As one can see from the title, this book is an introduction to the study of diophantine equations. The material is organized in two parts. The first part contains three chapters. Chapter 1 introduces the reader to the main elementary methods in solving diophantine equations such as decomposition, modular arithmetic, mathematical induction, Fermat's infinite descent. Chapter 2 presents some classical diophantine equations, including linear, pythagorean and some higher degree equations. Chapter 3 focuses on Pell's-type equations, serving again as an introduction to this special class of quadratic diophantine equations. Throughout Part I, each of the sections contains representative examples that illustrate the theoretical part.
Part II contains the complete solutions to all exercises featured in Part I. For several problems multiple solutions are included, along with useful comments and remarks. Many of the selected exercises and problems are original or have been give original solutions.
The book is intended for undergraduates, high school students and their teachers, mathematical contest (including Olympiad and Putnam) participants, as well as any person interested in essential mathematics. Află mai mult
This book gathers these methods and tricks into a compact guide. It starts from the very basics, but covers almost all elementary methods known today (aside from a few very advanced ones). At higher levels of competition, good contestants are pretty much expected to know every technique listed here - meaning that there do appear inequalities that fit every presented category. Conversely, the vast majority of inequalities that do appear may be assigned to one of the sections of the book.
That being said, the present work is not meant as a textbook, or a comprehensive training material. It is very compact and fast-paced. There are a few examples for every subject, but to fully assimilate said subject, one needs to look elsewhere for more practice problems. The reader who truly starts ”from scratch” may quickly find him/herself overwhelmed by the pace and the increasing difficulty of the content; may this entirely normal impression not deter the student from pushing through!
Rather, the book is meant as crash course introduction to all facets of the subject, and as a reference guide. The readerwill get acquainted with all themethods, but must practice each tool on his/her own, using other sources. The one truth about competitive math is that solving problems is the most important part of learning. This holds just as true for inequalities and fortunately there are a lot of them out there to be used for training purposes. Află mai mult
The name of Vasile Cirtoaje, in mathematical terms, actually equals with inequalities. Many problems from the book, their majority I would say, are obtained by the author himself. If you will carefully read the book, you may find that your skills in solving inequalities were considerably improved. However, one needs not to read every single page of this material; the chapters are independent and you may even open the book somewhere, arbitrarily, and try to solve an inequality or find its solution from the book. Also, we must add that the book is full of up to date problems, since the author is an active member of the Mathlinks Site Forum on the Internet. Last, but not least, one has to remark the outstanding tenacity and enthusiasm of the author in solving inequalities; definitely, he is a passionate of this realm of elementary mathematics. And this book is neither more, nor less than a work of a master.
Bulgaria is a country with long traditions in mathematical competitions. There are numerous regional competitions connected with important dates in Christian calendar or in Bulgarian history. These competitions range in format and difficulty and give opportunity to all students in lower and secondary school to test their abilities in problem solving. Great many of them being fascinated by problem solving in such competitions start working hard in order to acquire new knowledge in mathematics.
The most important and prestigious national competitions in Bulgaria are Winter Mathematical Competitions, Spring Mathematical Competitions and National Olympiad. The organization of these competitions is responsibility of the Ministry of Education and Science, the Union of Bulgarian Mathematicians and the local organizers. The problems for the competitions are prepared by so called Team of extra curricula research - a specialized body of the Union of Bulgarian Mathematicians.
This book contains all problems for grades 8 to 12 from the above mentioned national competitions in the period 2003-2006. The problems from all selection tests for BMO and IMO are also included. Most of the problems are regarded as difficult IMO type problems. The book is intended for undergraduates, high school students and teachers who are interested in olympiad mathematics. Află mai mult
In the pages that follow, we present a large variety of problems involving such inequalities, questions that became famous in (mathematical) competitions or journals because of their beauty. The most important prerequisite for benefiting from this book is the desire to master the craft of discovery and proof. The formal requirements are quite modest. Anyone who knows basic inequalities such as the ones of Cauchy-Schwarz, Holder, Schur, Chebyshev or Bernoulli is well prepared for almost everything to be found here. The student who is not that experienced will also be exposed in the first part to a wide combination of moderate and easy problems, ideas, techniques, and all the ingredients leading to a good preparation for mathematical contests. Some of the problems we chose to discuss are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities. Nevertheless, the book develops many results which are rarely seen, and even experienced readers are likely to find material that is challenging and informative.
To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions. Still, as George Polya and the others have taught us, there are principles of problem solving. With practice and good coaching we can all improve our skills. Just like singers, actors, or pianists, we have a path toward a deeper mastery of our craft. Află mai mult
Culegerea se adreseaza elevilor talentati si pasionati de "problem solving" punand la indemana acestora un valoros material. Subiectele date la etapa locala, judeteana, nationala sau la concursurile de matematica sunt insotite de solutii detaliate. Lucrarea se inscrie in colectia "Olimpiade", binecunoscuta in lista publicatiilor editurii GIL, dar este pentru prima data cand autorii sunt toti studenti la Facultatea de Matematica si Informatica din cadrul Universitatii Bucuresti. Află mai mult