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New, Newer, and Newest Inequalities

Titu Andreescu and Marius Stanean
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Frederic Morneau-Guerin
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New, Newer and Newest Inequalities – a problem book from Titu Andreescu and Marius Stanean – is intended as a sequel to 116 Algebraic Inequalities From the Awesome Math Year-Round Program (2018) and 118 Inequalities for Mathematics Competitions (2019). 
Much like the previous two works also published by XYZ Press – a company founded in 2008 by famous head coach of the U.S. International Mathematical Olympiad team Dr. Titu Andreescu – this book was designed to help aspiring candidates hone their training to potentially take part in mathematics competitions.
In the first part, which includes around 140 pages, readers will first come across the classical inequalities such as the Cauchy-Schwarz inequality and the Arithmetic Mean–Geometric Mean Inequality as well as well-known methods such as the mathematical induction method and the Lagrange multiplier method. Next, somewhat less familiar inequalities are introduced such as Abel’s, Newton’s, Maclaurin’s, and Blundon’s inequalities. Less usual methods are also covered, such as the “mixing” and “stronger mixing” variables methods. The explanation of the mathematical content, which is succinct but sufficient and clearly written, is accompanied by 100 examples to illustrate the point. 
The second part of the book, which features around 200 pages, is devoted to problems. These are divided into two categories: there are first 65 introductory problems (that are nevertheless tough) and then, 65 more advanced problems (a substantial number of which are of appropriate difficulty to be Olympiad questions that were prepared by the two authors). Interestingly, at least one complete and detailed solution is offered for each of the problems. Nevertheless, some important lessons are presented implicitly, and readers are left to take them on by themselves. This book, which has been specifically designed for that purpose, will prove a particularly useful resource for instructors and students training to take part in national and international mathematics competitions. However, the routine problems to gradually build familiarity and incorporate the underlying ideas are far too few in number and the explanations are much too brief for this book to be a valuable read for a novice learner. Moreover, this work does not include any visual representation to support understanding. 
Lastly, it should be noted that this is another work from XYZ Press with shortcomings that are often brought up, namely the lack of an index, bibliography or a list of suggested works covering the topic in depth to situate the inequalities presented in the context that reveals their full meaning and significance. 


Frederic Morneau-Guerin is a professor in the Department of Education at Universite TELUQ. He holds a Ph.D. in abstract harmonic analysis.