An Excursion Through Elementary Mathematics, Volume I: Real Numbers and Functions

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 first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, 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 some of 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.

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 is 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.