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This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.

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"Basic (elementary) inequalities and their application.- Inequalities between means, (with two and three variables).- Geometric (triangle) inequalities.- Bernoulli´s inequality, the Cauchy-Schwarz inequality, Chebishev´s inequality, Surányi´s inequality.- Inequalities between means (general case).- Points of incidence in applications of the AM-GM inequality.- The rearrangement inequality.- Convexity, Jensen´s inequality.- Trigonometric substitutions and their application for proving algebraic inequalities.- The most usual forms of trigonometric substitutions.- Characteristic examples, using trigonometric substitutions.- Hölder´s inequality, Minkowski´s inequality and their generalizations.- Generalizations of the Cauchy-Schwarz inequality, Chebishev´s inequality and the mean inequalities.- Newton´s inequality, Maclaurin´s inequality.- Schur´s inequality, Muirhead´s inequality.- Two theorems from differential calculus, and their applications for proving inequalities.- One method of proving symmetric inequalities with three variables.- Method for proving symmetric inequalities with three variables defined on set of real numbers.- Abstract concreteness method (ABC method).- Sum of Squares (S.O.S - method).- Strong mixing variables method (S.M.V Theorem).- Lagrange multipliers method.

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From the reviews:

"The book is aimed at the more advanced students about to enter university and is particularly useful to candidates, and their trainers, for mathematical competitions such as the Olympiads. The themes and methods are introduced in 19 chapters of the book, which include exercises and their solutions." (Peter Shiu, The Mathematical Gazette, Vol. 98 (541), March, 2014)
"This volume collects problems of various degrees of difficulty in the field of elementary inequalities. This book is intended as a valuable source for the training of high-school students in view of mathematical Olympiads. School teachers will also gain benefit from this book. ... It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level. ... this book is a ´must have´ for a university´s library, and I recommend it highly to its ´ideal audience´." (Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1233, 2012)

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Dipl. Math. Zdravko Cvetkovski, European University-Skopje, R. Macedonia, Informatics Department.

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