In the mathematical problem-solving literature, there are few works that focus on trigonometry and explore its vast applications. This book responds to the need of those who would like to expand their knowledge of trigonometry, to develop a solid understanding in trigonometry, and gain the necessary skills to solve trigonometry problems. The book presents complex trigonometric concepts in a simplified manner, making it suitable for both beginners and experienced problem solvers. It contains many examples and problems taken from various competitions and journals, helping readers to appreciate the relevance of trigonometry in today's mathematical contests. The book covers both the geometric and the algebraic aspects of trigonometry, providing a thorough understanding of basic trigonometric principles, identities, equations, and problem-solving techniques.

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.

AwesomeMath Summer Program started in 2006. Since then until 2021 there have been 48 admission tests featuring a total of 510 problems. The vast majority of the problems were created by Dr. Titu Andreescu, Co-founder and Director of AwesomeMath. The problems are original, carefully designed, and cover all four traditional areas of competition math: Algebra, Geometry, Number Theory, and Combinatorics. The problems and solutions are divided in two volumes. Volume I focuses on the years since the start of the summer program in 2006 through 2014. Volume II includes the years 2015 to 2021, inclusively. Each volume starts with the statements of the test problems. Complete and enhanced solutions to all problems are then presented, numerous problems having multiple solutions.

AwesomeMath Summer Program started in 2006. Since then until 2021 there have been 48 admission tests featuring a total of 510 problems. The vast majority of the problems were created by Dr. Titu Andreescu, Co-founder and Director of AwesomeMath. The problems are original, carefully designed, and cover all four traditional areas of competition math: Algebra, Geometry, Number Theory, and Combinatorics. The problems and solutions are divided in two volumes. Volume I focuses on the years since the start of the summer program in 2006 through 2014. Volume II includes the years 2015 to 2021, inclusively. Each volume starts with the statements of the test problems. Complete and enhanced solutions to all problems are then presented, numerous problems having multiple solutions.

The evolving landscape that makes up modern mathematical competitions has exposed contestants to polynomials more than ever before. Recently, almost all advanced mathematical competitions have at least one problem on polynomials. Despite this rising importance, only a few problem books bring attention to this topic. Thus, the vast universe that polynomials encapsulates should be more thoroughly investigated. This book casts light on the topic of polynomials from numerous angels. The authors present important theoretical facts in harmony with their showcased applications. There are 8 chapters, 252 solved examples, 105 end of chapter problems, all with detailed solutions, as well as 77 additional problems that further enhance the books' exposition.

The ubiquity of polynomials and their ability to characterize complex patterns let us better understand generalizations, theorems, and elegant paths to solutions that they provide. We strive to showcase the true beauty of polynomials through a well-thought collection of problems from mathematics competitions and intuitive lectures that follow the sub-topics. Thus, we present a view of polynomials that incorporates various techniques paired with the favorite themes that show up in math contests.