Ten Themes in Algebra for Mathematics Competitions

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.