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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.
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.
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.
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