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About the Authors

Titu Andreescu received his Ph.D. from the West University of Timisoara, Ro- mania. The topic of his dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as di- rector of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993– 2002), director of the Mathematical Olympiad Summer Program (1995–2002), and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certificate of Appreciation” from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Pro- gram in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions to numerous textbooks and problem books are recognized worldwide.
Dorin Andrica received his Ph.D in 1992 from “Babes¸-Bolyai” University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes¸-Bolyai” since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and sci- entific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called “Andrica’s Conjecture.” He has been a regular faculty member at the Canada–USA Mathcamps between 2001–2005 and at the AwesomeMath Summer Program (AMSP) since 2006.
Zuming Feng received his Ph.D. from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997–2006), was the deputy leader of the USA IMO Team (2000–2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999–2002). He has been a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathe- matical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP) since 2006. He received the Edyth May Sliffe Award for Distinguished High School Mathe- matics Teaching from the MAA in 1996 and 2002.
Titu Andreescu Dorin Andrica Zuming Feng

104 Number Theory Problems
From the Training of the USA IMO Team

Birkha¨user Boston • Basel • Berlin
Titu Andreescu
The University of Texas at Dallas
Department of Science/Mathematics Education Richardson, TX 75083
Dorin Andrica
“Babes¸-Bolyai” University Faculty of Mathematics 3400 Cluj-Napoca Romania

Zuming Feng
Phillips Exeter Academy Department of Mathematics Exeter, NH 03833

Cover design by Mary Burgess.

Mathematics Subject Classification (2000): 00A05, 00A07, 11-00, 11-XX, 11Axx, 11Bxx, 11D04

Library of Congress Control Number: 2006935812

ISBN-10: 0-8176-4527-6 e-ISBN-10: 0-8176-4561-6
ISBN-13: 978-0-8176-4527-4 e-ISBN-13: 978-0-8176-4561-8

Printed on acid-free paper.

§c 2007 Birkha¨user Boston /
All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkha¨user Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden.
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