Preface

Mathematical Olympiads boast a tradition spanning over a century. The inaugural mathematical competitions took place in Eastern Europe (Hungary and Romania) by the late 19th century. In 1959, the inaugural International Mathematical Olympiad occurred in Romania, with seven countries and 52 students in attendance. Fast forward to 2010, and the IMO was hosted in Kazakhstan, featuring 97 participating countries and 517 students.

It is evident that the interest among young students in mathematics and mathematical competitions has reached unprecedented levels. A glance at various mathematical forums online reveals tens of thousands of registered users engaging in millions of posts.

Contemplating the creation of this book, we pondered its intended audience. Should it cater to the novice student taking initial strides into the world of mathematical problem-solving, the more seasoned reader well-versed in competitions, or perhaps the educator seeking a valuable resource to aid students in preparing for mathematical contests? Opting for a comprehensive approach, we have taken on the challenge of addressing all potential readers.

Consequently, we have curated Olympiad problems spanning various levels of difficulty. Some are approachable yet far from mere exercises, while others pose significant challenges, even for seasoned Olympiad experts.

The majority of the problems originate from diverse mathematical competitions such as the International Mathematical Olympiad, The Tournament of the Towns, national Olympiads, and regional Olympiads. While some problems are original creations of the authors, others are drawn from mathematical folklore.

Organized into three chapters—Algebra, Geometry and Trigonometry, and Number Theory and Combinatorics—the problems mirror the classification used at the International Mathematical Olympiad.

Within each chapter, problems are grouped by topic into self-contained sections. These sections commence with fundamental facts, followed by a curated selection of problems and in-depth discussions of their solutions. Concluding each section are proposed problems, with their comprehensive solutions presented in the second part of the book.

We encourage the beginning reader to carefully examine the problems solved at the beginning of each section and try to solve the proposed problems before examining the solutions provided at the end of the book. As for the advanced reader, our advice is to try finding alternative solutions and generalizations of the proposed problems.

In the second edition of the book, we added two new sections in Chaps. 1 and 3, and more than 60 new problems with complete solutions.

Titu Andreescu
Bogdan Enescu
University of Texas at Dallas
“B.P. Hasdeu” National College