This book is a translation from Russian of Part II of the book Mathematics Through Problems: From Olympiads and Math Circles to Profession. Part I, Algebra, was recently published in the same series. Part III, Combinatorics, will be published soon. The main goal of this book is to develop important parts of mathematics through problems. The authors tried to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover and recreate much of elementary mathematics and start edging into more sophisticated topics such as projective and affine geometry, solid geometry, and so on, thus building a bridge between standard high school exercises and more intricate notions in geometry. Definitions and/or references for material that is not standard in the school curriculum are included. To help students that might be unfamiliar with new material, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions. The book is based on classes taught by the authors at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Author(s): Alexey A. Zaslavsky, Mikhail B. Skopenkov
Series: MSRI Mathematical Circles Library
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 177
Tags: geometry problems; math olympiads;
Cover
Title page
Foreword
Problems, exercises, circles, and olympiads
Why this book, and how to use it
English-language references
Introduction
What this book is about and who it is for
Learning by solving problems
Parting words By A.Ya.Kanel-Belov
Olympiads and mathematics
Research problems for high school students
How this book is organized
Resources and literature
Acknowledgments
Numbering and notation
Notation
References
Chapter 1. Triangle
1. Carnot’s principle (1) By V.Yu.Protasov and A.A.Gavrilyuk
Suggestions, solutions, and answers
2. The center of the inscribed circle (2) By V.Yu.Protasov
Suggestions, solutions, and answers
3. The Euler line By V.Yu.Protasov
Suggestions, solutions, and answers
4. Carnot’s formula (2*) By A.D.Blinkov
Suggestions, solutions, and answers
5. The orthocenter, orthotriangle, and nine-point circle (2) By V.Yu.Protasov
Suggestions, solutions, and answers
6. Inequalities involving triangles (3*) By V.Yu.Protasov
Suggestions, solutions, and answers
7. Bisectors, heights, and circumcircles (2) By P.A.Kozhevnikov
Suggestions, solutions, and answers
8. \enquote{Semi-inscribed} circle (3*) By P.A.Kozhevnikov
Main series of problems—1
Main series of problems—2
Supplementary problems—1
Supplementary problems—2
Suggestions, solutions, and answers
9. The generalized Napoleon’s theorem (2*) By P.A.Kozhevnikov
Introductory problems
Formulation and proof of the generalized Napoleon’s theorem
Suggestions, solutions, and answers
10. Isogonal conjugation and the Simson line (3*) By A.V.Akopyan
Suggestions, solutions, and answers
Additional reading
Chapter 2. Circle
1. The simplest properties of a circle (1) By A.D.Blinkov
Suggestions, solutions, and answers
2. Inscribed angles (1) By A.D.Blinkov and D.A.Permyakov
Suggestions, solutions, and answers
3. Inscribed and circumscribed circles (2) By A.A.Gavrilyuk
Suggestions, solutions, and answers
4. The radical axis (2) By I.N.Shnurnikov and A.I.Zasorin
5. Tangency (2) By I.N.Shnurnikov and A.I.Zasorin
6. Ptolemy’s and Casey’s Theorems (3*) By A.D.Blinkov and A.A.Zaslavsky
6.A. Ptolemy’s Theorem
6.B. Casey’s Theorem
Suggestions, solutions, and answers
Chapter 3. Geometric transformations
1. Applications of transformations (1) By A.D.Blinkov
Suggestions, solutions, and answers
2. Classification of isometries of the plane (2) By A.B.Skopenkov
Hints
3. Classification of isometries of space (3*) By A.B.Skopenkov
Hints
4. An application of similarity and homothety (1) By A.D.Blinkov
Suggestions, solutions, and answers
5. Rotational homothety (2) By P.A.Kozhevnikov
5.A. Introductory problems involving cyclists
5.B. Main problems
5.C. Additional problems
Suggestions, solutions, and answers
6. Similarity (1) By A.B.Skopenkov
7. Dilation to a line (2) By A.Ya.Kanel-Belov
Suggestions, solutions, and answers
8. Parallel projection and affine transformations (2) By A. B. Skopenkov
Suggestions, solutions, and answers
9. Central projection and projective transformations (3) By A. B. Skopenkov
10. Inversion (2) By A. B. Skopenkov
Additional reading
Chapter 4. Affine and projective geometry
1. Mass points (2) By A.A.Gavrilyuk
Suggestions, solutions, and answers
2. The cross-ratio (2) By A.A.Gavrilyuk
Suggestions, solutions, and answers
3. Polarity (2) By A.A.Gavrilyuk and P.A.Kozhevnikov
Fundamental properties and introductory problems
Main problems
Additional problems
Suggestions, solutions, and answers
Additional reading
Chapter 5. Complex numbers and geometry (3) By A.A.Zaslavsky
1. Complex numbers and elementary geometry
Suggestions, solutions, and answers
2. Complex numbers and Möbius transformations
Additional problems
Suggestions, solutions, and answers
Additional reading
Chapter 6. Constructions and loci
1. Loci (1) By A.D.Blinkov
Suggestions, solutions, and answers
2. Construction and loci problems involving area (1) By A.D.Blinkov
Suggestions, solutions, and answers
3. Construction toolbox (2) By A.A.Gavrilyuk
Suggestions, solutions, and answers
4. Auxiliary constructions (2*) By I.I.Shnurnikov
Suggestions, solutions, and answers
Additional reading
Chapter 7. Solid geometry
1. Drawing (2) By A.B.Skopenkov
Suggestions, solutions, and answers
2. Projections (2) By M.A.Korchemkina
2.A. Projections of figures constructed from cubes
2.B. Trajectories
3. Regular polyhedra (3)
3.A. Inscribed and circumscribed polyhedra By A.Ya.Kanel-Belov
Suggestions, solutions, and answers
3.B. Symmetries By A.B.Skopenkov
4. Higher-dimensional space (4*) By A.Ya.Kanel-Belov
4.A. Simplest polyhedra in higher-dimensional space By Yu.M.Burman and A.Ya.Kanel-Belov
4.B. Multi-dimensional volumes
4.C. Volumes and intersections
4.D. Research problems
4.E. Partitions into parts of smaller diameter By A.M.Raigorodsky
Suggestions, solutions, and answers
Additional reading
Chapter 8. Miscellaneous geometry problems
1. Geometric optimization problems (2) By A.D.Blinkov
Suggestions, solutions, and answers
2. Area (2) By A.D.Blinkov
Suggestions, solutions, and answers
3. Conic sections (3*) By A.V.Akopyan
Suggestions, solutions, and answers
4. Curvilinear triangles and non-Euclidean geometry (3*) By M.B.Skopenkov
Additional problems
Suggestions, solutions, and answers
Additional reading
Bibliography
Index
Back Cover
