Иностранная литература, посвященная олимпиадной тематике

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Иностранная литература, посвященная олимпиадной тематике

Большинство книг можно найти на сайте gen.lib.rus.ec.

• Andreescu T., Kedlaya K., Zeitz P. Mathematical contests 1995-1996: Olympiad problems and solutions from around the world - MAA, 1997, 202 p.
• Andreescu T., Kedlaya K. Mathematical olympiads 1996-1997: Olympiad problems and solutions from around the world - MAA, 1998, 182 p.
• Andreescu T., Kedlaya K. Mathematical Olympiads 1997-1998: Problems and Solutions from Around the World - AMC, 1999
• Andreescu T., Cirtoaje V., Dospinescu G., Lascu M. Old and New Inequalities - GIL, 2000, 125 p.
• Andreescu T., Feng Z. Mathematical Olympiads 1999-2000: Problems and Solutions from around the World - MAA, 2001, 344 p.
• Andreescu T., Feng Z., Lee Jr G. Mathematical Olympiads 2000-2001: Problems and Solutions from around the World - MAA, 2002, 271 p.
• Andreescu T., Feng Z. 102 Combinatorial Problems from the training of USA IMO team - Birkhäuser, 2003, p. 115
• Andreescu T., Andrica D. 360 Problems for Mathematical Contests - 2003, GIL Publishing house, 2003, 280 p.
• Andreescu T., Feng Z. USA & International Mathematical Olympiads 2003 - MAA, 2004, 101 p.
• Andreescu T., Feng Z. A path to combinatorics for undergraduates: counting strategies - Birkhäuser, 2004, 228 p
• Andreescu T., Mushkarov O., Stoyanov L. Geometric Problems on Maxima and Minima - Birkhauser, 2005
• Andreescu T., Feng Z. 103 Trigonometry Problems. From the Training of the USA IMO Team - Birkhäuser, 2005 - 214 pp
• Andreescu T., Feng Z., Loh P. USA and International Mathematical Olympiads 2004 - MAA, 2005, 61 p.
• Andreescu T., Andrica D., Feng Z. 104 Number Theory Problems. From the Training of the USA IMO Team - Birkhäuser, 2007 - 204 pp
• Andreescu T., Boreico I. Functional Equations (Electronic Edition) - 2007, 140 p.
• Andreescu T., Gelca R. Putnam and Beyond - Springer, 2007, 815 p.
• Andreescu T., Gelca R. Mathematical Olympiad Challenges (2nd) - Birkhäuser, 2008, 291 p.
• Andreescu T., Dospinescu G. Problems from the book [2nd Edition ed.] - XYZ Press, 2008, 554 p.
• Andreescu T., Andrica D. Number theory: Structures, examples, and problems - Birkhäuser, 2009, 412 p.
• Andreescu T., Andrica D., Cucurezeanu I. An introduction to Diophantine equations: A problem-based approach - Birkhauser, 2010, 345 p.
• Andreescu T., Enescu B. Mathematical Olympiad Treasures (2nd) - Birkhäuser, 2011, 262 p.

• Djukic D., Jankovic V., Matic I., Petrovic N. The Imo Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004 [1 ed.] (Problem Books in Mathematics) - Springer, 2006, 755 pp
• Djukic D., Jankovic V., Matic I., Petrovic N. The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 [2 ed.] (Problem Books in Mathematics) - Springer, Berlin, 2011, 825 pp

• Gronau H.-D. 50th IMO - 50 Years of International Mathematical Olympiads - Springer, 2011, 241 pp

• Jiagu X. Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol. 1 (Mathematical Olympiad Series Vol. 6) - World Scientific Publishing Company, 2010, 184 pp,
• Jiagu X. Lecture Notes on Mathematical Olympiad Courses: For Junior Section, Vol. 2 (Mathematical Olympiad Series Vol. 6) - World Scientific Publishing Company, 2010, 192 pp,

• Liu A. Chinese Mathematics Competitions and Olympiads, Book 1: 1981-1993 (Enrichment Series, Volume 13) - AMT Publishing, 1998, 194 pp
• Liu A. Chinese Mathematics Competitions and Olympiads, Book 2: 1993-2001 (Enrichment Series, Volume 22) - AMT Publishing, 2005, 174 pp

• Manfrino R.B. Inequalities: A Mathematical Olympiad Approach - Birkhauser, 2009

• Posamentier A.S., Salkind C.T. Challenging Problems in Geometry - Dover Books on Mathematics, 1996
• Posamentier A.S., Salkind C.T. Challenging Problems in Algebra - Dover Books on Mathematics, 1996

• Sahoo P. K. , Kannappan P. Introduction to Functional Equations - Chapman and Hall/CRC, 2011, 465 Pages
• Slinko A. USSR Mathematical Olympiads 1989-1992 (Enrichment Series, Volume 11) - 1997

• Xiong B., Lee P.Y. Mathematical Olympiad in China: Problems and Solutions - World Scientific Publishing Company, 2007, 276 pp
• Xiong B., Lee P.Y. Mathematical Olympiad in China (2007-2008): Problems and Solutions - World Scientific Publishing Company, 2009, 194 pp

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Дополнительно
КНИЖНЫЕ ПОЛКИ СООБЩЕСТВА EEK.DIARY, в том числе:
Литература по подготовке к математическим олимпиадам (часть I)
Литература по подготовке к математическим олимпиадам (часть II)

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Топики, посвященные олимпиадам разных стран:
Математическая олимпиада Аргентины
Математические соревнования в Германии
Греческая математическая олимпиада
Конкурс Georg Mohr (Дания)
Всеукраинская математическая олимпиада
Математический конкурс Франции
Испанская математическая олимпиада
Математическая олимпиада в Италии
Математическая олимпиада в Хорватии
Шведские математические соревнования
Польская математическая олимпиада
Математическая олимпиада Чехии и Словакии
Математические состязания в Бельгии
Математические олимпиады в Бангладеш
Математическая олимпиада Мексики
Олимпиада Португалии по математике
Scottish Mathematical Challenge
Ирландская математическая олимпиада
Niels Henrik Abel Mathematics Contest (Норвегия)
Математическая олимпиада в Голландии
Математическая олимпиада в Румынии
Математическая олимпиада в Сербии
Математические соревнования в Венгрии
Математическая олимпиада в Сирии
Математическая олимпиада в Великобритании
Олимпиада в Перу
Таиландская олимпиада
Математическая олимпиада в Бразилии

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[mpl]

23.09.2010 в 02:54

Пишет  Robot:

Mathematical Olympiad in China

Chinese Mathematics Competitions and Olympiads, Book 1: 1981-1993 (Enrichment Series, Volume 13)
By Andy Liu
Publisher: AMT Publishing
Number Of Pages: 194
Publication Date: 1998
ISBN-10 / ASIN: 1876420006
ISBN-13 / EAN: 9781876420000
Product Description:
This book contains the problems and solutions of two contests: the Chinese National High School Competition from 1981-82 to 1992-93, and the Chinese Mathematical Olympiad from 198586 to 199293. China has an outstanding record in the International Mathematical Olympiad, and the book contains the problems which were used to identify the team candidates and select the Chinese teams. The problems are meticulously constructed, many with distinctive flavour. They come in all levels of difficulty, from the relatively basic to the most challenging.
Скачать Djvu, 400dpi, seacharble, single pages, cover, backcover. Size 2.1 mb rar'd
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Chinese Mathematics Competitions and Olympiads, Book 2: 1993-2001 (Enrichment Series, Volume 22)
By Andy Liu
Publisher: AMT Publishing
Number Of Pages: 174
Publication Date: 2005
ISBN-10 / ASIN: 1876420162
ISBN-13 / EAN: 9781876420161
Product Description:
This book is a continuation of the earlier volume (1981-1993) and covers the years 1993 to 2001. China has an outstanding record in the International Mathematical Olympiad, and the book contains the problems which were used to identify the team candidates and select the Chinese teams. The problems are meticulously constructed, many with distinctive flavour. They come in all levels of difficulty, from the relatively basic to the most challenging.
Скачать DjVu, 400dpi, searchable, single pages, cover (color), backcover (color). Size 2.11mb rar'd mediafire.com || ifolder.ru

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23.09.2010 в 02:54

Пишет  Robot:

