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This book describes mini-courses in a Mathematical "Circle," i.e., an organization that discovers and nurtures young mathematical talents through meaningful extra-curricular activities. This is the third volume in a trilogy describing in particular the S.M.A.R.T. Circle project, which was founded in Edmonton, Canada in 1981. The acronym S.M.A.R.T. stands for Saturday Mathematical Activities, Recreations & Tutorials.
This book, Volume III, consists of mini-courses and explains what actually takes place in the Circle. Volume I describes how to run a Circle, and Volume II, consisting of student projects, addresses the purpose of the Circle. All three volumes provide a wealth of resources (mathematical problems, quizzes and games, together with their solutions). The books will be of interest to self-motivated students who want to conduct independent research, teachers who work with these students, and teachers who are currently running or planning to run Mathematical Circles of their own.

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Preface.- Acknowledgement.- Table of Contents.- Part I. Geometric Topics.- Chapter 1. Area and Dissection.- Section 1. Qualitative and Quantitative Treatments of Area.- Section 2. The Bolyai-Gerwin Theorem and Pythagoras´ Theorem.- Section 3. Dissection Problems.- Chapter 2. Projective Geometry.- Section 1. Synthetic Approach.- Section 2. Metric Approach.- Section 3. Analytic Approach.- Chapter 3. Conic Sections.- Section 1. Loci.- Section 2. The Parabola.- Section 3. Ellipses and Hyperbolas.- Chapter 4. Inversive Geometry.- Section 1. Inversion.- Section 2. Applications to Euclidean Geometry.- Section 3. Mohr-Mascheroni Constructions.- Chapter 5. Convexity.- Section 1. Figures.- Section 2. Convex Figures.- Section 3. Figures of Constant Width.- Part II. Other Topics.- Chapter 6. Balancing Problems.- Section 1. Identifying Fake Coins.- Section 2. Other Problems.- Section 3. Other Balances.- Chapter 7. Graph Theory.- Section 1. Basic Concepts.- Section 2. Trees.- Section 3. Directed Graphs.- Chapter 8. Beanstalks.- Section 1. Red and Blue Beanstalks.- Section 2. Infinite Beanstalks.- Section 3. Beansprouts.- Chapter 9. Inequalities.- Section 1. The Rearrangement Inequality.- Section 2. The Majorization Inequality.- Section 3. Trigonometric Inequalities.- Chapter 10. Polynomial Equations.- Section 1. Complex Numbers.- Section 2. Cubic Equations.- Section 3. Quartic Equations.

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Andy Liu is an established author with eleven books to his credit. He is Professor Emeritus of the Department of Mathematical and Statistical Sciences at the University of Alberta, Canada. He has won numerous international awards in Mathematics teaching and outreach, as have several of his former students. He was the leader of the Canadian team to the International Mathematical Olympiad in 2000 (South Korea) and in 2003 (Japan), acts as Vice President of the Tournament of Towns and ran the S.M.A.R.T. Circle for over 30 years.

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