Train Your Brain: Challenging Yet Elementary Mathematics

by Bogumil Kaminski, Pawel Pralat

Overview

This accessible book helps readers to see the bigger picture of advanced mathematics. The book contains carefully selected, challenging problems in an easy-to-follow, step-by-step process. Neither prior preparation nor any mathematical sophistication is required.

The authors guide the reader to “train their brain” to think and express themselves in a rigorous, mathematical way, and to extract facts, analyze the problem, and identify main challenges. A firm foundation in a diverse range of topics is presented. Moreover, the authors show how to draw appropriate, true conclusions. Computer support is used to better intuition into discussed problems.

The book is designed for self-study. It can be used to bridge the gap between introductory calculus/linear algebra courses and more advanced courses offered at universities. It improves the ability to read, write, and think in a rigorous, mature mathematical fashion. The reader will develop a deeper understanding in preparation to succeed in more advanced course work.

Features
•The authors employ a six-step process:

1.SOURCE
2.PROBLEM
3.THEORY
4.SOLUTION
5.REMARK
6.EXERCISES

•An Appendix introduces programming in Julia

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.

Product Details

ISBN-13:	9780367564872
Publisher:	CRC Press
Publication date:	12/31/2020
Series:	Textbooks in Mathematics
Pages:	338
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

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.

Table of Contents

Introduction ix

1 Inequalities 1

1.1 Convexity and Concavity 2

1.2 Arithmetic-Geometric Inequality 6

1.3 Mathematical Induction 10

1.4 Bernoulli's Inequality 13

1.5 Euler's Number 15

1.6 Asymptotics 19

1.7 Cauchy-Schwarz Inequality 25

1.8 Probability 29

1.9 Geometry 32

2 Equalities and Sequences 35

2.1 Combining Equalities 39

2.2 Extremal Values 43

2.3 Solving via Inequalities 47

2.4 Trigonometric Identities 51

2.5 Number of Solutions 58

2.6 Sequence Invariants 63

2.7 Solving Sequences 66

3 Functions, Polynomials, and Functional Equations 73

3.1 Vieta's Formulas 76

3.2 Functional Equations, Exploration 79

3.3 Functional Equations, Necessary Conditions 84

3.4 Polynomials with Integer Coefficients 87

3.5 Unique Representation of Polynomials 92

3.6 Polynomial Factorization 97

3.7 Polynomials and Number Theory 100

4 Combinatorics 103

4.1 Enumeration 105

4.2 Tilings 109

4.3 Counting 114

4.4 Extremal Graph Theory 117

4.5 Probabilistic Methods 122

4.6 Probability 129

4.7 Combinations of Geometrical Objects 135

4.8 Pigeonhole Principle 138

4.9 Generating Functions 144

5 Number Theory 149

5.1 Greatest Common Divisors 152

5.2 Modular Arithmetic 157

5.3 Factorization 161

5.4 Fermat's Little Theorem and Euler's Theorem 163

5.5 Rules of Divisibility 169

5.6 Remainders 171

5.7 Aggregation 173

5.8 Equations 176

6 Geometry 181

6.1 Circles 182

6.2 Congruence 187

6.3 Similarity 189

6.4 Menelaus's Theorem 193

6.5 Parallelograms 197

6.6 Power of a Point 199

6.7 Areas 203

6.8 Thales' Theorem 207

7 Hints 211

7.1 Inequalities 211

7.2 Equalities and Sequences 213

7.3 Functions, Polynomials, and Functional Equations 214

7.4 Combinatorics 215

7.5 Number Theory 218

7.6 Geometry 220

8 Solutions 223

8.1 Inequalities 223

8.2 Equalities and Sequences 237

8.3 Functions, Polynomials, and Functional Equations 256

8.4 Combinatorics 271

8.5 Number Theory 296

8.6 Geometry 312

Further Reading 321

Index 323