Mathematical Olympiad in China: Problems and Solutions
By Bin Xiong, Yee Lee Peng
Publisher: World Scientific Publishing Company
Number Of Pages: 276
Publication Date: 2007-06-20
ISBN-10 / ASIN: 9812707891
ISBN-13 / EAN: 9789812707895
Binding: Paperback
Product Description:
The International Mathematical Olympiad (IMO) is a competition for high school students. China has taken part in IMO twenty times since 1985 and has won the top ranking for countries thirteen times, with a multitude of golds for individual students. The 6 students China sent every year were selected from 20 to 30 students among approximately 130 students who take part in the China Mathematical Competition during the winter months. This volume comprises a collection of original problems with solutions that China used to train their Olympiad team in the years from 2003 to 2006.
Скачать Djvu RAR OCR & bookmarks 1.97 MB mediafire.com || ifolder.ru
Скачать pdf 9812707891.zip (7.52 MB) mediafire.com || ifolder.ru

Mathematical Olympiad in China (2007-2008): Problems and Solutions
by: Xiong Bin, Lee Peng Yee
en | World Scientific Publishing Company
Number Of Pages: 194
Publication Date: 2009-05-21
ISBN-10 / ASIN: 9814261149
ISBN-13 / EAN: 9789814261142
Product Description:
The International Mathematical Olympiad (IMO) is a competition for high school students. China has taken part in the IMO 21 times since 1985 and has won the top ranking for countries 14 times, with a multitude of golds for individual students. The six students China has sent every year were selected from 20 to 30 students among approximately 130 students who took part in the annual China Mathematical Competition during the winter months. This volume comprises a collection of original problems with solutions that China used to train their Olympiad team in the years from 2006 to 2008. Mathematical Olympiad problems with solutions for the years 2002 2006 appear in an earlier volume, Mathematical Olympiad in China.
Скачать Djvu& 600dpi+OCR & bookmarks 1.64 MB mediafire.com || ifolder.ru
Скачать ocr'ed optimized pdf, bookmark'ed+paginated 2,2 мб mediafire.com || ifolder.ru

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28.09.2010 в 19:55

Пишет  pemac:

Задачник ММО - Международная математическая олимпиада

The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004 (Международная математическая олимпиада) The IMO Compendium is a book that contains all problems ever shortlisted for the International Mathematical Olympiad (IMO) with solutions, and all of the available longlisted problems. To the best of our knowledge, such a huge collection of IMO related problems has never been assembled before. Автори: Dusan Djukic, Vladimir Jankovic, Ivan Matic, Nikola Petrovic www.onlinedisk.ru/file/882015/

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24.01.2011 в 20:28

Пишет  pemac:

50th IMO - 50 Years of International Mathematical Olympiads Springer | 2011 | 241 pages | DJVU | 11 MB n July 2009 Germany hosted the 50th International Mathematical Olympiad (IMO). For the very first time the number of participating countries exceeded 100, with 104 countries from all continents. Celebrating the 50th anniversary of the IMO provides an ideal opportunity to look back over the past five decades and to review its development to become a worldwide event. This book is a report about the 50th IMO as well as the IMO history. A lot of data about all the 50 IMOs are included. We list the most successful contestants, the results of the 50 Olympiads and the 112 countries that have ever taken part. It is impressive to see that many of the world’s leading research mathematicians were among the most successful IMO participants in their youth. Six of them gave presentations at a special celebration: Bollobás, Gowers, Lovász, Smirnov, Tao and Yoccoz. This book is aimed at students in the IMO age group and all those who have interest in this worldwide leading competition for highschool students. Скачать http://ifolder.ru

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04.03.2012 в 20:38

Пишет  Robot:

Добавления на Книжные полки
Еще одна книга олимпиадной тематики, добавленная  kostyaknop

USSR Mathematical Olympiads 1989-1992 (Enrichment Series, Volume 11) by Arkadii M. Slinko (ENG) The All-Union Olympiad of the USSR, before the breakup of the USSR was perhaps the most famous of all National Mathematical. Olympiads. There was great competition among students even to compete, and the standard and range of questions was arguably higher than that found in the International Mathematical Olympiad. Arkadii Slinko was highly involved in these last few years of the event and in this book he not only provides all the questions, but also presents highly expository solutions. This book is a must for teachers and all students aspiring to the highest Olympiad Challenges. Сборник задач и решений Всесоюзных олимпиад 1989-1992 гг. на английском языке, выпущенный в Австралии. За книгу большое спасибо  kostyaknop Скачать (pdf/rar, 7.57 Мб) ifolder.ru || kknop.com

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08.02.2010 в 22:04

Пишет  Alidoro:

Олимпиадные задачи на английском языке

Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov Geometric Problems on Maxima and Minima, 9780817635176 (0817635173), Birkhauser, 2005 Оригинал с аннотацией: old.book.pdfchm.net/geometric-problems-on-maxim... Скачать без регистрации: ifolder.ru/16333574 или www.mediafire.com/?hsxo9tm31uecr6k

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08.02.2010 в 22:04

Пишет  Alidoro:

Олимпиадные задачи на английском языке

Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado Inequalities: A Mathematical Olympiad Approach 9783034600491 (3034600496), Birkhauser, 2009 Олимпиадные задачи с решениями, немного теории и упражнений. Оригинал с аннотацией здесь: book.pdfchm.net Отсюда книгу можно скачать без регистрации: ifolder.ru или mediafire.com PDF 1,5 Mb

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Posamentier A.S., Salkind C.T. Challenging Problems in Algebra - Dover Books on Mathematics, 1996 libgen.info

08.02.2010 в 22:04

Пишет  Alidoro:

Олимпиадные задачи на английском языке

Posamentier A.S., Salkind C.T. Challenging Problems in Geometry - Dover Books on Mathematics, 1996 Оригинал book.pdfchm.net Скачать без регистрации ifolder.ru или mediafire.com 7 Mb

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23.09.2010 в 02:54

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Mathematical Olympiad in China

Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol 1,2
Product Description:
Olympiad mathematics is not a collection of techniques of solving mathematical problems but a system for advancing mathematical education. This book is based on the lecture notes of the mathematical Olympiad training courses conducted by the author in Singapore. Its scope and depth not only covers and exceeds the usual syllabus, but introduces a variety concepts and methods in modern mathematics. In each lecture, the concepts, theories and methods are taken as the core. The examples are served to explain and enrich their intension and to indicate their applications. Besides, appropriate number of test questions is available for reader's practice and testing purpose. Their detailed solutions are also conveniently provided. The examples are not very complicated so that readers can easily understand. There are many real competition questions included which students can use to verify their abilities. These test questions are from many countries, e.g. China, Russia, USA, Singapore, etc. In particular, the reader can find many questions from China, if he is interested in understanding mathematical Olympiad in China. This book serves as a useful textbook of mathematical Olympiad courses, or as a reference book for related teachers and researchers.

Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol 1 (Mathematical Olympiad Series)
By Xu Jiagu
Publisher: World Scientific Publishing Company
Number Of Pages: 184
ISBN-10 / ASIN: 9814293547
ISBN-13 / EAN: 9789814293549
Product Description:
Operations on Rational Numbers
Linear Equations of Single Variable
Multiplication Formulae
Absolute Value and Its Applications
Congruence of Triangles
Similarity of Triangles
Divisions of Polynomials
Solutions to Testing Questions

Скачать vector searchable PDF, 1.019 MB, with cover, bookmarks and pagination
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Lecture Notes on Mathematical Olympiad Courses: For Junior Section, Vol. 2 (Mathematical Olympiad Series)
By Xu Jiagu
Publisher: World Scientific Publishing Company
Number Of Pages: 192
Publication Date: 2009-12-11
ISBN-10 / ASIN: 9814293555
ISBN-13 / EAN: 9789814293556
Product Description:
Congruence of Integers
Decimal Representation of Integers
Pigeonhole Principle
Linear Inequality and System of Linear Inequalities
Inequalities with Absolute Values
Geometric Inequalities
Solutions to Testing Questions
Скачать vector searchable PDF, 1.024 MB , with cover, bookmarks and pagination
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Andreescu T., Kedlaya K., Zeitz P. Mathematical contests 1995-1996: Olympiad problems and solutions from around the world - MAA, 1997, 202 p. This book is a supplement to Mathematical Olympiads 1995: Olympiad Problems from Around the World, published by the American Mathematics Competitions. It contains solutions to the problems from 28 national and regional contests featured in the earlier pamphlet, together with selected problems (without solutions) from national and regional contests given during the academic year 1995-1996. Скачать (djvu, 1 Mb) libgen.info

