An Excursion through Elementary Mathematics, Volume I

1. Front Matter

   Pages i-xiii

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2. The Set of Real Numbers

   • Antonio Caminha Muniz Neto

   Pages 1-18

3. Algebraic Identities, Equations and Systems

   • Antonio Caminha Muniz Neto

   Pages 19-59

4. Elementary Sequences

   • Antonio Caminha Muniz Neto

   Pages 61-86

5. Induction and the Binomial Formula

   • Antonio Caminha Muniz Neto

   Pages 87-110

6. Elementary Inequalities

   • Antonio Caminha Muniz Neto

   Pages 111-141

7. The Concept of Function

   • Antonio Caminha Muniz Neto

   Pages 143-200

8. More on Real Numbers

   • Antonio Caminha Muniz Neto

   Pages 201-244

9. Continuous Functions

   • Antonio Caminha Muniz Neto

   Pages 245-273

10. Limits and Derivatives

    • Antonio Caminha Muniz Neto

    Pages 275-345

11. Riemann’s Integral

    • Antonio Caminha Muniz Neto

    Pages 347-445

12. Series of Functions

    • Antonio Caminha Muniz Neto

    Pages 447-510

13. Back Matter

    Pages 511-652

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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.

• Functions
• Real numbers
• Real functions
• Elementary algebra
• Problem solving
• Mathematics Olympiad
• IMO
• matrix theory

“This book develops several basic concepts in elementary mathematics and which are needed in the preparation of mathematical Olympiads around the world. … Throughout the book, the author puts great emphasis on explaining the issues discussed. … The presentation is pedagogical and coherent throughout with emphasis on concepts and underlying ideas … . The author has succeeded in crafting a comprehensive book to give a well-rounded picture of the role of real numbers and functions in elementary mathematics.” (Teodora Liliana Rădulescu, zbMATH 1371.97002, 2017)

• Mathematics, Universidade Federal do Ceará, Fortaleza, Brazil

  Antonio Caminha Muniz Neto

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 2001International 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.

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