The aim of this book is to present some ideas, methods and topics in elementary combinatorial geometry. Even if most of the book can be understood without any mathematical background, so it is accessible for 12-14 years old children too, we recommend it to high school students and college students as an introduction to a few topics in combinatorial (or convex) geometry (counting problems, pigeonhole principle, Helly's theorem, Sperner lemma). Our approach is basically an elementary one, but it is useful to know some combinatorial techniques and some basic notions. These notions appear in the subject index, so they can be identified easily.
The main purpose was not to create an exhaustive collection, but to offer a quick and short overview, to present a few properties which have surprising applications also in higher mathematics (the Sperner lemma) and to show some interesting ways of generalizations. So this book wants to be a kind of bridge between elementary problems and university courses. Află mai mult
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
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