Contents
1 1995 Contests: Problems and Solutions 3
1.1 Austria 3
1.2 Bulgaria 6
1.3 China 13
1.4 Czech and Slovak Republics 19
1.5 France 23
1.6 Germany 27
1.7 Greece 31
1.8 Hungary 34
1.9 India 42
1.10 Iran 48
1.11 Ireland 54
1.12 Israel 60
1.13 Japan 65
1.14 Korea 70
1.15 Poland 79
1.16 Romania 83
1.17 Russia 96
1.18 Singapore 117
1.19 Taiwan 120
1.20 Turkey 124
1.21 United Kingdom 128
1.22 United States of America 132
1.23 Vietnam 138
1.24 Austrian-Polish Mathematics Competition 142
1.25 Balkan Mathematical Olympiad 149
1.26 Czech-Slovak Match 152
1.27 Iberoamerican Olympiad 156
1.28 UNESCO 160
2 1996 National Contests: Problems 163
2.1 Bulgaria 163
2.2 Canada 165
2.3 China 166
2.4 Czech and Slovak Republics 167
2.5 France 168
2.6 Germany 169
2.7 Greece 170
2.8 Iran 171
2.9 Ireland 173
2.10 Italy 175
2.11 Japan 176
2.12 Poland 177
2.13 Romania 178
2.14 Russia 181
2.15 Spain 184
2.16 Turkey 185
2.17 United Kingdom 186
2.18 United States of America 188
2.19 Vietnam 189
3 1996 Regional Contests: Problems 190
3.1 Asian Pacific Mathematics Olympiad 190
3.2 Austrian-Polish Mathematics Competition 191
3.3 Balkan Mathematical Olympiad 193
3.4 Czech-Slovak Match 194
3.5 Iberoamerican Olympiad 195
3.6 St. Petersburg City Mathematical Olympiad 197

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Andreescu T., Kedlaya K. Mathematical olympiads 1996-1997: Olympiad problems and solutions from around the world - MAA, 1998, 182 p. This book is a continuation Mathematical Olympiads 1995-1996: Olympiad Problems from Around the World, published by the American Mathematics Competitions. It contains solutions to the problems from 25 national and regional contests featured in the earlier pamphlet, together with selected problems (without solutions) from national and regional contests given during 1997. Скачать (pdf, 0.6 Mb) libgen.info

Contents
1 1996 National Contests: Problems and Solutions 3
1.1 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 17
1.5 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.9 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.10 Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.11 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.12 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.13 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.14 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.15 Spain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.16 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1.17 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 84
1.18 United States of America . . . . . . . . . . . . . . . . . . . 89
1.19 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2 1996 Regional Contests: Problems and Solutions 100
2.1 Asian Pacic Mathematics Olympiad . . . . . . . . . . . . . 100
2.2 Austrian-Polish Mathematics Competition . . . . . . . . . . 103
2.3 Balkan Mathematical Olympiad . . . . . . . . . . . . . . . . 108
2.4 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 110
2.5 Iberoamerican Olympiad . . . . . . . . . . . . . . . . . . . . 114
2.6 St. Petersburg City Mathematical Olympiad . . . . . . . . 118
3 1997 National Contests: Problems 131
3.1 Austria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.3 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.5 Colombia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.6 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 140
3.7 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.8 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.9 Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.10 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.11 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.12 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.13 Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.14 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.15 Korea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.16 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.17 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.18 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.19 South Africa . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.20 Spain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.21 Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.22 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.23 Ukraine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
3.24 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 167
3.25 United States of America . . . . . . . . . . . . . . . . . . . 168
3.26 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4 1997 Regional Contests: Problems 170
4.1 Asian Pacic Mathematics Olympiad . . . . . . . . . . . . . 170
4.2 Austrian-Polish Mathematical Competition . . . . . . . . . 171
4.3 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 173
4.4 Hungary-Israel Mathematics Competition . . . . . . . . . . 174
4.5 Iberoamerican Mathematical Olympiad . . . . . . . . . . . 175
4.6 Nordic Mathematical Contest . . . . . . . . . . . . . . . . . 177
4.7 Rio Plata Mathematical Olympiad . . . . . . . . . . . . . . 178
4.8 St. Petersburg City Mathematical Olympiad (Russia) . . . 179

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Andreescu T., Kedlaya K. Mathematical Olympiads 1997-1998: Problems and Solutions from Around the World - AMC, 1999 This book is a continuation of Mathematical Olympiads 1996-1997: Olympiad Problems from Around the World, published by the American Mathematics Competitions. It contains solutions to the problems from 34 national and regional contests featured in the earlier book, together with selected problems (without solutions) from national and regional contests given during 1998. Скачать (pdf, 0.8 Mb) libgen.info

Contents
1 1997 National Contests: Solutions 3
1.1 Austria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 Colombia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 34
1.7 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.8 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.9 Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.10 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.11 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.12 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.13 Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.14 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.15 Korea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.16 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1.17 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.18 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.19 South Africa . . . . . . . . . . . . . . . . . . . . . . . . . . 105
1.20 Spain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1.21 Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.22 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.23 Ukraine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1.24 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 127
1.25 United States of America . . . . . . . . . . . . . . . . . . . 130
1.26 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2 1997 Regional Contests: Solutions 141
2.1 Asian Pacic Mathematics Olympiad . . . . . . . . . . . . . 141
2.2 Austrian-Polish Mathematical Competition . . . . . . . . . 145
2.3 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 149
2.4 Hungary-Israel Mathematics Competition . . . . . . . . . . 153
2.5 Iberoamerican Mathematical Olympiad . . . . . . . . . . . 156
2.6 Nordic Mathematical Contest . . . . . . . . . . . . . . . . . 161
2.7 Rio Plata Mathematical Olympiad . . . . . . . . . . . . . . 163
2.8 St. Petersburg City Mathematical Olympiad (Russia) . . . 166
3 1998 National Contests: Problems 180
3.1 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.2 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.3 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.4 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 185
3.5 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3.6 India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.7 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
3.8 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3.9 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.10 Korea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.11 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.12 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.13 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
3.14 Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.15 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
3.16 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 209
3.17 United States of America . . . . . . . . . . . . . . . . . . . 211
3.18 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4 1998 Regional Contests: Problems 213
4.1 Asian Pacic Mathematics Olympiad . . . . . . . . . . . . . 213
4.2 Austrian-Polish Mathematics Competition . . . . . . . . . . 214
4.3 Balkan Mathematical Olympiad . . . . . . . . . . . . . . . . 216
4.4 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 217
4.5 Iberoamerican Olympiad . . . . . . . . . . . . . . . . . . . . 218
4.6 Nordic Mathematical Contest . . . . . . . . . . . . . . . . . 219
4.7 St. Petersburg City Mathematical Olympiad (Russia) . . . 220

[mpl]

Andreescu T., Cirtoaje V., Dospinescu G., Lascu M. Old and New Inequalities - GIL, 2000, 125 p. This work blends together classic inequality results with brand new problems, some of which devised only a few days ago. What could be special about it when so many inequality problem books have already been written? We strongly believe that even if the topic we plunge into is so general and popular our book is very different. Of course, it is quite easy to say this, so we will give some supporting arguments. This book contains a large variety of problems involving inequalities, most of them difficult, questions that became famous in competitions because of their beauty and difficulty. Скачать (pdf, Мб) libgen.info

Contents
Preface 3
Chapter 1. Problems 7
Chapter 2. Solutions 25
Glossary 121
Further reading 127

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Andreescu T., Feng Z. Mathematical Olympiads 1999-2000: Problems and Solutions from around the World - MAA, 2001, 344 p. Скачать

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Andreescu T., Feng Z., Lee Jr G. Mathematical Olympiads 2000-2001: Problems and Solutions from around the World - MAA, 2002, 271 p. Скачать

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
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
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 earlier book, together with selected problems (without solutions) 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 wishes to improve his or her performance on the USA Math Olympiad (USAMO) and Team Selection Test (TST). Some of the problems are comparable to the USAMO in that they came from national contests.
Others are harder, as some countries first have a national Olympiad, and later one or more exams to select a team for the IMO. And some 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.
As a rule of thumb, most contests allow time ranging between one-half to one full hour per problem.
The problems themselves should provide much enjoyment for all those fascinated by solving challenging mathematics questions.

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Andreescu T., Feng Z. 102 Combinatorial Problems from the training of USA IMO team - Birkhäuser, 2003, p. 115 Скачать

Preface
This book contains one hundred and two highly selected problems used in the training and testing of the USA International Mathematical Olympiad (IMO) team. It is not a collection of very difficult, impenetrable questions. Instead, the book gradually builds students combinatorial skills and techniqties. This work aims to broaden students view of mathematics and better prepare them for possible participation in various mathcmatical competitions. It provides in depth enrichment in important areas of combinatorics by reorganizing and enhancing students problem solving tactics and strategies. The book further stimulates students interest for future study of mathematics.

