Train Your Brain

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. Features The authors employ a six-step process: SOURCE PROBLEM THEORY SOLUTION REMARKS EXERCISES 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.

List of Figures

Introduction

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

Further Reading

Index