Problems for the mathematical olympiads: From the first team selection test to the IMO
by Andrei Neguț
Copyright: 2005
Pages: 158
ISBN-10: 9739417523
ISBN-13: 9789739417525
Decription (From back cover):
This is a text on elementary mathematics written by a very promising young
mathematician who is at his first experience of this kind. The author, Andrei
Negut, now a student at Princeton, was, for years, the winner of high-school
national and International Mathematical competitions.
This is an old dream of his, to gather and comment some of the most beautiful
problems in mathematics he was thinking about during his intense work and
learning.
The result is this wonderful book that contains beside a list of problems in
classical fields of mathematics (algebra, geometry, combinatorics) that
Andrei loved the most, a lot of original and sometimes even wonderful
solutions. The text is well written. The explanations are complete and
proving a deep intuition of the author.
I recommend the book to anyone interested professionally or as a hobby in
problem solving. Students preparing for mathematical competitions will find
a good training material.
I must confess that the feeling that the book represents the beginning of an
important career in mathematics, was anytime present when reading the text.
Radu Gologan
University "Politehnica" Bucharest and Institute of Mathematics
Coordinator of the Romanian Mathematical Olympiads professors.
FOREWORD:
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.
There are many such collections of problems that are directed at IMO participants. Yet in my view there are two things which individualize this book: the first of these is in the selected problems. These are some of the most beautiful problems I encountered in my four-year preparation for such high-level math competitions, and they have been presented to me by many great professors and instructors over the years. They are neither boring nor tedious, but require a certain amount of insight and ingenuity, which I find to be the necessary quality of any 'beautiful' math problem. Moreover, I have tried not to add here very well-known problems (such as problems from past IMO's or from other important contests), as these very likely become known to every student in the first year of his Olympiad career. Instead, there is a smaller chance that the reader might already know the problems presented here, and the basis of any good preparation is to work as many new problems as possible. Any one of these problems could be given at an IMO, and I hope that this book might help them come to light.
I have branded each problem with one of 3 degrees of difficulty: E for easy, M for medium and D for difficult (the reader can find the category of a problem at the beginning of its solution). But these are just relative, as an E problem is of the level of a problem 1 at an IMO, an M problem is similar to what one should expect from problem 2 and D could be a problem 3. Therefore, a novice in the Olympiad world should not feel frustrated if he has trouble with an E problem, because on an absolute scale it can be quite difficult. Such problems are far beyond the level of regular school work.
The second thing 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. I have also presented a few of the concepts I have employed throughout the book in an appendix. Thus I can state that in my opinion, a potential IMO participant needs two things: ingenuity on one hand and a firm grasp on all these 'toys' and tricks on the other. Which of them is more important, I do not know yet, and I can only guess.
I would like to thank the people who invented these wonderful problems. While most of the solutions are my own work, the problems are not mine and I am in great debt to their creators. Since these problems were mainly taken from my notes and papers, their exact origin is unknown to me, and I have replaced the name of their authors by the symbols ***. It is not a fitting homage, and I apologize for this.
But because I have encountered these problems during my own preparation as an IMO participant, they have also become a part of me. Each one of them is associated to the person who showed it to me, the friends who told me their beautiful solutions, or the contests in which I have or have not solved them. I would like to thank all of the professors who helped me and who made me into what I am, and though I can't name them all, it is people like Radu Gologan, Severius Moldoveanu, Dorela Fainisi, Dan Schwartz, Calin Popescu, Mihai Baluna, Bogdan Enescu, Dinu Serbanescu and Mircea Becheanu who have shown me the most beautiful and subtle art there is. I will also not forget all the comrades and friends that have passed through the Olympiad experience with me, but they will forgive me for not naming them. They know who they are. I also cannot forget my family, who has stood by me and given me that priceless moral support which is indifferent of how well or how badly I behaved in the competitions.
I would like to thank Mircea Lascu and the Gil Publishing House for supporting me and this book on the long journey to publishing, and Prof. Radu Gologan for a great deal of help and useful advice. I also want to thank Gabriel Kreindler, Andrei Stefanescu, Andrei Ungureanu and Adrian Zahariuc for providing some of the solutions present in this book.
I wish you the best of luck in all your endeavors, mathematical or not.
Andrei Neguț
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