Bøger af Titu Andreescu

Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme-value problems, examples, and solutions primarily through Euclidean geometry.Key features and topics:* Comprehensive selection of problems, including Greek geometry and optics, Newtonian mechanics, isoperimetric problems, and recently solved problems such as Malfatti's problem * Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning* Presentation and application of classical inequalities, including Cauchy--Schwarz and Minkowski's Inequality; basic results in calculus, such as the Intermediate Value Theorem; and emphasis on simple but useful geometric concepts, including transformations, convexity, and symmetry * Clear solutions to the problems, often accompanied by figures * Hundreds of exercises of varying difficulty, from straightforward to Olympiad-caliberWritten by a team of established mathematicians and professors, this work draws on the authors' experience in the classroom and as Olympiad coaches. By exposing readers to a wealth of creative problem-solving approaches, the text communicates not only geometry but also algebra, calculus, and topology. Ideal for use at the junior and senior undergraduate level, as well as in enrichment programs and Olympiad training for advanced high school students, this book's breadth and depth will appeal to a wide audience, from secondary school teachers and pupils to graduate students, professional mathematicians, and puzzleenthusiasts.

Casts light on the topic of polynomials from numerous angles. 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.

An in-depth book centred around Algebra, featuring beautiful and intriguing problems. All problems were created by the authors. The book showcases the heart of Algebra not as a textbook, but rather as a collection of the most essential concepts and their applications.

A sequel to the 116 Algebraic Inequalities from the AwesomeMath Year-round Program and 118 Inequalities for Mathematics Competitions. The book delves into other elementary techniques but also powerful methods and generalizations for constrained optimization in the theory of inequalities.

The problems in the book were selected from the 1975 to 1986 Romanian Team Selection Tests. The problems are not only accompanied by their solutions, but also many have more than one solution and comments in order to make them clearer or to get connections to other problems.

This book reveiws the last two decades of computational techniques and progress in the classical theory of quadratic diophantine equations. Presents important quadratic diophantine equations and applications, and includes excellent examples and open problems.

Featuring a problem-solving approach to linear algebra, this work aims to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigen vectors, and orthogonality. It also emphasizes applications and connections to fields such as biology, economics, computer graphics, cryptography, and political science.

Features selected techniques in the field. Inequalities from each section are ordered increasingly by the number of variables and the degree of difficulty. Each problem has at least one complete solution, and many problems have multiple solutions, useful in developing the necessary array of mathematical machinery for competitions.

Building bridges between classical results and contemporary nonstandard problems, this highly relevant work embraces important topics in analysis and algebra from a problem-solving perspective.

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.

As a sequel to 113 Geometric Inequalities from the AwesomeMath Summer Program, this book extends the themes discussed in the former book and broadens a problem-solver's competitive arsenal.

Help your students to think critically and creatively through team-based problem solving instead of focusing on testing and outcomes.Professionals throughout the education system are recognizing that standardized testing is holding students back. Schools tend to view children as outcomes rather than as individuals who require guidance on thinking critically and creatively. Awesome Math focuses on team-based problem solving to teach discrete mathematics, a subject essential for success in the STEM careers of the future. Built on the increasingly popular growth mindset, this timely book emphasizes a problem-solving approach for developing the skills necessary to think critically, creatively, and collaboratively.In its current form, math education is a series of exercises: straightforward problems with easily-obtained answers. Problem solving, however, involves multiple creative approaches to solving meaningful and interesting problems. The authors, co-founders of the multi-layered educational organization AwesomeMath, have developed an innovative approach to teaching mathematics that will enable educators to:* Move their students beyond the calculus trap to study the areas of mathematics most of them will need in the modern world* Show students how problem solving will help them achieve their educational and career goals and form lifelong communities of support and collaboration* Encourage and reinforce curiosity, critical thinking, and creativity in their students* Get students into the growth mindset, coach math teams, and make math fun again* Create lesson plans built on problem based learning and identify and develop educational resources in their schoolsAwesome Math: Teaching Mathematics with Problem Based Learning is a must-have resource for general education teachers and math specialists in grades 6 to 12, and resource specialists, special education teachers, elementary educators, and other primary education professionals.

Includes problem-solving tactics and practical test-taking techniques that provide enrichment and preparation for various math competitions. This title provides comprehensive introduction to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry.

This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations.

This unique approach to combinatorics is centered around unconventional, essay-type combinatorial examples, followed by a number of carefully selected, challenging problems and extensive discussions of their solutions.

Covers such topics as: 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.

Problems illustrating important mathematical techniques with solutions and accompanying essays.

Covers telescoping sums and products in algebra and trigonometry; the use of complex numbers and de Moivre's Formula; Abel's summation formula; mathematical induction; combinatorial identities; and multiplicative functions and the use of Mobius function. The theory is presented together with rich examples.

Trigonometry finally receives the attention it deserves in this stand-alone book. The theory chapter is an invaluable pedagogical resource with lots of examples and guided exercises. The subsequent chapters offer a collection of carefully selected introductory through advanced problems and solutions intended to enhance the problem-solving skills of the reader.

Part of the Math Leads for Mathletes series, this book provides more challenging units for young math problem solvers and many others! The book draws on the authors' experience working with young mathletes and on the collective wisdom of mathematics educators around the world to help parents and mentors challenge and teach their aspiring math problem solvers.

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.

Presents a collection of carefully selected geometry problems designed for passionate geometers and students preparing for the IMO. This book presents a multitude of beautiful synthetic solutions that are meant to give a sense of how one should think about difficult geometry problems. Each problem comes with at least two such solutions and with additional remarks.

Serves as a very good resource and teaching material for anyone who wants to discover the beauty of Induction and its applications, from novice mathematicians to Olympiad-driven students and professors teaching undergraduate courses. The authors explore 10 different areas of mathematics, including topics that are not usually discussed in an Olympiad-oriented book on the subject.

Takes familiar ideas and extends them to a rich variety of problems. The topics covered span algebra, geometry, number theory, and even a few elements of mathematical analysis. The "landscapes" presented provide a "view" into areas that are not typically encountered in great depth in standard coursework but nonetheless have profound implications.

This book starts with simple arithmetic inequalities and builds to sophisticated inequality results such as the Cauchy-Schwarz and Chebyshev inequalities. Nothing beyond high school algebra is required of the student. The exposition is lean. Most of the learning occurs as the student engages in the problems posed in each chapter. And the learning is not "linear".

Explores the theory and techniques involved in proving algebraic inequalities. To expand the reader's mathematical toolkit, the authors present problems from journals and contests from around the world. Inequalities are an essential topic in Olympiad problem solving, and this title is an instructive resource for students striving for success at national and international competitions.

Provides an introduction to central topics in elementary algebra from a problem-solving point of view. This introductory book uses both simple and challenging examples which shed light on essential algebraic strategies and techniques, as well as their application in diverse meaningful problems.

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