Inequalities A Mathematical Olympiad Approach

Inequalities A Mathematical Olympiad Approach

Radmila Bulajich Manfrino
José Antonio Gémez Ortega
Rogelio Valdez Delgado

lnequalities
A Mathematical Olympiad Approach

Contents

Introduction vii
1 Numerical Inequalities 1
1.1 Order in the real numbers . . . . . . . . . . . . 1
1.2 The quadratic function ax2 + 2bx + c . . . 4
1.3 A fundamental inequality,
arithmetic mean-geometric mean . . . . . . 7
1.4 A wonderful inequality:
The rearrangement inequality . . . . . . . . 13
1.5 Convex functions . . . . . . . . . . . . . . . 20
1.6 A helpful inequality . . . . . . . . . . . . 33
1.7 The substitution strategy . . . .. . . . . . 39
1.8 Muirhead’s theorem . . . . . . . . . . . . 43

2 Geometric Inequalities 51
2.1 Two basic inequalities . . . . . . . . . . . . . . . 51
2.2 Inequalities between the sides of a triangle . 54
2.3 The use of inequalities in the geometry of the triangle .59
2.4 Euler’s inequality and some applications . . 66
2.5 Symmetric functions of a, b and c . . . . . . . . 70
2.6 Inequalities with areas and perimeters . . . . 75
2.7 Erdos-Mordell Theorem . . . . . . . . . . . . . . . 80
2.8 Optimization problems . . . . . . . . . . . . . . . . 88

3 Recent Inequality Problems 101

4 Solutions to Exercises and Problems 117
4.1 Solutions to the exercises in Chapter 1 . .. . . 117
4.2 Solutions to the exercises in Chapter 2. . . . . 140
4.3 Solutions to the problems in Chapter 3. . . . . 162

Notation 205