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An Excursion through Elementary Mathematics, Volume III

Discrete Mathematics and Polynomial Algebra

  • Textbook
  • © 2018

Overview

Authors:
  1. Antonio Caminha Muniz Neto

    1. Universidade Federal do Ceará, Fortaleza, Brazil

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  • 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

Part of the book series: Problem Books in Mathematics (PBM)

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Table of contents (22 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Elementary Counting Techniques

    • Antonio Caminha Muniz Neto
    Pages 1-32

  3. More Counting Techniques

    • Antonio Caminha Muniz Neto
    Pages 33-66

  4. Generating Functions

    • Antonio Caminha Muniz Neto
    Pages 67-93

  5. Existence of Configurations

    • Antonio Caminha Muniz Neto
    Pages 95-123

  6. A Glimpse on Graph Theory

    • Antonio Caminha Muniz Neto
    Pages 125-156

  7. Divisibility

    • Antonio Caminha Muniz Neto
    Pages 157-191

  8. Diophantine Equations

    • Antonio Caminha Muniz Neto
    Pages 193-207

  9. Arithmetic Functions

    • Antonio Caminha Muniz Neto
    Pages 209-220

  10. Calculus and Number Theory

    • Antonio Caminha Muniz Neto
    Pages 221-242

  11. The Relation of Congruence

    • Antonio Caminha Muniz Neto
    Pages 243-268

  12. Congruence Classes

    • Antonio Caminha Muniz Neto
    Pages 269-281

  13. Primitive Roots and Quadratic Residues

    • Antonio Caminha Muniz Neto
    Pages 283-315

  14. Complex Numbers

    • Antonio Caminha Muniz Neto
    Pages 317-345

  15. Polynomials

    • Antonio Caminha Muniz Neto
    Pages 347-361

  16. Roots of Polynomials

    • Antonio Caminha Muniz Neto
    Pages 363-394

  17. Relations Between Roots and Coefficients

    • Antonio Caminha Muniz Neto
    Pages 395-416

  18. Polynomials Over \(\boldsymbol {\mathbb R}\)

    • Antonio Caminha Muniz Neto
    Pages 417-433

  19. Interpolation of Polynomials

    • Antonio Caminha Muniz Neto
    Pages 435-450

  20. On the Factorisation of Polynomials

    • Antonio Caminha Muniz Neto
    Pages 451-476
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Keywords

About this book

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.

Authors and Affiliations

  • Universidade Federal do Ceará, Fortaleza, Brazil

    Antonio Caminha Muniz Neto

About the author

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.

Bibliographic Information

  • Book Title: An Excursion through Elementary Mathematics, Volume III

  • Book Subtitle: Discrete Mathematics and Polynomial Algebra

  • Authors: Antonio Caminha Muniz Neto

  • Series Title: Problem Books in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-77977-5

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing AG, part of Springer Nature 2018

  • Hardcover ISBN: 978-3-319-77976-8Published: 26 April 2018

  • Softcover ISBN: 978-3-030-08590-2Published: 24 January 2019

  • eBook ISBN: 978-3-319-77977-5Published: 17 April 2018

  • Series ISSN: 0941-3502

  • Series E-ISSN: 2197-8506

  • Edition Number: 1

  • Number of Pages: XII, 648

  • Number of Illustrations: 24 b/w illustrations

  • Topics: Discrete Mathematics, Field Theory and Polynomials

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