This book presents topics in algebra and analysis with useful techniques for solving mathematical Olympiad problems. Most of the examples, exercises and problems in the book originate from the mathematical Olympiad contests around the world. The material is divided in ten chapters. The first four chapters are related to basic material that can be used in the first stages of any mathematical contests at high school level. The next four chapters comprise more advanced topics which are required for the international mathematical competitions and can be also useful for students in their first year of university. The final chapters provide a comprehensive list of problems which were presented at national and international contests in recent years, and solutions to all exercises and problems. in the book.