The present book is a collection of all problems proposed at Balkan Mathematical Olympiads (BMO), also containing some of the problems discussed during these contests by the jury. It is not only a collection of elementary problems but a page of history - the history of BMO, one of the first reginoal mathematical competition for highschool students.
As it is already accepted, International Mathematical Olympiads (IMO), which was created in 1959, had an important contribution to development of mathematical education of young students all around the world. They extended the area of problems used in mathematical education of young people and also improved their quality. Many mathematicians realized how important is to use some parts of their research problems as elementary educational problems.
All problems are presented with complete solutions. Many problems have several alternative solutions and we also present some extensions. The reader can follow the real increasing of the quality of problems as an illustration of the improvement of mathematical preparation of students. An additional preparatory addendum, containing notions and classical useful results has been added to the end of the book. Află mai mult
This book is intended to help students preparing for all rounds of Mathematical Olympiads or any other significant mathematics contest. Teachers will also find this work useful in training young talented students. Our experience as contestants was a great asset in preparing this book. To this we added our vast personal experience from the other side of the "barricade", as creators of problems and members of numerous contest committees.
The book is organized in six chapters: algebra, number theory, geometry, trigonometry, analysis and comprehensive problems. In addition, other fields of mathematics found their place in this book, for example, combinatorial problems can be found in the last chapter, and problems involving complex numbers are included in the trigonometry section. Moreover, in all chapters of this book the serious reader can find numerous challenging inequality problems. All featured problems are interesting, with an increased level of difficulty; some of them are real gems that will give great satisfaction to any math lover attempting to solve or even extend them. 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
This book consists of a number of math problems, all of which are meant primarily as preparation for competitions such as the International Mathematical Olympiad. They are therefore of IMO level, and require only elementary notions of math; however, since the International Mathematical Olympiad is perhaps the most difficult exam in elementary mathematics, any participant should have with him a good knowledge and grasp of what he is dealing with. This book is not meant to teach elementary math at an IMO level, but to help a prospective participant train and enhance his understanding of these concepts.
The second this that is important about this book is the solutions. Any good participant at an IMO needs to know not so much theory as tricks to be employed in elementary problems. It is far less useful as far as IMO's are concerned (and far more difficult) for a student to learn multivariable calculus and Lagrange multipliers than to know how to apply geometrical inversion. That is why I emphasize on all these methods, lemmas and propositions in my solutions, and I have often sacrificed succinctness of a proof to the educational value of presenting one of 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.