Inequalities: A Mathematical Olympiad Approach

✍ Scribed by Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 217
• Edition: 1
• Category: Library

✦ Synopsis

This book presents classical inequalities and specific inequalities which are particularly useful for attacking and solving optimization problems. Most of the examples, exercises and problems that appear in the book originate from Mathematical Olympiad contests around the world. The material is divided into four chapters. In Chapter 1 algebraic inequalities are presented, starting with the basic ones and ending with more sophisticated techniques; Chapter 2 deals with geometric inequalities and Chapter 3 comprises a comprehensive list of recent problems that appeared in those contests during the last 14 years. Finally, hints and solutions to all exercises and problems are given in Chapter 4.

✦ Table of Contents

Cover......Page 1
Inequalities: A Mathematical Olympiad Approach......Page 2
Copyright......Page 3
Introduction......Page 4
Contents......Page 6
1.1 Order in the real numbers......Page 8
1.2 The quadratic function ax^2+bx+c......Page 11
1.3 A fundamental inequality, arithmetic mean-geometric mean......Page 14
1.4 A wonderful inequality: The rearrangement inequality......Page 20
1.5 Convex functions......Page 27
1.6 A helpful inequality......Page 40
1.7 The substitution strategy......Page 46
1.8 Muirhead’s theorem......Page 50
2.1 Two basic inequalities......Page 58
2.2 Inequalities between the sides of a triangle......Page 61
2.3 The use of inequalities in the geometry of the triangle......Page 66
2.4 Euler’s inequality and some applications......Page 73
2.5 Symmetric functions of a, b and c......Page 77
2.6 Inequalities with areas and perimeters......Page 82
2.7 Erdos-Mordell Theorem......Page 87
2.8 Optimization problems......Page 95
3. Recent Inequality Problems......Page 108
4.1 Solutions to the exercises in Chapter 1......Page 124
4.2 Solutions to the exercises in Chapter 2......Page 147
4.3 Solutions to the problems in Chapter 3......Page 169
Notation......Page 212
Bibliography......Page 214
Index......Page 216