You are here

Geometric Problems on Maxima and Minima

Titu Andreescu, Oleg Mushkarov, and Luchezar Stoyanov
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Henry Ricardo
, on

As an avid problem solver with a strong interest in inequalities, particularly algebraic and analytic inequalities, I am delighted to supplement my repertoire with the techniques illustrated in this volume. Geometric extremal problems such as those treated in this volume are dense in the problem sections of many journals (e.g., Crux Mathematicorum with Mathematical Mayhem).

The book contains hundreds of problems, classical and modern, all with hints or complete solutions. Solution methods include the use of geometrical transformations, algebraic inequalities (AGM, Cauchy-Schwarz,…), and calculus (Extreme Value Theorem, Fermat’s Theorem,…). The Glossary is very helpful, as is the bibliography of books and journal articles, although I was surprised to find Ivan Niven’s Maxima and Minima Without Calculus missing from the list of references.

Over the years, Titu Andreescu and various collaborators have used their experiences as teachers and as Olympiad coaches to produce a series of excellent problem solving manuals, as a quick MAA Reviews search on "Andreescu" will reveal. The present volume continues that tradition and should appeal to a wide audience ranging from advanced high school students to professional mathematicians. Here’s a sample of four problems that caught my eye:

Cut two nonintersecting circles from a triangle such that the sum of their areas is maximal.

Of all quadrilaterals inscribed in a given half-disk find the one of maximum area.

Consider n2 arbitrary points in a unit square. Show that there exists a broken line with vertices at these points whose length is not greater than 2n.

Show that one can cover a unit square by means of any finite collection of squares of total area 4. 
[Problem 26 in D. J. Newman’s A Problem Seminar shows that the covering can be accomplished with a finite collection of total area 3 and that this is optimal.]

Henry Ricardo ( is Professor of Mathematics at Medgar Evers college of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in 2002.

 Preface.- Methods for Finding Geometric Extrema.- Selected Types of Geometric Extremum Problems.- Miscellaneous.- Hints and Solutions to the Exercises.- Notation.- Glossary of Terms.- Bibliography.