Contents
♦ PREFACE.............................................iii
♦ INTRODUCTION ......................................v
♦ ACKNOWLEDGEMENTS ...........................viii
♦ ABBREVIATIONS AND NOTATIONS ................x
♦ 1. INTRODUCTORY PROBLEMS .....................1
♦ 2. ADVANCED PROBLEMS .........................11
♦ 3. SOLUTIONS TO INTRODUCTORY PROBLEMS .22
♦ 4. SOLUTIONS TO ADVANCED PROBLEMS ......
♦ GLOSSARY .........................................128
♦ FURTHER READING

[mpl]

Andreescu T., Andrica D. 360 Problems for Mathematical Contests - 2003, GIL Publishing house, 2003, 280 p. Скачать

Contents
FOREWORD ....................................................................3
FROM THE AUTHORS .........................................................5
Chapter 1. ALGEBRA .........................................................7
Problems ....................................................................®
Solutions ...................................................................17
Chapter 2. NUMBER THEORY ...............................................47
Problems ...................................................................49
Solutions ...................................................................57
Chapter 3. GEOMETRY ......................................................85
Problems ...................................................................®7
Solutions ...................................................................
Chapter 4. TRIGONOMETRY ...............................................137
Problems....................i.............................................139
Solutions ..................................................................147
Chapter 5. MATHEMATICAL ANALYSIS ...................................179
Problems ..................................................................181
Solutions ..................................................................
Chapter 6. COMPREHENSIVE PROBLEMS ................................221
Problems ..................................................................223
Solutions ..................................................................235

FROM THE AUTHORS
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.
All the featured problems are supposed to be original. They are the fruit of our collaboration for the last 30 years with several elementary mathematics journals from all over the world. Many of these problems were used in contests throughout these years, from the first round to the international level. It is possible that some problems are already known, but this is not critical. The important thing is that an educated - to a certain extent - reader will find in this book problems that bring something new and will teach new ways of dealing with key mathematics concepts, a variety of methods, tactics, and strategies.
The problems are divided in chapters, although this division is not firm, for some of the problems require background in several fields of mathematics.
Besides the traditional fields; algebra, geometry, trigonometry and analysis, we devoted an entire chapter to number theory, because many contest problems require knowledge in this field.
The comprehensive problems in the last chapter are also intended to help undergraduate students participating in mathematics contests hone their problem solving skills. Students and teachers can find here ideas and questions that can be interesting topics for mathematics circles.
Due to the difficulty level of the problems contained in this book, we deemed it appropriate to give a very clear and complete presentation of all solutions. In many cases, alternative solutions are provided.
As a piece of advice to all readers, we suggest that they try to find their own solutions to the problems before reading the given ones. Many problems can be solved in multiple ways and pertain to Interesting extensions.
This edition is significantly different from the 2002 Romanian edition. It features more recent problems, enhanced solutions, along with references for all published problems.
We wish to extend our gratitude to everyone who influenced in one way or another the final version of this book.
We will gladly receive any observation from the readers.

[mpl]

Andreescu T., Feng Z. USA & International Mathematical Olympiads 2003 - MAA, 2004, 101 p. Скачать

Contents
Preface ix
Acknowledgments xi
Abbreviations and Notations xiii
Introduction xv
1 TheProblems 1
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 TeamSelection Test . . . . . . . . . . . . . . . . . . . . . 3
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Hints 7
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 TeamSelection Test . . . . . . . . . . . . . . . . . . . . . 8
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Formal Solutions 11
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 TeamSelection Test . . . . . . . . . . . . . . . . . . . . . 32
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 ProblemCredits 61
5 Glossary 63
6 FurtherReading 69
7 Appendix 75
1 2002 Olympiad Results . . . . . . . . . . . . . . . . . . . 75
2 2002 Olympiad Results . . . . . . . . . . . . . . . . . . . 77
3 2001 Olympiad Results . . . . . . . . . . . . . . . . . . . 79
4 2000 Olympiad Results . . . . . . . . . . . . . . . . . . . 80
5 1999 Olympiad Results . . . . . . . . . . . . . . . . . . . 82
6 1999–2003 Cumulative IMO Results . . . . . . . . . . . 83
About the Authors 85

Preface
This book is intended to help students preparing to participate in the USA Mathematical Olympiad (USAMO) in the hope of representing the United States at the International Mathematical Olympiad (IMO). The USAMO is the third stage of the selection process leading to participation in the IMO. The preceding examinations are the AMC 10 or the AMC 12 (which replaced the American High School Mathematics Examination) and the American Invitational Mathematics Examination (AIME). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence.
The top 12 USAMO students are invited to attend the Mathematical Olympiad Summer Program (MOSP) regardless of their grade in school. Additional MOSP invitations are extended to the most promising nongraduating USAMO students, as potential IMO participants in future years.
During the first days of MOSP, IMO-type exams are given to the top 12 USAMO students with the goal of identifying the six members of the USA IMO Team. The Team Selection Test (TST) simulates an actual IMO, consisting of six problems to be solved over two 4 1/2 hour sessions. The 12 equally weighted problems (six on the USAMO and six on the TST) determine the USA Team.
The Mathematical Olympiad booklets have been published since 1976. Copies for each year through 1999 can be ordered from the Mathematical Association of American (MAA) American Mathematics Competitions (AMC). This publication, Mathematical Olympiads 2000, Mathematical Olympiads 2001, and Mathematical Olympiads 2002 are published by the MAA. In addition, various other publications are useful in preparing for the AMC-AIME-USAMO-IMO sequence (see Chapter 6, Further Reading). For more information about the AMC examinations, or to order Mathematical Olympiad booklets from previous years, please write to
Steven Dunbar, MAA Director for K–12 Programs
American Mathematics Competitions
University of Nebraska-Lincoln
1740 Vine Street
Lincoln, NE 68588-0658,
or visit the AMC web site at www.unl.edu/amc.

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Andreescu T., Feng Z. 103 Trigonometry Problems. From the Training of the USA IMO Team - Birkhäuser, 2005 - 214 pp Скачать ifolder.ru/31380904

Contents
Preface vii
Acknowledgments ix
Abbreviations and Notation xi
1 Trigonometric Fundamentals 1
Definitions of Trigonometric Functions in Terms of Right Triangles 1
ThinkWithin the Box 4
You’ve Got the Right Angle 6
Think Along the Unit Circle 10
Graphs of Trigonometric Functions 14
The Extended Law of Sines 18
Area and Ptolemy’s Theorem 19
Existence, Uniqueness, and Trigonometric Substitutions 23
Ceva’s Theorem 28
Think Outside the Box 33
Menelaus’s Theorem 33
The Law of Cosines 34
Stewart’s Theorem 35
Heron’s Formula and Brahmagupta’s Formula 37
Brocard Points 39
Vectors 41
The Dot Product and the Vector Form of the Law of Cosines 46
The Cauchy–Schwarz Inequality 47
Radians and an Important Limit 47
Constructing Sinusoidal Curves with a Straightedge 50
Three Dimensional Coordinate Systems 51
Traveling on Earth 55
Where AreYou? 57
De Moivre’s Formula 58
2 Introductory Problems 63
3 Advanced Problems 73
4 Solutions to Introductory Problems 83
5 Solutions to Advanced Problems 125
Glossary 199
Further Reading 211

Preface
This book contains 103 highly selected problems used in the training and testing of the U.S. International Mathematical Olympiad (IMO) team. It is not a collection of very difficult, impenetrable questions. Instead, the book gradually builds students’ trigonometric skills and techniques. The first chapter provides a comprehensive introduction to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry. This chapter can serve as a textbook for a course in trigonometry. This work aims to broaden students’ view of mathematics and better prepare them for possible participation in various mathematical competitions. It provides in-depth enrichment in important areas of trigonometry by reorganizing and enhancing problem-solving tactics and strategies. The book further stimulates interest for the future study of mathematics.
In the United States ofAmerica, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), theAmerican Invitational Mathematics Examination (AIME), and the United States of America Mathematical Olympiad (USAMO). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately 50 very promising students who have risen to the top in the American Mathematics Competitions.
The six students representing the United States ofAmerica in the IMO are selected on the basis of their USAMO scores and further testing that takes place during MOSP.
Throughout MOSP, full days of classes and extensive problem sets give students thorough preparation in several important areas of mathematics. These topics include combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, functional equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations, and classical inequalities.
Olympiad-style exams consist of several challenging essay problems. Correct solutions often require deep analysis and careful argument. Olympiad questions can seem impenetrable to the novice, yet most can be solved with elementary high school mathematics techniques, cleverly applied.
Here is some advice for students who attempt the problems that follow.
• Take your time! Very few contestants can solve all the given problems.
• Try to make connections between problems. An important theme of this work is that all important techniques and ideas featured in the book appear more than once!
• Olympiad problems don’t “crack” immediately. Be patient. Try different approaches. Experiment with simple cases. In some cases, working backwards from the desired result is helpful.
• Even if you can solve a problem, do read the solutions. They may contain some ideas that did not occur in your solutions, and they may discuss strategic and tactical approaches that can be used elsewhere. The solutions are also models of elegant presentation that you should emulate, but they often obscure the tortuous process of investigation, false starts, inspiration, and attention to detail that led to them. When you read the solutions, try to reconstruct the thinking that went into them. Ask yourself, “What were the key ideas? How can I apply these ideas further?”
• Go back to the original problem later, and see whether you can solve it in a different way. Many of the problems have multiple solutions, but not all are outlined here.
• Meaningful problem-solving takes practice. Don’t get discouraged if you have trouble at first. For additional practice, use the books on the reading list.

