The book contains selected problems aimed for high school students that are interested in competing in math competitions or simply for people of all ages and backgrounds who want to expand their knowledge and to challenge themselves with interesting questions. The problems are mostly selected from an extensive collection of problems from Polish Mathematical Olympics and many appear here in English for the first time. Each chapter consists of many sections devoted to a collection of related topics. Each of these sections starts with a problem followed by the necessary background (definitions and theorems used), careful and detailed solution, and discussion of possible generalizations.
Table of Contents
List of Figures
Chapter 1: Inequalities
Chapter 2: Equalities and Sequences
Chapter 3: Functions, Polynomials, and Functional Equations
Chapter 4: Combinatorics
Chapter 5: Number Theory
Chapter 6: Geometry
Chapter 7: Hints
Chapter 8: Solutions
The book contains carefully selected, challenging problems to prepare readers for rigorous mathematics. Neither prior preparation nor any mathematical sophistication is required. The authors guide the readers to think and express themselves in a rigorous, mathematical way, to extract facts, analyze the problem, and identify main challenges. Moreover, they show how to draw appropriate, true conclusions and to see a big picture.
Readers are provided with a firm foundation in a diverse range of topics. Computer support is used to build a better intuition into discussed problems. The book can be used to bridge the gap between introductory calculus/linear algebra courses and more advanced courses that are offered at universities. It improves the ability to read, write, and think in a rigorous, mature mathematical fashion. It provides a solid foundation of various topics that would be useful for more advanced courses.
The content of this book is also suitable for high school students that are interested in competing in math competitions or simply for people of all ages and backgrounds who want to expand their knowledge and to challenge themselves with interesting questions.
The authors employ a six-step process:
All exercises are followed by hints and solutions given at the end of the book.
Additionally, the book is enhanced by a "Julia language companion" available on-line that contains an introduction to the Julia language programming and detailed explanations of the codes used in the book.