Inequalities

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.

Introduction	p. vii
Numerical Inequalities	p. 1
Order in the real numbers	p. 1
The quadratic function ax2 + 2bx + c	p. 4
A fundamental inequality, arithmetic mean-geometric mean	p. 7
A wonderful inequality: The rearrangement inequality	p. 13
Convex functions	p. 20
A helpful inequality	p. 33
The substitution strategy	p. 39
Muirhead's theorem	p. 43
Geometric Inequalities	p. 51
Two basic inequalities	p. 51
Inequalities between the sides of a triangle	p. 54
The use of inequalities in the geometry of the triangle	p. 59
Euler's inequality and some applications	p. 66
Symmetric functions of a, b and c	p. 70
Inequalities with areas and perimeters	p. 75
Erdos-Mordell Theorem	p. 80
Optimization problems	p. 88
Recent Inequality Problems	p. 101
Solutions to Exercises and Problems	p. 117
Solutions to the exercises in Chapter 1	p. 117
Solutions to the exercises in Chapter 2	p. 140
Solutions to the problems in Chapter 3	p. 162
Notation	p. 205
Bibliography	p. 207
Index	p. 209
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