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Ten Themes in Algebra for Mathematics Competitions (Paperback)
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Ten Themes in Algebra for Mathematics Competitions (Paperback)
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This book contains ten frequently recurring themes in algebraic
problems. Each chapter starts with a brief introduction that
includes examples useful for the reader to grasp the main ideas
needed to solve the proposed problems. The first chapter deals with
quadratic functions and underscores the use of the discriminant and
the relations involving the roots of a quadratic trinomial and its
coefficients. The second chapter emphasizes that every square of a
real number is non-negative. This simple property leads to numerous
applications also encountered in subsequent chapters. Chapter 3
focuses on several inequalities, including the most famous
inequality in the world of mathematical Olympiads: the
Cauchy-Schwarz Inequality. Chapter 4 is devoted to problems related
to minima and maxima of algebraic expressions. These problems can
also be approached using the techniques studied in the previous
chapter. The fifth chapter is about a beautiful identity involving
the cubes of three numbers and the triple of their product and you
will see that this identity has numerous interesting applications.
Chapter 6 deals with complex numbers. Some definitions and useful
results are given to assist the reader in solving the proposed
problems. The seventh chapter features Lagrange's Identity, which
has various unexpected applications, including those involving
problems related to number theory. Chapter 8 focuses on the
so-called Sophie Germain's Identity. Here, too, you will find
problems in which the application of this identity will be anything
but obvious. Chapter 9 looks at expressions of the form t + k/t and
meaningful applications. Finally, the last chapter is about the
fifth-degree polynomials x^5 + x + \ pm 1 and assorted non-routine
problems. Solutions to all proposed problems are provided in the
second part of the book: there is a corresponding solution chapter
for each of the ten chapters in the first part.
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