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105 Algebra Problems from the AwesomeMath Summer Program
Titu Andreescu, University of Texas at Dallas, Richardson, TX
A publication of XYZ Press.
cover
XYZ Series
2013; 200 pp; hardcover
Volume: 2
ISBN-10: 0-9799269-5-5
ISBN-13: 978-0-9799269-5-2
List Price: US$49.95
Member Price: US$39.96
Order Code: XYZ/2
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The main purpose of this book is to provide an introduction to central topics in elementary algebra from a problem-solving point of view. While working with students who were preparing for various mathematics competitions or exams, the author observed that fundamental algebraic techniques were not part of their mathematical repertoire. Since algebraic skills are not only critical to algebra itself but also to numerous other mathematical fields, a lack of such knowledge can drastically hinder a student's performance. Taking the above observations into account, the author has put together this introductory book using both simple and challenging examples which shed light upon essential algebraic strategies and techniques, as well as their application in diverse meaningful problems. This work is the first volume in a series of such books.

The featured topics from elementary and classical algebra include factorizations, algebraic identities, inequalities, algebraic equations and systems of equations. More advanced concepts such as complex numbers, exponents and logarithms, as well as other topics, are generally avoided. Nevertheless, some problems are constructed using properties of complex numbers which challenge and expose the reader to a broader spectrum of mathematics. Each chapter focuses on specific methods or strategies and provides an ample collection of accompanying problems that graduate in difficulty and complexity. In order to assist the reader with verifying mastery of the theoretical component, 105 problems are included in the last sections of the book, of which 52 are introductory and 53 are advanced.

All problems come together with solutions, many employing several approaches and providing the motivation behind the solutions offered.

A publication of XYZ Press. Distributed in North America by the American Mathematical Society.

Readership

Middle and high school students interested in mathematics competition preparation.

Reviews

"In case you are thinking about participating in a mathematical contest in the future and need a good place to start this might be one of the best choices for you. ..."

-- Yair Mulian

"... This book is clearly meant for students who enjoy working on such problems and learning new tricks to solve problems. I strongly recommend this book for such a student. Andresscu's books are all top notch and this is no exception."

-- Blake Thornton

"This is an excellent collection of problems, solutions, and narrative that would be great as self-study for motivated high school students, or for a math circle leader to use as a resource for a problem solving class. I'm very glad that I worked through the material. Content-wise, this book is top-notch! ..."

-- Japheth Wood

Table of Contents

  • Preface
  • Completing the square and quadratic equations
  • Factorizations and algebraic identities
  • Factoring expressions involving \(a-b, b-c, c-a\)
  • Factoring \(a^{3} + b^{3} + c^{3}-3 abc\)
  • AM-GM and Hölder's inequality
  • Lagrange's identity and the Cauchy-Schwarz inequality
  • Making linear combinations
  • Fixed points and monotonicity
  • The floor function
  • Taking advantage of symmetry
  • Introductory problems
  • Advanced problems
  • Solutions to introductory problems
  • Solutions to advanced problems
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