Venkatachala B.J. Inequalities. An Approach through Problems

New York: Springer, 2018. — 523 p. — (Texts and Readings in Mathematics 49). — ISBN: 978-981-10-8732-5.The International Mathematical Olympiad(IMO), which started as a simple contest among seven communist block countries in Europe in 1959, has now encompassed the whole world. With nearly 100 countries participating in this mega event, this has acquired a true international character. Mathematical olympiad has effused new enthusiasm in the last few generations of young students and really talented young minds have started getting attracted to the rare beauty of mathematics. Along with it, new ideas have emerged and many intricate problems woven around these ideas have naturally been discovered. This has enriched basic mathematics, strengthened the foundations of elementary mathematics and has posed challenging problems to the younger generation.
In turn, high school mathematics has undergone a profound change. Even though the concept of inequalities is old, the mathematical olympiad movement has generated new problems based on these inequalities and newer applications of these old inequalities. There are classical inequalities like the Arithmetic mean-Geometric mean inequality and the Cauchy-Schwarz inequality which are very old and which have innumerable applications. The main purpose of this book is to give a comprehensive presentation of inequalities and their use in mathematical olympiad problems. This book is also intended for those students who would like to participate in mathematical olympiad. Hopefully, this will fill a little vacuum that exists in the world of books. I have not touched on integral inequalities; they are not a part of olympiad mathematics.
The book is divided into six chapters. The first chapter describes all the classical inequalities which are useful for students who are interested in mathematical olympiad and similar contests. I have tried to avoid too much of theory; many results are taken for granted whenever a need for some advanced mathematics is required(especially results from calculus). Rather, I have put more emphasis on problems. The second chapter gives many useful techniques for deriving more inequalities. Here again the stress is on the application of different methods to problems. An important class of inequalities, called geometric inequalities, is based on geometric structures, like, triangles and quadrilaterals. Some important geometric inequalities are derived in the third chapter. In the fourth chapter, the problems(mainly taken from olympiad contests) whose solution(s) involve application of inequalities are discussed. The fifth chapter is simply a large collection of problems from various contests round the world and some problems are also taken from several problem journals. The reader is advised to try these problems on his own before looking into their possible solutions, which are discussed in the sixth chapter