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Inequalities

A Mathematical Olympiad Approach

Radmila Bulajich Manfrino José Antonio Gómez Ortega Rogelio Valdez Delgado

Birkhọuser Basel · Boston · Berlin

2000 Mathematical Subject Classification 00A07; 26Dxx, 51M16

Library of Congress Control Number: 2009929571

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ISBN 978-3-0346-0049-1 Birkhọuser Verlag, Basel – Boston – Berlin

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ISBN 978-3-0346-0049-1 e-ISBN 978-3-0346-0050-7

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Autors:

Radmila Bulajich Manfrino Rogelio Valdez Delgado Facultad de Ciencias Universidad Autónoma Estado de Morelos Av. Universidad 1001 Col. Chamilpa 62209 Cuernavaca, Morelos México e-mail: bulajich@uaem.mx valdez@uaem.mx

José Antonio Gómez Ortega Departamento de Matemàticas Facultad de Ciencias, UNAM Universidad Nacional Autónoma de México Ciudad Universitaria 04510 México, D.F. México e-mail: jago@fciencias.unam.mx

Introduction

This book is intended for the Mathematical Olympiad students who wish to pre- pare for the study of inequalities, a topic now of frequent use at various levels of mathematical competitions. In this volume we present both classic inequalities and the more useful inequalities for confronting and solving optimization prob- lems. An important part of this book deals with geometric inequalities and this fact makes a big difference with respect to most of the books that deal with this topic in the mathematical olympiad. The book has been organized in four chapters which have each of them a different character. Chapter 1 is dedicated to present basic inequalities. Most of them are numerical inequalities generally lacking any geometric meaning. How- ever, where it is possible to provide a geometric interpretation, we include it as we go along. We emphasize the importance of some of these inequalities, such as the inequality between the arithmetic mean and the geometric mean, the Cauchy- Schwarz inequality, the rearrangement inequality, the Jensen inequality, the Muir- head theorem, among others. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad prob- lems. We also emphasize how the substitution strategy is used to deduce several inequalities. The main topic in Chapter 2 is the use of geometric inequalities. There we ap- ply basic numerical inequalities, as described in Chapter 1, to geometric problems to provide examples of how they are used. We also work out inequalities which have a strong geometric content, starting with basic facts, such as the triangle inequality and the Euler inequality. We introduce examples where the symmetri- cal properties of the variables help to solve some problems. Among these, we pay special attention to the Ravi transformation and the correspondence between an inequality in terms of the side lengths of a triangle a, b, c and the inequalities that correspond to the terms s, r and R, the semiperimeter, the inradius and the circumradius of a triangle, respectively. We also include several classic geometric problems, indicating the methods used to solve them. In Chapter 3 we present one hundred and twenty inequality problems that have appeared in recent events, …