An Excursion through Elementary Mathematics, Volume III

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more.
As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level.
The book also explores someof the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.

Combines an in-depth overview of the theory with problems presented at several Mathematical Olympiads around the world Offers a comprehensive course on problem-solving techniques Presents a coherent development of mathematical ideas and methods behind problem solving Brings several classical, relevant results of various fields in mathematics

Auteur
Antonio Caminha M. Neto received his PhD from the Federal University of Ceará, Brazil in 2004. In the same year he joined the University as a Professor of Mathematics, where he is now a member of the Differential Geometry Research Group. The author of several research papers, Caminha was distinguished by a CNPq Research Grant on Differential Geometry. He is also an Affiliate Member of the Brazilian Academy of Sciences. Prior to his academic career, Caminha was himself an Olympic competitor, who has placed 4th in the 1990 Brazilian Mathematical Olympiad. Subsequently, as a high school teacher in the 1990s, he coached Brazilian students in preparation for various mathematical competitions, from regional meets to the Iberoamerican Mathematical Olympiad and the International Mathematical Olympiad, where several of them were medalists. He was also a Leader of the Brazilian Team at the 1996 and 1999 South Cone Mathematical Olympiad, and Deputy Leader of the Brazilian Team at the 1995 and 2001 International Mathematical Olympiads. In 2012, Caminha published a six-volume book collection entitled Topics in Elementary Mathematics with the Brazilian Mathematical Society, which gave rise to this book. He also published a book on selected topics on Differential Geometry, especially the Bochner method and harmonic maps.

Contenu

Chapter 01- Elementary Counting Techniques.- Chapter 02- More Counting Techniques.- Chapter 03- Generating Functions.- Chapter 04- Existence of Configurations.- Chapter 05- A Glimpse on Graph Theory.- Chapter 06- Divisibility.- Chapter 07- Diophantine Equations.- Chapter 08- Arithmetic Functions.- Chapter 09- Calculus and Number Theory.- Chapter 10- The Relation of Congruence.- Chapter 11- Congruence Classes.- Chapter 12- Primitive Roots and Quadratic Residues.- Chapter 13- Complex Numbers.- Chapter 14- Polynomials. Chapter 15- Roots of Polynomials.- Cahpter 16- Relations Between Roots and Coefficients.- Chapter 17- Polynomials over R.- Chapter 18- Interpolation of Polynomials.- Chapter 19- On the Factorization of Polynomials.- Chapter 20- Algebraic and Transcendental Numbers.- Chapter 21- Linear Recurrence Relations.- Chapter 22- Hints and Solutions.