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The Mathematical Olympiad examinations, covering the USA Mathematical Olympiad (USAMO) and the International Mathematical Olympiad (IMO), have been published annually by the MAA American Mathematics Competition since 1976.
The IMO is the world mathematics championship for high school students. It takes place every year in a different country. The IMO competitions help to discover, challenge, and encourage mathematically gifted young people all over the world.
The USAMO and the Team Selection Test (TST) are the last two stages of the selection process leading to selection of the US team in the IMO. The preceding examinations are the AMC 10 or AMC 12 and the American Invitational Mathematics Examination (AIME). Participation in the AIME, USAMO, and the TST is by invitation only, based on performance in the preceding exams of the sequence.
In addition to presenting their carefully written solutions to the problems presented here, the editors have provided remarkable solutions developed by the examination committees, contestants, and experts, during or after the contests. They also provide a comprehensive guide to other materials on advanced problem solving.
This collection of excellent problems and beautiful solutions is a valuable companion for students who wish to develop their interest in mathematics outside the school curriculum and to deepen their knowledge of mathematics.
Abbreviations and Notations xiii
Chapter 1 The Problems 1
1 USAMO 1
2 Team Selection Test 3
3 IMO 5
Chapter 2 Hints 7
1 USAMO 7
2 Team Selection Test 7
3 IMO 7
Chapter 3 Formal Solutions 9
1 USAMO 9
2 Team Selection Test 29
3 IMO 44
Chapter 4 Problem Credits 75
Chapter 5 Glossary 77
Chapter 6 Further Reading 85
Chapter 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
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 non-graduating 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 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).