![]() ![]() Later (1868) it was proved that the two systems were equally consistent and as consistent as the real number system. In the late 19th century (approximately 1823), three different mathematicians (Bolyai, Lobachevsky and Gauss) proved independently that there was a different system that could be used that assumed the 5th postulate was incorrect. Some really great proofs were created by mathematicians trying to prove the parallel postulate. Mathematicians kept trying to prove that the 5th postulate (commonly known as the parallel postulate) could be proved from the first four postulates and thus was unnecessary. He also published a book on selected topics on Differential Geometry, especially the Bochner method and harmonic maps.There was a big debate for hundreds of years about whether you really needed all 5 of Euclid's basic postulates. 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 was also a Leader of the Brazilian Team at the 19 South Cone Mathematical Olympiad, and Deputy Leader of the Brazilian Team at the 19 International Mathematical Olympiads. 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. Prior to his academic career, Caminha was himself an Olympic competitor, who has placed 4th in the 1990 Brazilian Mathematical Olympiad. He is also an Affiliate Member of the Brazilian Academy of Sciences. The author of several research papers, Caminha was distinguished by a CNPq Research Grant on Differential Geometry. 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. Neto received his PhD from the Federal University of Ceará, Brazil in 2004. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.Īntonio Caminha M. 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. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. Examples and problems are discussed only if they are helpful as applications of the theory. ![]() It starts from the most basic theoretical principles, without being either too general or too axiomatic. This second volume covers Plane Geometry, Trigonometry, Space Geometry, Vectors in the Plane, Solids and much more.Īs 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 expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. ![]()
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