Geometric problems on maxima and minima
Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov, Titu AndreescuKey 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-caliber
Written 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 puzzle enthusiasts.
01Methods for Finding Geometric Extrema......Page 10
02Selected Types of Geometric Extremum Problems......Page 72
03Miscellaneous......Page 104
04Hints and Solutions to the Exercises......Page 114
back-matter......Page 264
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Combinatorics of Permutations (Discrete Mathematics and Its Applications)
There are 650 articles with the word permutation in the title whose primary classification is combinatorics, but, until now, there have been no books addressing the topic. The very first book to be published on the subject, Combinatorics of Permutations contains a comprehensive, up to date treatment of the subject. Covering both enumerative and external combinatorics, this book can be used as either a graduate text or as a reference for professional mathematicians. The book includes many applications from computer science, molecular biology, probabilistic methods, and pattern avoidance, and the numerous exercises show readers a fairly comprehensive list of recent results from the field.
Introduction to Geometry, 2nd Edition
This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises. This unabridged paperback edition contains complete coverage, ranging from topics in the Euclidean plane to affine geometry, projective geometry, differential geometry and topology.
Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle. (Cosimo Classics)
Felix Klein, Wooster Woodruff Beman, David Eugene Smith
This short book, first published in 1897, addresses three geometry puzzles that have been passed down from ancient times. Written for high school students, this book aims to show a younger audience why math should matter and to make the problems found in math intriguing. Klein presents for his readers an investigation of the possibility or impossibility of finding solutions for the following problems in light of mathematics available to him: duplication of the cube trisection of an angle quadrature of the circle Mathematicians and students of the history of math will find this an intriguing work. German mathematician FELIX KLEIN (1849-1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis and Elementary Mathematics from an Advanced Standpoint: Geometry.DjVulibre:DjVu2TIFF2PDF,loss of text data and larger file sizes.Calibre:produces poorly formatted PDFs and may encounter parsing errorsDjVuToy:1. Maintain the original compression rate of DjVu files.2. Preserve the OCR metadata of DjVu documents.3. The software is very lightweight and completely free.4. Only for Windows DjVuToy Download:https://url26.ctfile.com/f/41895126-925895940-3a0fec?p=9167Password:9167
Geometry and Topology
Geometry Provides A Whole Range Of Views On The Universe, Serving As The Inspiration, Technical Toolkit And Ultimate Goal For Many Branches Of Mathematics And Physics. This Book Introduces The Ideas Of Geometry, And Includes A Generous Supply Of Simple Explanations And Examples. The Treatment Emphasises Coordinate Systems And The Coordinate Changes That Generate Symmetries. The Discussion Moves From Euclidean To Non-euclidean Geometries, Including Spherical And Hyperbolic Geometry, And Then On To Affine And Projective Linear Geometries. Group Theory Is Introduced To Treat Geometric Symmetries, Leading To The Unification Of Geometry And Group Theory In The Erlangen Program. An Introduction To Basic Topology Follows, With The Möbius Strip, The Klein Bottle And The Surface With G Handles Exemplifying Quotient Topologies And The Homeomorphism Problem. Topology Combines With Group Theory To Yield The Geometry Of Transformation Groups,having Applications To Relativity Theory And Quantum Mechanics. A Final Chapter Features Historical Discussions And Indications For Further Reading. With Minimal Prerequisites, The Book Provides A First Glimpse Of Many Research Topics In Modern Algebra, Geometry And Theoretical Physics. The Book Is Based On Many Years' Teaching Experience, And Is Thoroughly Class-tested. There Are Copious Illustrations, And Each Chapter Ends With A Wide Supply Of Exercises. Further Teaching Material Is Available For Teachers Via The Web, Including Assignable...
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Each chapter devoted to single type of problem with accompanying commentary and set of practice problems. Amateur puzzlists, students of mathematics and geometry will enjoy this rare opportunity to match wits with civilization’s great mathematicians and witness the invention of modern mathematics.<br
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A fascinating journey into the mind-bending world of prime numbers Cicadas of the genus Magicicada appear once every 7, 13, or 17 years. Is it just a coincidence that these are all prime numbers? How do twin primes differ from cousin primes, and what on earth (or in the mind of a mathematician) could be sexy about prime numbers? What did Albert Wilansky find so fascinating about his brother-in-law's phone number? Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions. In Prime Numbers: The Most Mysterious Figures in Math, you'll meet the world's most gifted mathematicians, from Pythagoras and Euclid to Fermat, Gauss, and Erd?o?s, and you'll discover a host of unique insights and inventive conjectures that have both enlarged our understanding and deepened the mystique of prime numbers. This comprehensive, A-to-Z guide covers everything you ever wanted to know--and much more that you never suspected--about prime numbers, including: * The unproven Riemann hypothesis and the power of the zeta function * The "Primes is in P" algorithm * The sieve of Eratosthenes of Cyrene * Fermat and Fibonacci numbers * The Great Internet Mersenne Prime Search * And much, much more
Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World (MAA Problem Book Series)
Edited By Titu Andreescu And Zuming Feng
This volume contains a large range of problems, with and without solutions, taken from 25 national and regional mathematics olympiads from around the world, and the problems are drawn from several years' contests. In many cases, more than one solution is given to a single problem in order to highlight different problem-solving strategies. The collection is intended as practice for students preparing for these competitions. Teachers and general readers looking for interesting problems will find also it very useful.