[mpl]

Andreescu T., Feng Z., Loh P. USA and International Mathematical Olympiads 2004 - MAA, 2005, 61 p. Скачать

Contents

Acknowledgments
Preface
Abbreviations and Notations
Introduction
1 The Problems
USAMO
Team Selection Test..........................................3
IMO..........................................................5
2 Hints 7
USAMO......................................................7
Team Selection Test..........................................7
IMO..........................................................7
3 Formal Solutions 9
USAMO......................................................9
Team Selection Test..........................................29
IMO..........................................................44
4 Problem Credits 75
5 Glossary 77
6 Further Reading 85
7 Appendix 91
1 2004 Olympiad Results......................................91
2 2003 Olympiad Results......................................93
3 2002 Olympiad Results......................................95
4 2001 Olympiad Results......................................97
5 2000 Olympiad Results......................................98
6 2000-2004 Cumulative IMO Results ......................100

Preface
This book is intended to help students preparing to participate in the USA Mathematical Olympiad (USAMO) in the hope of representing the United States at the International Mathematical Olympiad (IMO). The USAMO is the third stage of the selection process leading to participation in the IMO. The preceding examinations arc the AMC 10 or the AMC 12 (which rcplaccd the American High School Mathematics Examination) and the American Invitational Mathematics Examination (AIME). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence.
The lop 12 USAMO students are invited to attend the Mathematical Olympiad Summer Program (MOSP) regardless of their grade in school. Additional MOSP invitations are extended to the most promising nongraduating USAMO students, as potential IMO participants in future years. During the first days of MOSP, IMO-type exams arc given to the top 12 USAMO students with the goal of identifying the six members of the USA IMO Team. The Team Selection Test (TST) simulates an actual IMO, consisting of six problems to be solved over two 4 1/2 hour sessions. The 12 equally weighted problems (six on the USAMO and six on the TST) determine the USA Team.
The Mathematical Olympiad booklets have been published since 1976. Copies for each year through 1999 can be ordered from the Mathematical Association of America's (MAA) American Mathematics Competitions (AMC). This publication, as well as Mathematical Olympiads 2000. Mathematical Olympiads 2001, Mathematical Olympiads 2002, and Mathematical Olympiads 2003 are published by the MAA. In addition, various other publications are useful in preparing for the AMC-AIME-USAMO-IMO sequence (see Chapter 6. Further Reading). For more information about the AMC examinations, or to order Mathematical Olympiad booklets from previous years, please write to
Steven Dunbar. MAA Director for К-12 Programs
American Mathematics Competitions
University of Nebraska-Lincoln
1740 Vine Street
Lincoln. NE 68588-0658,
or visit the AMC web site at www.unl.edu/amc.

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Andreescu T., Andrica D., Feng Z. 104 Number Theory Problems. From the Training of the USA IMO Team - Birkhäuser, 2007 - 204 pp Скачать ifolder.ru/31381089

Contents
Preface vii
Acknowledgments ix
Abbreviations and Notation xi
1 Foundations of Number Theory 1
Divisibility 1
Division Algorithm 4
Primes 5
The Fundamental Theorem of Arithmetic 7
G.C.D. 11
Euclidean Algorithm 12
B´ezout’s Identity 13
L.C.M. 16
The Number of Divisors 17
The Sum of Divisors 18
Modular Arithmetics 19
Residue Classes 24
Fermat’s Little Theorem and Euler’s Theorem 27
Euler’s Totient Function 33
Multiplicative Function 36
Linear Diophantine Equations 38
Numerical Systems 40
Divisibility Criteria in the Decimal System 46
Floor Function 52
Legendre’s Function 65
Fermat Numbers 70
Mersenne Numbers 71
Perfect Numbers 72
2 Introductory Problems 75
3 Advanced Problems 83
4 Solutions to Introductory Problems 91
5 Solutions to Advanced Problems 131
Glossary 189
Further Reading 197
Index 203

Preface
This book contains 104 of the best problems used in the training and testing of the U.S. International Mathematical Olympiad (IMO) team. It is not a collection of very difficult, and impenetrable questions. Rather, the book gradually builds students’ number-theoretic skills and techniques. The first chapter provides a comprehensive introduction to number theory and its mathematical structures. This chapter can serve as a textbook for a short course in number theory. This work aims to broaden students’ view of mathematics and better prepare them for possible participation in various mathematical competitions. It provides in-depth enrichment in important areas of number theory by reorganizing and enhancing students’ problem-solving tactics and strategies. The book further stimulates students’ interest for the future study of mathematics.
In the United States of America, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), the American Invitational Mathematics Examination (AIME), and the United States of America Mathematical Olympiad (USAMO). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately fifty very promising students who have risen to the top in the American Mathematics Competitions. The six students representing the United States of America in the IMO are selected on the basis of their USAMO scores and further testing that takes place during MOSP. Throughout MOSP, full days of classes and extensive problem sets give students thorough preparation in several important areas of mathematics. These topics include combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, functional equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations, and classical inequalities.
Olympiad-style exams consist of several challenging essay problems. Correct solutions often require deep analysis and careful argument. Olympiad questions can seem impenetrable to the novice, yet most can be solved with elementary high school mathematics techniques, when cleverly applied.
Here is some advice for students who attempt the problems that follow.
• Take your time! Very few contestants can solve all the given problems.
• Try to make connections between problems. An important theme of this work is that all important techniques and ideas featured in the book appear more than once!
• Olympiad problems don’t “crack” immediately. Be patient. Try different approaches. Experiment with simple cases. In some cases, working backward from the desired result is helpful.
• Even if you can solve a problem, do read the solutions. They may contain some ideas that did not occur in your solutions, and they may discuss strategic and tactical approaches that can be used elsewhere. The solutions
are also models of elegant presentation that you should emulate, but they often obscure the tortuous process of investigation, false starts, inspiration, and attention to detail that led to them. When you read the solutions, try to
reconstruct the thinking that went into them. Ask yourself, “What were the key ideas? How can I apply these ideas further?”
• Go back to the original problem later, and see whether you can solve it in a different way. Many of the problems have multiple solutions, but not all are outlined here.
• Meaningful problem solving takes practice. Don’t get discouraged if you have trouble at first. For additional practice, use the books on the reading list.

