2.1 Basic Theorems and Techniques.- 2.2 Polynomial Equations.- 2.3 Algebraic Equations and Inequalities.- 2.4 The Classical Inequalities.- 3 Combinatorics.- 3.1 What is Combinatorics?.- 3.2 Basics of Counting.- 3.3 Recurrence Relations.- 3.4 Generating Functions.- 3.5 The Inclusion-Exclusion Principle.- 3.6 The Pigeonhole Principle.- 3.7 Combinatorial Averaging.- 3.8 Some Extremal Problems.- Hints and Answers for Selected Exercises.- General References.- List of Symbols.