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 problems. 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 apply 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 symmetrical 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, covering all levels, from the national and up to the regional and international olympiad competitions. vi Introduction In Chapter 4 we provide solutions to each of the two hundred and ten exercises in Chapters 1 and 2, and to the problems presented in Chapter 3. Most of the solutions to exercises or problems that have appeared in international mathematical competitions were taken from the official solutions provided at the time of the competitions. This is why we do not give individual credits for them. Some of the exercises and problems concerning inequalities can be solved using different techniques, therefore you will find some exercises repeated in different sections. This indicates that the technique outlined in the corresponding section can be used as a tool for solving the particular exercise. The material presented in this book has been accumulated over the last fifteen years mainly during work sessions with the students that won the national contest of the Mexican Mathematical Olympiad. These students were developing their skills and mathematical knowledge in preparation for the international competitions in which Mexico participates. We would like to thank Rafael Mart ́ınez Enr ́ıquez, Leonardo Ignacio Mart ́ınez Sandoval, David Mireles Morales, Jes ́us Rodr ́ıguez Viorato and Pablo Sober ́on Bravo for their careful revision of the text and helpful comments for the improvement of the writing and the mathematical content.
