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

Springer Science & Business Media, Apr 5, 2007 - Mathematics - 204 pages
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.

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### Contents

 Foundations of Number Theory 1 Division Algorithm 4 Primes 5 The Fundamental Theorem of Arithmetic 7 GCD 11 Euclidean Algorithm 12 Bezouts Identity 13 LCM 16
 Numerical Systems 40 Divisibility Criteria in the Decimal System 46 Floor Function 52 Legendres Function 65 Fermat Numbers 70 Mersenne Numbers 71 Perfect Numbers 72 Introductory Problems 75

 The Number of Divisors 17 The Sum of Divisors 18 Modular Arithmetic 19 Residue Classes 24 Fermats Little Theorem and Eulers Theorem 27 Eulers Totient Function 33 Multiplicative Function 36 Linear Diophantine Equations 38