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Inequalities: A Mathematical Olympiad Approach

R. B. Manfrino, J. A. Gómez Ortega, and R. Valdez Delgado
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Henry Ricardo
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Based on fifteen years of preparing students for the Mexican Mathematical Olympiad and the International Mathematical Olympiad, this book presents a calculus-free introduction to inequalities and optimization problems with many interesting examples and exercises.

The book starts with properties of order in the real numbers and the triangle inequality, and then continues through such mainstays of mathematical competitions as the arithmetic-geometric mean inequality and the inequalities of Cauchy-Schwarz, Hölder, Minkowski, Jensen, Nesbitt, and Schur. The careful treatment of convexity includes a discussion of Popoviciu’s relatively recent (1965) inequality for convex functions, a very useful weapon in the problem solver’s arsenal: If f is a real-valued convex function on an interval I, then for a, b, c in I,

The first chapter concludes with Muirhead’s theorem and a large number of its applications.

Chapter 2 starts with basic inequalities concerning triangles and goes on to develop Euler’s theorem and inequality involving the circumcenter (O), incenter (I), circumradius (R), and inradius (r) of a triangle: OI2 = R2 –2Rr and R ≥ 2r. Area and perimeter inequalities follow. This chapter ends with geometric optimization problems–for example, several solutions to the Fermat-Steiner problem and Heron’s shortest path problem. (See my previous review for related material.)

Chapter 3 consists of 120 recent (1995-2008) national and international Olympiad problems, a mixture of numerical and geometric inequalities. In Chapter 4, the authors present solutions or hints to all exercises and problems appearing in the book. There is a twenty-one item bibliography (in which I was surprised to find a Spanish edition of the Courant and Robbins expository gem: ¿Qué son las Mathemáticas?).

Most books on Olympiad-level competitions have sections on inequalities, but the book under review focuses on this genre of problems in a particularly attractive and effective way, providing good practice material. I recommend this softcover volume to anyone interested in mathematical competition preparation. I also suggest looking at the book Inequalities: An Approach Through Problems by B. J. Venkatachala, a slightly more comprehensive treatment of inequalities, but still at the Olympiad level.

Henry Ricardo ( has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.

Introduction.- 1 Numerical Inequalities.- 1.1 Order in the real numbers.- 1.2 The quadratic function ax2 + 2bx + c.- 1.3 A fundamental inequality, arithmetic mean-geometric mean.- 1.4. A wonderful inequality: the rearrangement inequality.- 1.5 Convex functions.- 1.6 A helpful inequality.- 1.7 The substitutions strategy.- 1.8 Muirhead's theorem.- 2 Geometric Inequalities.- 2.1 Two basic inequalities.- 2.2 Inequalities between the sides of a triangle.- 2.3 The use of inequalities in the geometry of the triangle.- 2.4 Euler's inequality and some applications.- 2.5 Symmetric functions of a, b and c.- 2.6 Inequalities with areas and perimeters. 2.7 Erdös-Mordell theorem.- 2.8 Optimization problems.- 3 Recent Inequality Problems.- 4 Solutions to Exercises and Problems.- Bibliography.- Index.