This book is a must-have book for coaches and players who take part in math competitions. The first edition of this book was published in 1993, and the second edition was published in 2002. It has been welcomed by teachers who are engaged in mathematics competition counseling and students who are interested in making a difference in mathematics Olympics, and has become an authoritative reference book for domestic mathematics competition research. Professor Shan Dun, the author of this book, is good at the study of number theory and combinatorial mathematics, and is also a famous expert in "Olympic Mathematics" in China. He has been the coach of IMO (International Mathematical Olympiad) China Training Team for many times, and has made great contributions to China becoming a powerful IMO country. This book is revised on the basis of the first two editions. There are 50 lectures in the book, including number theory, algebra, geometry, combination, problem solving and problems in proposition discussion. The focus of this book is not to add more knowledge, but to encourage students to use existing knowledge to solve problems. In recent years, many new achievements have been made in the research of mathematical competitions at home and abroad. In view of this situation, the author has greatly revised the book with the attitude of Excellence. I believe this book will also be welcomed by readers!