Problems for the Mathematical Olympiads (From the First Team Selection Test to the IMO)

160 Pages·2005·5.913 MB·English

by  Andrei Negut

ANDREI NEGUT
'
Problems
For the Mathematical
Ol~mpiads
F"ROM 1+1£ FfRVI IEAM 5£L£CllON 1£51
10 1+1£ IMO
ANDREINEGUT
PROBLEMS
FOR THE MATHEMATICAL
OLYMPIADS
FROM THE FIRST TEAM SELECTION
TEST TO THE IMO
GIL Publishing House
© GIL Publishing House
Problems for the Mathematical Olympiads-From the first Team
Selection Test to the IMO
Author: Andrei Negu(
ISBN 973-9417-52-3
Copyright © 2005 by GIL. All rights reserved.
National Library of Romania CIP Description
NEGUT,ANDREI
Problems for the Mathematical
Olympiads-From the first Team Selection Test to
the IMO I Andrei Negut -Zalau: GIL, 2005
p.;cm.
Bibliogr.
ISBN 973-9417-52-3
51(075.35)(079.1)
GIL Publishing House
P.O. Box 44, Post Office 3, 450200, Zalau, Romania,
tel. (+ 40) 0260/616314
fax.: (+ 40) 0260/616414
e-mail: [email protected]
www.gil.ro
To my parents Livia and Nicu, and my brother Radu

Contents
FOREWORD ....................................................................... 7
I. Problems
Chapter 1. GEOMETRY ........................................................ 9
Chapter 2. NUMBER THEORY ............................................... 17
Chapter 3. COMBINATORICS ................................................ 23
Chapter 4. ALGEBRA ......................................................... 31
II. Solutions
Chapter 1. GEOMETRY ....................................................... 39
Chapter 2. NUMBER THEORY ............................................... 75
Chapter 3. COMBINATORICS ................................................ 95
Chapter 4. ALGEBRA ........................................................ 121
III. APPENDIX 1: USEFUL FACTS ............................................. 145
IV. APPENDIX 2: SOURCES OF PROBLEMS .................................. 158

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 !mow not so much theory as tricks to be employed in
elementaiy problems. It is far less useful as far as IMO's are concerned (and far more
difficult) for a student to learn multi variable calculus and Lagrange multipliers than to
Foreword
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 Negut
Chapter 1
GEOMETRY