Geometric Problems on Maxima and Minima

ISBN-13: 9780817635176 / Angielski / Miękka / 2005 / 280 str.

Titu Andreescu; Oleg Mushkarov; Luchezar Stoyanov

Opis

Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry. Key features and topics: * Comprehensive selection of problems, including Greek geometry and optics, Newtonian mechanics, isoperimetric problems, and recently solved problems such as Malfatti s problem * Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning * Presentation and application of classical inequalities, including Cauchy--Schwarz and Minkowski s Inequality; basic results in calculus, such as the Intermediate Value Theorem; and emphasis on simple but useful geometric concepts, including transformations, convexity, and symmetry * Clear solutions to the problems, often accompanied by figures * Hundreds of exercises of varying difficulty, from straightforward to Olympiad-caliber Written by a team of established mathematicians and professors, this work draws on the authors experience in the classroom and as Olympiad coaches. By exposing readers to a wealth of creative problem-solving approaches, the text communicates not only geometry but also algebra, calculus, and topology. Ideal for use at the junior and senior undergraduate level, as well as in enrichment programs and Olympiad training for advanced high school students, this book s breadth and depth will appeal to a wide audience, from secondary school teachers and pupils to graduate students, professional mathematicians, and puzzle enthusiasts."

Recenzje

From the reviews:

"As an avid problem solver with a strong interest in inequalities...I am delighted to supplement my repertoire with the techniques illustrated in this volume.... The book contains hundreds of problems, classical and modern, all with hints or complete solutions.... 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.... The present volume continues that tradition and should appeal to a wide audience ranging from advanced high school students to professional mathematicians."   -MAA

"The whole exposition of the book is kept at a sufficiently elementary level, so that it can be understood by high-school students. Apart from trying to be comprehensive in terms of types of problems and techniques for their solutions, the authors have tried to offer various different levels of difficulty making the book possible to use by people with different interests in mathematics, different abilities, and of different age groups." -V. Oproiu, Analele Stiintifice

"This excellent book, Geometric problems on maxima and minima, deals not only with these famous problems, but well over a hundred other such problems, many of which were completely novel and new to me. ... This book will certainly greatly appeal to highschool students, mathematics teachers, professional mathematicians, and puzzle enthusiasts. I would regard it as absolutely essential reading for students preparing for mathematics competitions around the world." (Michael de Villiers, The Mathematical Gazette, Vol. 92 (525), 2008)

Spis treści

Methods for Finding Geometric Extrema.- Selected Types of Geometric Extremum Problems.- Miscellaneous.- Hints and Solutions to the Exercises.