As you have seen in "Secrets in Inequalities. Volume 1" and also from the title, this volume is about advanced inequalities. The main idea which let us start this project was a book about inequalities. There are many, we have known but we want them organized in a different way. This book will be very well understood by those who have already read Volume 1, and more than this it will be the continuation of the nominated book just like the second part of a trip, a trip in the world of inequalities.
Remarkably, the method of using classical inequalities is left behind the first four methods, because we realize that you can flexibly use this method only after you have already developed a larger horizon in the field of inequalities; and this is true: a solid background in a field is usually needed to find the simplest and the most natural solution to a problem. You will find here good modern approaches to prove inequalities: the mixing variables method (in its general and special forms), the method of analyzing squares, the contradiction method, the method of induction (and again, the above mentioned method of using classical inequalities). As you are already used from previous parts, you have lots of beautiful and hard problems to exercise your skills in using all these methods. 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.
This work blends together classic inequality results with brand new problems, some of which devised only a few days ago. What could be special about it when so many inequality problem books have already been written? We strongly believe that even if the topic we plunge into is so general and popular our book is very different. Of course, it is quite easy to say this, so we will give some supporting arguments. This book contains a large variety of problems involving inequalities, most of them difficult, questions that became famous in competitions because of their beauty and difficulty.
Some of the problems we chose to present are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities. Anyone will find here a challenge to prove his or her skills. If we have not convinced you, then please take a look at the last problems and hopefully you will agree with us. 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
It is true that there are very many books on inequalities and you have all the right to be bored and tired of them. But we tell you that this is not the case with this one. Just read the proof of Nesbitt's Inequality in the very beginning of the material, and you will understand exactly what we mean.
Every topic is described through various and numerous examples taken from many sources, especially from math contests around the world, from recent contests and recent books, of from (more or less) specialized sites on the Internet, which makes the book very lively and interesting to read for those who are involved in such activities, students and teachers from all over the world.
Don't let the problems overwhelm you, though they are quite impressive problems, study applications of the first five basic inequalities mentioned above, plus the Abel formula, symmetric inequalities and the derivative method. Now relax with the AM-GM inequality - the foundational brick of inequalities. Află mai mult
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