Number Theory Problems

About the Authors
Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. 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 director 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 theAwesomeMath 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 Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions tonumerous 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 numerousmathematics textbooks, problem books, articles and scientific 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 boardsof 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. Heteaches 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 USAMathematical 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 Mathematics 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

Birkh¨ user aBoston • Basel • Berlin

Titu Andreescu The University of Texas at Dallas Department of Science/Mathematics Education Richardson, TX 75083 U.S.A. titu.andreescu@utdallas.edu Zuming Feng Phillips Exeter Academy Department of Mathematics Exeter, NH 03833 U.S.A. zfeng@exeter.edu

Dorin Andrica “Babes-Bolyai” University ¸ Faculty of Mathematics 3400 Cluj-Napoca Romania dorinandrica@yahoo.comCover 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 ISBN-13: 978-0-8176-4527-4 Printed on acid-free paper.
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