[mpl]

Andreescu T., Boreico I. Functional Equations (Electronic Edition) - 2007, 140 p. 17 Chapters and 199 Problems With Solution Скачать

Content:
Chapter 1:Contructive Problems 1
Chapter 2:Binary and other base 19
Chapter 3: Constructing functions by iterations 21
Chapter 4: Approximating with linear functions 24
Chapter 5: Extremal element method 26
Chapter 6:Multiplicative Cauchy Equation 28
Chapter 7:Substitutions 30
Chapter 8:Fixed Point 42
Chapter 9:Additive Cauchy Equation 44
Chapter 10:Polynomial Equation 55
Chapter 11:Interation and Recurrence Relation 65
Chapter 12:Polynomial Recurrence and Continuity 73
Chapter 13:Odd and Even Part of Function 78
Chapter 14:Symmetrization and Addition Variable 80
Chapter 15:Functional Inequality 82
Chapter 16: Miscellaneous 84
Chapter 17:Solution To All Problems

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Andreescu T., Gelca R. Putnam and Beyond - Springer, 2007, 815 p. Скачать

Contents
Preface . xi
A Study Guide . xv
1 Methods of Proof . 1
1.1 Argument by Contradiction . 1
1.2 Mathematical Induction . 3
1.3 The Pigeonhole Principle . 11
1.4 Ordered Sets and Extremal Elements . 14
1.5 Invariants and Semi-Invariants . 19
2 Algebra . 25
2.1 Identities and Inequalities . 25
2.1.1 Algebraic Identities . 25
2.1.2 x2 ? 0. 28
2.1.3 The Cauchy–Schwarz Inequality . 32
2.1.4 The Triangle Inequality . 36
2.1.5 The Arithmetic Mean–Geometric Mean Inequality . 39
2.1.6 Sturm’s Principle . 42
2.1.7 Other Inequalities . 45
2.2 Polynomials . 45
2.2.1 AWarmup . 45
2.2.2 Viete’s Relations . 47
2.2.3 The Derivative of a Polynomial . 52
2.2.4 The Location of the Zeros of a Polynomial . 54
2.2.5 Irreducible Polynomials . 56
2.2.6 Chebyshev Polynomials . 58
2.3 Linear Algebra . 61
2.3.1 Operations with Matrices . 61
2.3.2 Determinants . 63
2.3.3 The Inverse of a Matrix . 69
2.3.4 Systems of Linear Equations . 73
2.3.5 Vector Spaces, Linear Combinations of Vectors, Bases . 77
2.3.6 Linear Transformations, Eigenvalues, Eigenvectors . 79
2.3.7 The Cayley–Hamilton and Perron–Frobenius Theorems . 83
2.4 Abstract Algebra . 87
2.4.1 Binary Operations . 87
2.4.2 Groups . 90
2.4.3 Rings . 95
3 Real Analysis . 97
3.1 Sequences and Series . 98
3.1.1 Search for a Pattern . 98
3.1.2 Linear Recursive Sequences . 100
3.1.3 Limits of Sequences . 104
3.1.4 More About Limits of Sequences . 111
3.1.5 Series . 117
3.1.6 Telescopic Series and Products . 120
3.2 Continuity, Derivatives, and Integrals . 125
3.2.1 Limits of Functions . 125
3.2.2 Continuous Functions . 128
3.2.3 The Intermediate Value Property . 131
3.2.4 Derivatives and Their Applications . 134
3.2.5 The Mean Value Theorem . 138
3.2.6 Convex Functions . 142
3.2.7 Indefinite Integrals . 147
3.2.8 Definite Integrals . 150
3.2.9 Riemann Sums . 153
3.2.10 Inequalities for Integrals . 156
3.2.11 Taylor and Fourier Series . 159
3.3 Multivariable Differential and Integral Calculus . 167
3.3.1 Partial Derivatives and Their Applications . 167
3.3.2 Multivariable Integrals . 174
3.3.3 The Many Versions of Stokes’ Theorem . 179
3.4 Equations with Functions as Unknowns . 185
3.4.1 Functional Equations . 185
3.4.2 Ordinary Differential Equations of the First Order . 191
3.4.3 Ordinary Differential Equations of Higher Order . 195
3.4.4 Problems Solved with Techniques of Differential Equations . 198
4 Geometry and Trigonometry . 201
4.1 Geometry . 201
4.1.1 Vectors . 201
4.1.2 The Coordinate Geometry of Lines and Circles . 206
4.1.3 Conics and Other Curves in the Plane. 212
4.1.4 Coordinate Geometry in Three and More Dimensions . 219
4.1.5 Integrals in Geometry . 225
4.1.6 Other Geometry Problems . 228
4.2 Trigonometry . 231
4.2.1 Trigonometric Identities . 231
4.2.2 Euler’s Formula . 235
4.2.3 Trigonometric Substitutions . 238
4.2.4 Telescopic Sums and Products in Trigonometry . 242
5 Number Theory . 245
5.1 Integer-Valued Sequences and Functions . 245
5.1.1 Some General Problems . 245
5.1.2 Fermat’s Infinite Descent Principle . 248
5.1.3 The Greatest Integer Function . 250
5.2 Arithmetic . 253
5.2.1 Factorization and Divisibility . 253
5.2.2 Prime Numbers . 254
5.2.3 Modular Arithmetic. 258
5.2.4 Fermat’s Little Theorem . 260
5.2.5 Wilson’s Theorem . 264
5.2.6 Euler’s Totient Function . 265
5.2.7 The Chinese Remainder Theorem . 268
5.3 Diophantine Equations . 270
5.3.1 Linear Diophantine Equations . 270
5.3.2 The Equation of Pythagoras . 274
5.3.3 Pell’s Equation . 276
5.3.4 Other Diophantine Equations . 279
6 Combinatorics and Probability . 281
6.1 Combinatorial Arguments in Set Theory and Geometry . 281
6.1.1 Set Theory and Combinatorics of Sets . 281
6.1.2 Permutations . 283
6.1.3 Combinatorial Geometry . 286
6.1.4 Euler’s Formula for Planar Graphs . 289
6.1.5 Ramsey Theory . 291
6.2 Binomial Coefficients and Counting Methods . 294
6.2.1 Combinatorial Identities . 294
6.2.2 Generating Functions . 298
6.2.3 Counting Strategies . 302
6.2.4 The Inclusion–Exclusion Principle . 308
6.3 Probability . 310
6.3.1 Equally Likely Cases . 310
6.3.2 Establishing Relations Among Probabilities . 314
6.3.3 Geometric Probabilities . 318
Solutions
Methods of Proof . 323
Algebra . 359
Real Analysis . 459
Geometry and Trigonometry . 603
Number Theory . 673
Combinatorics and Probability . 727
Index of Notation . 791
Index . 795

Preface
A problem book at the college level. A study guide for the Putnam competition. A bridge between high school problem solving and mathematical research. Afriendly introduction to fundamental concepts and results. All these desires gave life to the pages that follow.
The William Lowell Putnam Mathematical Competition is the most prestigious mathematics competition at the undergraduate level in the world. Historically, this annual event began in 1938, following a suggestion of William Lowell Putnam, who realized the merits of an intellectual intercollegiate competition. Nowadays, over 2500 students from more than 300 colleges and universities in the United States and Canada take part in it. The name Putnam has become synonymous with excellence in undergraduate mathematics.
Using the Putnam competition as a symbol, we lay the foundations of higher mathematics from a unitary, problem-based perspective. As such, Putnam and Beyond is a journey through the world of college mathematics, providing a link between the stimulating problems of the high school years and the demanding problems of scientific investigation. It gives motivated students a chance to learn concepts and acquire strategies, hone their skills and test their knowledge, seek connections, and discover real world applications. Its ultimate goal is to build the appropriate background for graduate studies, whether in mathematics or applied sciences.
..

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Andreescu T., Gelca R. Mathematical Olympiad Challenges (2nd) - Birkhäuser, 2008, 291 p. Скачать

