Winning Solutions

This book provides the mathematical tools and problem-solving experience needed to successfully compete in high-level problem solving competitions. Each section presents important background information and then provides a variety of worked examples and exercises to help bridge the gap between what the reader may already know and what is required for high-level competitions. Answers or sketches of the solutions are given for all exercises.

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.