Official site with more information and practice: www.mathallstar.org.Competition math is not about complicated theorems and formulas. It is more about how to analyze in order to solve challenging problems. Thus, merely remembering hundreds of theorems and formulas is far from sufficient to win math competitions. Students must be able to think effectively.This book aims to help students understand some frequently used methods of proof and improve their ability to think effectively. Contents in this book are organized based on methodologies rather than specific subjects. Consequently, examples and practice problems presented in each chapter may cover many different subjects. For example, the chapter Symmetry contains problems relating to polynomial factorization, equation, counting, and so on. Despite of being belong to different subjects, all of these problems can be solved by exploiting their intrinsic symmetric properties. Readers should focus on learning how to identify and utilize such properties. This analysis skill is a critical one to develop in addition to specific subject math knowledge.All the methodologies discussed in this book are intuitive and easy to understand. Some of them may be taught in classroom such as mathematical induction and proof by contradiction. Others may be not. Regardlessly, all of them are powerful to solve many competition problems. In addition, mastering the art of thinking not only is helpful for improving students' contest performance during school years, but also has positive impacts on their future. For example, some problems in this book originate from various job interviews. Being able to think effectively in order to solve such interview and other practical problems is certainly beneficial to their future.More information can be found at www.mathallstar.org.