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| Euclidean Geometry in Mathematical Olympiads |
Description :
This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage.
Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures.
The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions.
This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class.
AVANT PROPOS
This book is an outgrowth of five years of participating in mathematical olympiads, where geometry flourishes in great vigor. The ideas, techniques, and proofs come from countless resources—lectures at MOP∗ resources found online, discussions on the Art of Problem Solving site, or even just late-night chats with friends. The problems are taken from contests around the world, many of which I personally solved during the contest, and even a couple of which are my own creations.
As I have learned from these olympiads, mathematical learning is not passive—the only way to learn mathematics is by doing. Hence this book is centered heavily around solving problems, making it especially suitable for students preparing for national or international
olympiads. Each chapter contains both examples and practice problems, ranging from easy exercises to true challenges.
Indeed, I was inspired to write this book because as a contestant I did not find any resources I particularly liked. Some books were rich in theory but contained few challenging problems for me to practice on. Other resources I found consisted of hundreds of
problems, loosely sorted in topics as broad as “collinearity and concurrence”, and lacking
any exposition on how a reader should come up with the solutions in the first place. I have thus written this book keeping these issues in mind, and I hope that the structure of the book reflects this.
I am indebted to many people for the materialization of this text. First and foremost, I thank Paul Zeitz for the careful advice he provided that led me to eventually publish this book. I am also deeply indebted to Chris Jeuell and Sam Korsky whose careful readings of the manuscript led to hundreds of revisions and caught errors. Thanks guys!
I also warmly thank the many other individuals who made suggestions and comments on early drafts. In particular, I would like to thank Ray Li, Qing Huang, and Girish Venkat for their substantial contributions, as well as Jingyi Zhao, Cindy Zhang, and Tyler Zhu, among many others. Of course any remaining errors were produced by me and I accept sole responsibility for them. Another special thanks also to the Art of Problem Solving fora,from which countless problems in this text were discovered and shared. I would also like to acknowledge Aaron Lin, who I collaborated with on early drafts of the book.
Finally, I of course need to thank everyone who makes the mathematical olympiads possible the students, the teachers, the problem writers, the coaches, the parents. Math contests not only gave me access to the best peer group in the world but also pushed me to limits that I never could have dreamed were possible. Without them, this book certainly could not have been written.
Titre :Euclidean Geometry in Mathematical Olympiads
auteur(s) : Evan Chen
size : 5.7 Mb
file type : PDF
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