104 Number Theory Problems

1. Front Matter

   Pages i-xii

   PDF

2. Foundations of Number Theory

   Pages 1-74

3. Introductory Problems

   Pages 75-82

4. Advanced Problems

   Pages 83-90

5. Solutions to Introductory Problems

   Pages 91-130

6. Solutions to Advanced Problems

   Pages 131-187

7. Back Matter

   Pages 189-204

   PDF

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 dif?cult, and impenetrable questions. Rather, the book gradually builds students’ number-theoretic skills and techniques. The ?rst 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 s- dents’ 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 - amination (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 ?fty very promising students who have risen to the top in the American Mathematics Competitions.

• Area
• DEX
• Excel
• International Mathematical Olympiad
• Mathematica
• Problemlösungsprozess
• bridge
• mathematics
• number theory
• presentation
• techniques
• themes
• training

From the reviews:

"In short, this book is a very valuable tool for any student/coach interested in preparing for mathematics competitions, especially the International Mathematical Olympiad.  College students participating in the Putnam competition might also find quite a few interesting problems. Moreover, any course in number theory could be supplemented with this book and could use some of the references included. Even research mathematicians working in number theory will find this book of value in their pursuits." -MAA Online

"The names of the authors sound familiar for teachers of mathematics and mathematicians who use books of these types … . I am sure about the success of this book. It is going to be a ‘bestseller’. It can be useful for high school students preparing for contests, and for teachers helping them all over the world. I am also reliant on being able to insert some excellent problems of the book into the syllabus of number theory courses at university level." (József Kosztolányi, Acta Scientiarum Mathematicarum, Vol. 73, 2007)

“The book starts with a gentle introduction to number theory. It serves for a training of the participants of the U. S. International Mathematical Olympiad. … The 104 problems are carefully selected. … The solutions are also carefully presented.” (J. Schoissengeier, Monatshefte für Mathematik, Vol. 156 (3), March, 2009)

• Department of Science/Mathematics Education, The University of Texas at Dallas, Richardson, USA

  Titu Andreescu

• Faculty of Mathematics, “Babeş-Bolyai” University, Cluj-Napoca, Romania

  Dorin Andrica

• Department of Mathematics, Phillips Exeter Academy, Exeter, USA

  Zuming Feng

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 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 Program 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 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 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 Mathematical 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.