Geometric Problems on Maxima and Minima
Andreescu, Titu; Mushkarov, Oleg; Stoyanov, Luchezar
2006, X, 264 p. 262 illus., Softcover ISBN: 0-8176-3517-3
A Birkhäuser book

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.

Written for:

Undergraduates, math instructors, AP high school students, high school mathematics enrichment program participants, Math Olympiad students/participants/coaches

103 Trigonometry Problems From the Training of the USA IMO Team
Andreescu, Titu, Feng, Zuming
2005, XII, 214 p., Softcover ISBN: 0-8176-4334-6
A Birkhäuser book

103 Trigonometry Problems contains carefully-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques.

Key features:

• Gradual progression in problem difficulty builds and strengthens mathematical skills and techniques
• Basic topics include trigonometric formulas and identities, their applications in the geometry of the triangle, trigonometric equations and inequalities, and substitutions involving trigonometric functions
• Problem-solving tactics and strategies, along with practical test-taking techniques, provide in-depth enrichment and preparation for possible participation in various mathematical competitions
• Comprehensive introduction (first chapter) to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry expose advanced students to college level material

103 Trigonometry Problems is a cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training.

Written for:

Students; instructors; mathematics coaches