Mathematical Olympiads 2000-2001: Problems and Solutions from around the World

271 Pages·2003·1.11 MB·English

by  Titu Andreescu

Mathematical
Olympiads
2000–2001
Problems and Solutions
From Around the World
Copyright Information
Mathematical
Olympiads
2000–2001
Problems and Solutions
From Around the World
Edited by
Titu Andreescu,
Zuming Feng,
and
George Lee, Jr.
Published and distributed by
The Mathematical Association of America
MAA PROBLEM BOOKS
SERIES INFORMATION
Contents
Preface ..........................................................ix
Acknowledgments ...............................................xi
1 2000 National Contests: Problems and Solutions 1
1.1 Belarus ..................................................2
1.2 Bulgaria ................................................12
1.3 Canada .................................................24
1.4 China ...................................................27
1.5 Czech and Slovak Republics .............................34
1.6 Estonia .................................................39
1.7 Hungary ................................................44
1.8 India ...................................................50
1.9 Iran ....................................................53
1.10 Israel ...................................................62
1.11 Italy ....................................................65
1.12 Japan ...................................................69
1.13 Korea ...................................................74
1.14 Mongolia ...............................................78
1.15 Poland ..................................................85
1.16 Romania ................................................91
1.17 Russia ..................................................99
1.18 Taiwan ................................................134
1.19 Turkey ................................................139
v
vi
1.20 United Kingdom .......................................147
1.21 United States of America ..............................149
1.22 Vietnam ...............................................157
2 2000 Regional Contests: Problems and Solutions
163
2.1 Asian Pacific Mathematical Olympiad .................164
2.2 Austrian-Polish Mathematics Competition .............170
2.3 Balkan Mathematical Olympiad .......................175
2.4 Mediterranean Mathematical Olympiad ................179
2.5 St. Petersburg City Mathematical Olympiad (Russia) ..182
3 2001 National Contests: Problems ............207
3.1 Belarus ................................................208
3.2 Bulgaria ...............................................210
3.3 Canada ................................................212
3.4 China .................................................213
3.5 Czech and Slovak Republics ............................215
3.6 Hungary ...............................................216
3.7 India ..................................................217
3.8 Iran ...................................................218
3.9 Japan .................................................220
3.10 Korea .................................................221
3.11 Poland ................................................223
3.12 Romania ..............................................224
3.13 Russia .................................................226
3.14 Taiwan ................................................232
3.15 United States of America ..............................233
3.16 Vietnam ...............................................234
4 2001 Regional Contests: Problems ............235
4.1 Asian Pacific Mathematical Olympiad .................236
4.2 Austrian-Polish Mathematics Competition .............237
4.3 Balkan Mathematical Olympiad .......................238
vii
4.4 Baltic Mathematics Competition .......................239
4.5 Czech-Slovak-Polish Match ............................240
Glossary .......................................................241
Classification of Problems ......................................256

Preface
This book is a continuation of Mathematical Olympiads 1999–2000:
Problems and Solutions From Around the World, published by the
Mathematical Association of America. It contains solutions to the
problems from 27 national and regional contests featured in the
earlierbook,togetherwithselectedproblems(withoutsolutions)from
national and regional contests given during 2001. In many cases
multiple solutions are provided in order to encourage students to
compare different problem-solving strategies.
This collection is intended as practice for the serious student who
wishestoimprovehisorherperformanceontheUSAMathOlympiad
(USAMO)andTeamSelectionTest(TST).Someoftheproblemsare
comparabletotheUSAMOinthattheycamefromnationalcontests.
Others are harder, as some countries first have a national Olympiad,
andlaterone or more exams to selecta team forthe IMO.Andsome
problems come from regional international contests (“mini-IMOs”).
Different nations have different mathematical cultures, so you will
find some of these problems extremely hard and some rather easy.
We have tried to present a wide variety of problems, especially from
those countries that have often done well at the IMO.
Each contest has its own time limit. We have not furnished this
information, because we have not always included complete exams.
Asaruleofthumb,mostcontestsallowtimerangingbetweenone-half
to one full hour per problem.
The problems themselves should provide much enjoyment for all
those fascinated by solving challenging mathematics questions.
ix
Acknowledgments
Thanks to the following participants of the Mathematical Olympiad
Summer Program who helped in preparing and proofreading so-
lutions: Reid Barton, Steve Byrnes, Gabriel Carroll, Kamaldeep
Gandhi, Stephen Guo, Luke Gustafson, Michael Hamburg, Daniel
Jerison, Daniel Kane, Kiran Kedlaya, Ian Le, Tiankai Liu, Po-Ru
Loh, Sean Markan, Alison Miller, Christopher Moore, Gregory Price,
MichaelRothenberg,InnaZakharevich,TonyZhang,andYanZhang.
Without their efforts this work would not have been possible.
Titu Andreescu Zuming Feng George Lee, Jr.
x