Contents
Foreword xi
Preface to the Second Edition xiii
Preface to the First Edition xv
I Problems 1
1 Geometry and Trigonometry 3
1.1 APropertyofEquilateralTriangles . . . . . . . . . . . . . . . . . . . 4
1.2 CyclicQuadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Powerof aPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Dissections of Polygonal Surfaces . . . . . . . . . . . . . . . . . . . 15
1.5 Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 GeometricConstructionsandTransformations . . . . . . . . . . . . . 25
1.7 ProblemswithPhysicalFlavor . . . . . . . . . . . . . . . . . . . . . 27
1.8 TetrahedraInscribedinParallelepipeds . . . . . . . . . . . . . . . . . 29
1.9 Telescopic Sums and Products in Trigonometry . . . . . . . . . . . . 31
1.10 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . 34
2 Algebra and Analysis 39
2.1 NoSquare IsNegative . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Look at the Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Telescopic Sums and Products in Algebra . . . . . . . . . . . . . . . 44
2.4 On an Algebraic Identity . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Systems ofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 TheAbelSummationFormula . . . . . . . . . . . . . . . . . . . . . 58
2.8 x+1/x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.10 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . . 66
3 Number Theory and Combinatorics 69
3.1 ArrangeinOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Squares andCubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Repunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Digits ofNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Diophantine Equations with the Unknowns as Exponents . . . . . . . 83
3.7 NumericalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.8 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.9 PellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.10 PrimeNumbersandBinomialCoefficients . . . . . . . . . . . . . . . 99
II Solutions 103
1 Geometry and Trigonometry 105
1.1 APropertyofEquilateralTriangles . . . . . . . . . . . . . . . . . . . 106
1.2 CyclicQuadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.3 Powerof aPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.4 Dissections of Polygonal Surfaces . . . . . . . . . . . . . . . . . . . 125
1.5 Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
1.6 GeometricConstructionsandTransformations . . . . . . . . . . . . . 145
1.7 ProblemswithPhysicalFlavor . . . . . . . . . . . . . . . . . . . . . 151
1.8 TetrahedraInscribedinParallelepipeds . . . . . . . . . . . . . . . . . 156
1.9 Telescopic Sums and Products in Trigonometry . . . . . . . . . . . . 160
1.10 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . 165
2 Algebra and Analysis 171
2.1 NoSquare isNegative . . . . . . . . . . . . . . . . . . . . . . . . . 172
2.2 Look at the Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.3 Telescopic Sums and Products in Algebra . . . . . . . . . . . . . . . 183
2.4 On an Algebraic Identity . . . . . . . . . . . . . . . . . . . . . . . . 188
2.5 Systems ofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2.6 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
2.7 TheAbelSummationFormula . . . . . . . . . . . . . . . . . . . . . 202
2.8 x+1/x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
2.9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
2.10 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . . 217
3 Number Theory and Combinatorics 223
3.1 ArrangeinOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.2 Squares andCubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.3 Repunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
3.4 Digits ofNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.5 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.6 Diophantine Equations with the Unknowns as Exponents . . . . . . . 246
3.7 NumericalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 252
3.8 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
3.9 PellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
3.10 PrimeNumbersandBinomialCoefficients . . . . . . . . . . . . . . . 270
Appendix A: Definitions and Notation 277
A.1 GlossaryofTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
A.2 GlossaryofNotation . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Foreword
Why Olympiads?
Working mathematicians often tell us that results in the field are achieved after long experience and a deep familiarity with mathematical objects, that progress is made slowly and collectively, and that flashes of inspiration are mere punctuation in periods of sustained effort.
The Olympiad environment, in contrast, demands a relatively brief period of intense concentration, asks for quick insights on specific occasions, and requires a concentrated but isolated effort. Yet we have found that participants in mathematics Olympiads have often gone on to become first-class mathematicians or scientists and have attached great significance to their early Olympiad experiences.
For many of these people, the Olympiad problem is an introduction, a glimpse into the world of mathematics not afforded by the usual classroom situation. A good Olympiad problem will capture in miniature the process of creating mathematics. It’s all there: the period of immersion in the situation, the quiet examination of possible
approaches, the pursuit of various paths to solution. There is the fruitless dead end, as well as the path that ends abruptly but offers new perspectives, leading eventually to the discovery of a better route. Perhaps most obviously, grappling with a good problem provides practice in dealing with the frustration of working at material that refuses to
yield. If the solver is lucky, there will be the moment of insight that heralds the start of a successful solution. Like a well-crafted work of fiction, a good Olympiad problem tells a story of mathematical creativity that captures a good part of the real experience and leaves the participant wanting still more.
And this book gives us more. It weaves together Olympiad problems with a common theme, so that insights become techniques, tricks become methods, and methods build to mastery. Although each individual problem may be a mere appetizer, the table is set here for more satisfying fare, which will take the reader deeper into mathematics than might any single problem or contest.
The book is organized for learning. Each section treats a particular technique or topic. Introductory results or problems are provided with solutions, then related problems are presented, with solutions in another section.
The craft of a skilled Olympiad coach or teacher consists largely in recognizing similarities among problems. Indeed, this is the single most important skill that the coach can impart to the student. In this book, two master Olympiad coaches have offered the results of their experience to a wider audience. Teachers will find examples
and topics for advanced students or for their own exercise. Olympiad stars will find practice material that will leave them stronger and more ready to take on the next challenge, from whatever corner of mathematics it may originate. Newcomers to Olympiads will find an organized introduction to the experience.
There is also something here for the more general reader who is interested in mathematics. Simply perusing the problems, letting their beauty catch the eye, and working through the authors’ solutions will add to the reader’s understanding. The multiple solutions link together areas of mathematics that are not apparently related. They often
illustrate how a simple mathematical tool—a geometric transformation, or an algebraic identity—can be used in a novel way, stretched or reshaped to provide an unexpected solution to a daunting problem.
These problems are daunting on any level. True to its title, the book is a challenging one. There are no elementary problems—although there are elementary solutions. The content of the book begins just at the edge of the usual high school curriculum. The calculus is sometimes referred to, but rarely leaned on, either for solution or for motivation.
Properties of vectors and matrices, standard in European curricula, are drawn upon freely. Any reader should be prepared to be stymied, then stretched. Much is demanded of the reader by way of effort and patience, but the reader’s investment is greatly repaid.
In this, it is not unlike mathematics as a whole.

[mpl]

Andreescu T., Dospinescu G. Problems from the book [2nd Edition ed.] - XYZ Press, 2008, 554 p. Скачать

Preface
What can a new book of problems in elementary mathematics possibly contribute to the vast existing collection of journals, articles, and books? This was our main concern when we decided to write this book. The inevitability of this question does not facilitate the answer, because after five years of writing and rewriting we still had something to add. It could be a new problem, a comment we considered pertinent, or a solution that escaped our rationale until this predictive moment, when we were supposed to bring it under the scrutiny of a specialist in the field.
A mere perusal of this book should be sufficient to identify its target audience: students and coaches preparing for mathematical Olympiads, national or international. It takes more effort to realize that these are not the only potential beneficiaries of this work. While the book is rife with problems collected from various mathematical competitions and journals, one cannot neglect the classical results of mathematics, which naturally exceed the level of time-constrained competitions. And no, classical does not mean easy! These mathematical beauties are more than just proof that elementary mathematics can produce jewels. They serve as an invitation to mathematics beyond competitions, regarded by many to be the "true mathematics". In this context, the audience is more diverse than one might think.
...


Contents
1 Some Useful Substitutions 1
1.1 Theory and examples 2
1.2 Problems for training 20
2 Always Cauchy-Schwarz... 25
2.1 Theory and examples 27
2.2 Problems for training 43
3 Look at the Exponent 47
3.1 Theory and examples 49
3.2 Problems for training 67
4 Primes and Squares 73
4.1 Theory and examples 75
4.2 Problems for training 89
5 T2's Lemma 93
5.1 Theory and examples 95
5.2 Problems for training 111
6 Some Classical Problems in Extremal Graph Theory 115
6.1 Theory and examples 117
6.2 Problems for training 128
7 Complex Combinatorics 131
7.1 Theory and examples 133
7.2 Problems for training 148
8 Formal Series Revisited 153
8.1 Theory and examples 155
8.2 Problems for training 173
9 A Brief Introduction to Algebraic Number Theory 179
9.1 Theory and examples 181
9.2 Problems for training 200
10 Arithmetic Properties of Polynomials 205
10.1 Theory and examples 207
10.2 Problems for training 227
11 Lagrange Interpolation Formula 233
11.1 Theory and examples 235
11.2 Problems for training 259
12 Higher Algebra in Combinatorics 263
12.1 Theory and examples 265
12.2 Problems for training 282
13 Geometry and Numbers 289
13.1 Theory and examples 291
13.2 Problems for training 309
14 The Smaller, the Better 313
14.1 Theory and examples 315
14.2 Problems for training 327
15 Density and Regular Distribution 333
15.1 Theory and examples 335
15.2 Problems for training 350
16 The Digit Sum of a Positive Integer 353
16.1 Theory and examples 355
16.2 Problems for training 369
17 At the Border of Analysis and Number Theory 375
17.1 Theory and examples 377
17.2 Problems for training 394
18 Quadratic Reciprocity 399
18.1 Theory and examples 401
18.2 Problems for training 419
19 Solving Elementary Inequalities Using Integrals 425
19.1 Theory and examples 427
19.2 Problems for training 445
20 Pigeonhole Principle Revisited 451
20.1 Theory and examples 453
20.2 Problems for training 473
21 Some Useful Irreducibility Criteria 479
21.1 Theory and examples 481
21.2 Problems for training 501
22 Cycles, Paths, and Other Ways 505
22.1 Theory and examples 507
22.2 Problems for training 519
23 Some Special Applications of Polynomials 523
23.1 Theory and examples 525
23.2 Problems for training 543
Bibliography 547
Index 553

[mpl]

Andreescu T., Andrica D. Number theory: Structures, examples, and problems - Birkhäuser, 2009, 412 p. One of the oldest and liveliest branches of mathematics, Number Theory, is noted for its theoretical depth and applications to other fields, including representation theory, physics, and cryptography. The forefront of Number Theory is replete with sophisticated and famous open problems; at its foundation, however, are basic, elementary ideas that can stimulate and challenge beginning students. This textbook takes a problem-solving approach to Number Theory, situating each theoretical concept within the framework of some examples or some problems for readers to solve. Скачать

