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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.
The book covers many classical topics in elementary algebra, including factoring, quadratic functions, irrational expressions, Vieta's relations, equations and systems of equations, inequalities, sums and products, and polynomials. Expanding upon the previous work in the series, 105 Problems in Algebra from the AwesomeMath Summer Program, this book features additional more advanced topics, including exponents and logarithms, complex numbers, and trigonometry. The special section on trigonometric substitutions and more explores seemingly algebraic problems with natural geometric and trigonometric interpretations. To give the reader practice with the strategies and techniques discussed in each of the chapters, the authors have included 108 diverse problems, of which 54 are introductory and 54 are advanced. Solutions to all of these problems are provided, in which different approaches are compared.
This book contains 106 geometry problems used in the AwesomeMath Summer Program to train and test top middle and high-school students from the U.S. and around the world. Just as the camp offers both introductory and advanced courses, this book also builds up the material gradually. The authors begin with a theoretical chapter where they familiarize the reader with basic facts and problem-solving techniques. Then they proceed to the main part of the work, the problem sections. The problems are a carefully selected and balanced mix which offers a vast variety of flavors and difficulties, ranging from AMC and AIME levels to high-end IMO problems. Out of thousands of Olympiad problems from around the globe, the authors chose those which best illustrate the featured techniques and their applications. The problems meet the authors' demanding taste and fully exhibit the enchanting beauty of classical geometry. For every problem, they provide a detailed solution and strive to pass on the intuition and motivation behind it. Many problems have multiple solutions.Directly experiencing Olympiad geometry both as contestants and instructors, the authors are convinced that a neat diagram is essential to efficiently solve a geometry problem. Their diagrams do not contain anything superfluous, yet emphasize the key elements and benefit from a good choice of orientation. Many of the proofs should be legible only from looking at the diagrams.
"102 Combinatorial Problems" consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Key features: * Provides in-depth enrichment in the important areas of combinatorics by reorganizing and enhancing problem-solving tactics and strategies * Topics include: combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, theory of equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations and classical inequalities The book is systematically organized, gradually building combinatorial skills and techniques and broadening the student's view of mathematics. Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas that are tangential to combinatorics.
This book aims to give students a chance to begin exploring some introductory to intermediate topics in combinatorics, a fascinating and accessible branch of mathematics centered around (among other things) counting various objects and sets. We include chapters featuring tools for solving counting problems, proof techniques, and more to give students a broad foundation to build on. The only prerequisites are a solid background in arithmetic, some basic algebra, and a love for learning math.
This book provides the mathematical tools and problem-solving experience needed to successfully compete in high-level problem solving competitions. Each section presents important background information and then provides a variety of worked examples and exercises to help bridge the gap between what the reader may already know and what is required for high-level competitions. Answers or sketches of the solutions are given for all exercises.
This book contains 107 geometry problems used in the AwesomeMath Year-Round Program. The problems offer additional challenges for those who have progressed through the 106 Geometry Problems from the AwesomeMath Summer Camp publication. The book begins with a theoretical chapter, where the authors review basic facts and familiarize the reader with some more advanced techniques. The authors then proceed to the main part of the work, the problem sections. The problems are a carefully selected and balanced mix which offers a vast variety of flavors and difficulties, ranging from AMC and AIME levels to high-end IMO problems. Out of thousands of Olympiad problems from around the globe the authors chose those which best illustrate the featured techniques and their applications. The problems meet the authors' demanding taste and fully exhibit the enchanting beauty of classical geometry. For every problem the authors provide a detailed solution and strive to pass on the intuition and motivation behind it. Numerous problems have multiple solutions.Directly experiencing Olympiad geometry both as contestants and instructors, the authors are convinced that a neat diagram is essential to efficiently solve a geometry problem. Their diagrams do not contain anything superfluous, yet emphasize the key elements and benefit from a good choice of orientation. Many of the proofs should be legible only from looking at the diagrams.
In this volume they present innumerable beautiful results, intriguing problems, and ingenious solutions. The problems range from elementary gems to deep truths.
This book covers the theoretical background of exponents and logarithms, as well as some of their important applications. Starting from the basics, the reader will gain familiarity with how the exponential and logarithmic functions work, and will then learn how to solve different problems with them. The authors give the readers the opportunity to test their understanding of the topics discussed by exposing them to 114 carefully chosen problems, whose full solutions can be found at the end of the book.