Mathematical Olympiads 1999-2000: Problems And Solutions From Around The World (maa Problem Book Series)
Titu Andreescu; Zuming Feng; Mathematical Association Of America
Contained here are solutions to challenging problems from algebra, geometry, combinatorics and number theory featured in the earlier book, together with selected questions (without solutions) from national and regional Olympiads given during the year 2000. Intended for the serious student/problem solver, these books can help to improve performance in the Mathematical Olympiad competition. However, for those not entering the competition, there is much to challenge any mathematician, even those with advanced degrees. Different nations have different mathematical cultures, so you will find that some of the questions are extremely difficult and some rather easy. There are a wide variety of problems especially from those countries that have often done well in the IMO. Anyone interested in mathematical problem solving will encounter some beautiful mathematics in the pages of this book. If you are up to a real challenge, take some of these problems on!
Mathematical Olympiads, 2000-2001: Problems and Solutions from Around the World (MAA Problem Book Series)
Titu Andreescu, Zuming Feng, George Lee Jr, Po-Ru Loh
This volume is a continuation of Mathematical Olympiads 1999-2000: Problems and Solutions From Around the World, republishing hundreds of mathematics problems and solutions from that book as well as selected problems (without solutions) from national and regional contests in 2001. The collection provides practice material for serious high-school level mathematicians who wish to prepare for the USA Math Olympiad (USAMO) and Team Selection Test (TST). The newly added 2001 problems were contributed by dozens of countries including Korea, Belarus, China, Poland, and Romania
102 Combinatorial Problems : From the Training of the USA IMO Team
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. By Titu Andreescu, Zuming Feng.
Challenging Problems in Geometry (Dover Books on Mathematics)
Alfred S. Posamentier, Charles T. Salkind
I liked this book very much. I solved every single problem in the book with two students that I tutor for International Math Olympiads and carefully read the hints and solutions proposed at the end of the book. They really teach how to "attack" geometry problems using simple stuff like angle chasing, drawing parallel lines etc. I cannot recommend this book more to the readers with some mathematical sophistication. I even have a suggestion for parents that have some sort of mathematical background (engineers, bankers, doctors, etc.): If you want to spend quality time with your children, get this book and enjoy solving the problems together. I cannot imagine a more amusing pastime. I am looking forward to seeing new titles from the authors.
USA and International Mathematical Olympiads, 2003
Titu Andreescu; Zuming Feng; Mathematical Association Of America
The Mathematical Olympiad examinations, covering the USA Mathematical Olympiad (USAMO) and the International Mathematical Olympiad (IMO), have been published annually since 1976 by the MAA American Mathematics Competitions. This is the fourth volume in that series published by the MAA in its Problem Book series. The IMO is a world mathematics competition for high school students that takes place each year in a different country. Students from all over the world participate in this competition. The USAMO and the Team Selection Test are the last two stages of the process that lead to the selection of the team representing the USA in the IMO. Problems and solutions from both of these competitions for the year 2003 are included in this volume. These Olympiad style exams consist of several challenging essay-type problems. Although a correct and complete solution to an Olympiad problem often requires deep analysis and careful argument, the problems require no more than a solid background in high school mathematics coupled with a dose of mathematical ingenuity. There are helpful hints provided for each of the problems. These hints often help lead the student to a solution of the problem. Complete solutions to each of the problems is also included, and many of the problems are presented together with a collection of remarkable solutions developed by the examination committees, contestants and experts, during or after the contest. For each problem with...
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Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. Each chapter covers a different aspect of Euclidean geometry, lists relevant theorems and corollaries, and states and proves many propositions. Includes more than 200 problems, hints, and solutions. 1968 edition.
102 Combinatorial Problems : From the Training of the USA IMO Team
Titu Andreescu, Zuming Feng (Auth.)
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. By Titu Andreescu, Zuming Feng.
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360 Problems for Mathematical Contests by Titu Andreescu (2003-01-01)
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Number, shape, and symmetry : an introduction to number theory, geometry, and group theory
Diane L. Herrmann And Paul J. Sally, Jr
Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors' successful work with undergraduate students at the University of Chicago, seventh to tenth grade mathematically talented students in the University of Chicago's Young Scholars Program, and elementary public school teachers in the Seminars for Endorsement in Science and Mathematics Education (SESAME). The first half of the book focuses on number theory, beginning with the rules of arithmetic (axioms for the integers). The authors then present all the basic ideas and applications of divisibility, primes, and modular arithmetic. They also introduce the abstract notion of a group and include numerous examples. The final topics on number theory consist of rational numbers, real numbers, and ideas about infinity. Moving on to geometry, the text covers polygons and polyhedra, including the construction of regular polygons and regular polyhedra. It studies tessellation by looking at patterns in the plane, especially those made by regular polygons or sets of regular polygons. The text also determines the symmetry groups of these figures and patterns, demonstrating how groups arise in both geometry and number theory. The book is suitable for pre-service or in-service training for elementary school teachers, general education...