Contents
Foreword 7
Acknowledgments 9
Notation 11
I STRUCTURES, EXAMPLES, AND PROBLEMS 13
1 Divisibility 15
1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 The greatest common divisor . . . . . . . . . . . . . . . . . 30
1.4 Odd and even . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5 Modular arithmetics . . . . . . . . . . . . . . . . . . . . . . 42
1.6 Chinese remainder theorem . . . . . . . . . . . . . . . . . . 47
1.7 Numerical systems . . . . . . . . . . . . . . . . . . . . . . . 50
1.7.1 Representation of integers in an arbitrary base . . . 50
1.7.2 Divisibility criteria in the decimal system . . . . . . 51
2 Powers of Integers 61
2.1 Perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 Perfect cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 `k^th` powers of integers, k ≥ 4 . . . . . . . . . . . . . . . . . . 72
3 Floor Function and Fractional Part 77
3.1 General problems . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Floor function and integer points . . . . . . . . . . . . . . . 83
3.3 An useful result . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Digits of Numbers 91
4.1 The last digits of a number . . . . . . . . . . . . . . . . . . 91
4.2 The sum of the digits of a number . . . . . . . . . . . . . . 94
4.3 Other problems involving digits . . . . . . . . . . . . . . . . 100
5 Basic Principles in Number Theory 103
5.1 Two simple principles . . . . . . . . . . . . . . . . . . . . . 103
5.1.1 Extremal arguments . . . . . . . . . . . . . . . . . . 103
5.1.2 Pigeonhole principle . . . . . . . . . . . . . . . . . . 105
5.2 Mathematical induction . . . . . . . . . . . . . . . . . . . . 108
5.3 Infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4 Inclusion-exclusion . . . . . . . . . . . . . . . . . . . . . . . 115
6 Arithmetic Functions 119
6.1 Multiplicative functions . . . . . . . . . . . . . . . . . . . . 119
6.2 Number of divisors . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Sum of divisors . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Euler’s totient function . . . . . . . . . . . . . . . . . . . . . 131
6.5 Exponent of a prime and Legendre’s formula . . . . . . . . 135
7 More on Divisibility 141
7.1 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . 141
7.2 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 147
7.3 The order of an element . . . . . . . . . . . . . . . . . . . . 150
7.4 Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 153
8 Diophantine Equations 157
8.1 Linear Diophantine equations . . . . . . . . . . . . . . . . . 157
8.2 Quadratic Diophantine equations . . . . . . . . . . . . . . . 161
8.2.1 Pythagorean equation . . . . . . . . . . . . . . . . . 161
8.2.2 Pell’s equation . . . . . . . . . . . . . . . . . . . . . 164
8.2.3 Other quadratic equations . . . . . . . . . . . . . . . 169
8.3 Nonstandard Diophantine equations . . . . . . . . . . . . . 171
8.3.1 Cubic equations . . . . . . . . . . . . . . . . . . . . . 171
8.3.2 High-order polynomial equations . . . . . . . . . . . 173
8.3.3 Exponential Diophantine equations . . . . . . . . . . 176
9 Some special problems in number theory 179
9.1 Quadratic residues. Legendre’s symbol . . . . . . . . . . . . 179
9.2 Special numbers . . . . . . . . . . . . . . . . . . . . . . . . 188
9.2.1 Fermat’s numbers . . . . . . . . . . . . . . . . . . . . 188
9.2.2 Mersenne’s numbers . . . . . . . . . . . . . . . . . . 191
9.2.3 Perfect numbers . . . . . . . . . . . . . . . . . . . . . 192
9.3 Sequences of integers . . . . . . . . . . . . . . . . . . . . . . 193
9.3.1 Fibonacci and Lucas sequences . . . . . . . . . . . . 193
9.3.2 Problems involving linear recursive relations . . . . . 197
9.3.3 Nonstandard sequences of integers . . . . . . . . . . 204
10 Problems Involving Binomial Coefficients 211
10.1 Binomial coefficients . . . . . . . . . . . . . . . . . . . . . . 211
10.2 Lucas’ and Kummer’s Theorems . . . . . . . . . . . . . . . 218
11 Miscellaneous Problems 223
II SOLUTIONS TO PROPOSED PROBLEMS 229
12 Divisibility 231
12.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . 237
12.3 The greatest common divisor . . . . . . . . . . . . . . . . . 242
12.4 Odd and even . . . . . . . . . . . . . . . . . . . . . . . . . . 247
12.5 Modular arithmetics . . . . . . . . . . . . . . . . . . . . . . 248
12.6 Chinese remainder theorem . . . . . . . . . . . . . . . . . . 251
12.7 Numerical systems . . . . . . . . . . . . . . . . . . . . . . . 253
13 Powers of Integers 261
13.1 Perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . 261
13.2 Perfect cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 270
13.3 `k^th` powers of integers, k ≥ 4 . . . . . . . . . . . . . . . . . . 272
14 Floor Function and Fractional Part 275
14.1 General problems . . . . . . . . . . . . . . . . . . . . . . . . 275
14.2 Floor function and integer points . . . . . . . . . . . . . . . 279
14.3 An useful result . . . . . . . . . . . . . . . . . . . . . . . . . 280
15 Digits of Numbers 283
15.1 The last digits of a number . . . . . . . . . . . . . . . . . . 283
15.2 The sum of the digits of a number . . . . . . . . . . . . . . 284
15.3 Other problems involving digits . . . . . . . . . . . . . . . . 288
16 Basic Principles in Number Theory 291
16.1 Two simple principles . . . . . . . . . . . . . . . . . . . . . 291
16.2 Mathematical induction . . . . . . . . . . . . . . . . . . . . 294
16.3 Infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4 Inclusion-exclusion . . . . . . . . . . . . . . . . . . . . . . . 301
17 Arithmetic Functions 305
17.1 Multiplicative functions . . . . . . . . . . . . . . . . . . . . 305
17.2 Number of divisors . . . . . . . . . . . . . . . . . . . . . . . 307
17.3 Sum of divisors . . . . . . . . . . . . . . . . . . . . . . . . . 309
17.4 Euler’s totient function . . . . . . . . . . . . . . . . . . . . . 311
17.5 Exponent of a prime and Legendre’s formula . . . . . . . . 313
18 More on Divisibility 319
18.1 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . 319
18.2 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 326
18.3 The order of an element . . . . . . . . . . . . . . . . . . . . 328
18.4 Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 330
19 Diophantine Equations 333
19.1 Linear Diophantine equations . . . . . . . . . . . . . . . . . 333
19.2 Quadratic Diophantine equations . . . . . . . . . . . . . . . 336
19.2.1 Pythagorean equations . . . . . . . . . . . . . . . . 336
19.2.2 Pell’s equation . . . . . . . . . . . . . . . . . . . . . 337
19.2.3 Other quadratic equations . . . . . . . . . . . . . . 340
19.3 Nonstandard Diophantine equations . . . . . . . . . . . . . 343
19.3.1 Cubic equations . . . . . . . . . . . . . . . . . . . . 343
19.3.2 High-order polynomial equations . . . . . . . . . . . 345
19.3.3 Exponential Diophantine equations . . . . . . . . . 347
20 Some special problems in number theory 351
20.1 Quadratic residues. Legendre’s symbol . . . . . . . . . . . . 351
20.2 Special numbers . . . . . . . . . . . . . . . . . . . . . . . . 354
20.2.1 Fermat’s numbers . . . . . . . . . . . . . . . . . . . 354
20.2.2 Mersenne’s numbers . . . . . . . . . . . . . . . . . . 356
20.2.3 Perfect numbers . . . . . . . . . . . . . . . . . . . . 357
20.3 Sequences of integers . . . . . . . . . . . . . . . . . . . . . . 357
20.3.1 Fibonacci and Lucas sequences . . . . . . . . . . . . 357
20.3.2 Problems involving linear recursive relations . . . . 360
20.3.3 Nonstandard sequences of integers . . . . . . . . . . 364
21 Problems Involving Binomial Coefficients 379
21.1 Binomial coefficients . . . . . . . . . . . . . . . . . . . . . . 379
21.2 Lucas’ and Kummer’s Theorems . . . . . . . . . . . . . . . 384
22 Miscellaneous Problems 387
Glossary 393
References 401
Index of Authors 407
Subject Index 409