Includes bibliographical references and index.

Chapters
Introduction
Chapter 1. Induction
Chapter 2. Combinatorics
Chapter 3. The whole numbers
Chapter 4. Geometric transformations
Chapter 5. Inequalities
Chapter 6. Graphs
Chapter 7. The pigeonhole principle
Chapter 8. Complex numbers and polynomials
Chapter 9. Rational approximations
Chapter 10. Mathematics and the computer
Instead of a conclusion: teaching how to look for solutions of problems, or fantasy in the manner of Pólya
Solutions of the supplementary problems

2009; 337 pp
MSC: Primary 00;
“It is difficult to define the genre of the book. It is not a problem book, nor a textbook, nor a ‘book for reading about mathematics’. It is most of all reminiscent of a good lecture course, from which a thoughtful student comes away with more than was actually spoken about in the lectures.”

—from the Preface by A. S. Merkurjev

If you are acquainted with mathematics at least to the extent of a standard high school curriculum and like it enough to want to learn more, and if, in addition, you are prepared to do some serious work, then you should start studying this book.

An understanding of the material of the book requires neither a developed ability to reason abstractly nor skill in using the refined techniques of mathematical analysis. In each chapter elementary problems are considered, accompanied by theoretical material directly related to them. There are over 300 problems in the book, most of which are intended to be solved by the reader. In those places in the book where it is natural to introduce concepts outside the high school syllabus, the corresponding definitions are given with examples. And in order to bring out the meaning of such concepts clearly, appropriate (but not too many) theorems are proved concerning them.

Unfortunately, what is sometimes studied at school under the name “mathematics” resembles real mathematics not any closer than a plucked flower gathering dust in a herbarium or pressed between the pages of a book resembles that same flower in the meadow besprinkled with dewdrops sparkling in the light of the rising sun.