Introduction to Math Olympiad Problems

"Introduction to Math Olympiad Problems aims to introduce high school students to all the necessary topics that frequently emerge in international Math Olympiad competitions. In addition to introducing the topics, the book will also provide several repetitive-type guided problems to help develop vital techniques in solving problems correctly and efficiently. The techniques employed in the book will help prepare students for the topics they will typically face in an Olympiad-style event, but also for future college mathematics courses in Discrete Mathematics, Graph Theory, Differential Equations, Number Theory and Abstract Algebra. Features Numerous problems designed to embed good practice in readers, and build underlying reasoning, analysis, and problem-solving skills. Suitable for advanced high school students preparing for Math Olympiad competitions."-- Provided by publisher This aims to introduce high school students to all the necessary topics that frequently emerge in international Math Olympiad competitions. In addition to introducing the topics, the book will also provide several repetitive–type guided problems to help develop vital techniques in solving problems correctly and efficiently. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Contents 6 Preface 10 Author Bio 12 Acknowledgments 14 Chapter 1: Introduction 16 1.1. Patterns and Sequences 16 1.2. Integers 18 1.3. Geometry 19 1.4. Venn Diagrams 23 1.5. Factorial and Pascal's Triangle 25 1.6. Graph Theory 26 1.7. Piecewise Sequences 27 1.8. Chapter 1 Exercises 28 Chapter 2: Sequences and Summations 32 2.1. Linear and Quadratic Sequences 35 2.2. Geometric Sequences 37 2.3. Factorial and Factorial–Type Sequences 38 2.4. Alternating and Piecewise Sequences 40 2.5. Formulating Recursive Sequences 41 2.6. Solving Recursive Sequences 44 2.7. Summations 46 2.8. Chapter 2 Exercises 49 Chapter 3: Proofs 52 3.1. Algebraic Proofs 52 3.2. Proof by Induction 55 3.3. Chapter 3 Exercises 57 Chapter 4: Integers' Characteristics 60 4.1. Consecutive Integers 60 4.2. Prime Factorization and Divisors 62 4.3. Perfect Squares 64 4.4. Integers' Ending Digits 66 4.5. Chapter 4 Exercises 68 Chapter 5: Pascal's Triangle Identities 70 5.1. Horizontally Oriented Identities 73 5.2. Diagonally Oriented Identities 77 5.3. Binomial Expansion 79 5.4. Chapter 5 Exercises 81 Chapter 6: Geometry 84 6.1. Triangular Geometry 84 6.1.1. Isosceles Triangles 86 6.1.2. 30–60–90 Triangles 89 6.1.3. 45–45–90 Triangles 91 6.1.4. Additional Right Triangles 92 6.2. Area and Perimeter Geometry 96 6.3. Geometry and Proportions 103 6.4. Chapter 6 Exercises 109 Chapter 7: Graph Theory 116 7.1. Degrees of Vertices and Cycles 116 7.2. Regular Graphs 124 7.3. Semi-Regular Graphs 125 7.4. Hamiltonian Cycles 127 7.5. Chapter 7 Exercises 132 Chapter 8: Answers to Chapter Exercises 136 8.1. Answers to Chapter 1 Exercises 136 8.2. Answers to Chapter 2 Exercises 138 8.3. Answers to Chapter 4 Exercises 139 8.4. Answers to Chapter 5 Exercises 140 8.5. Answers to Chapter 6 Exercises 141 8.6. Answers to Chapter 7 Exercises 141 Chapter 9: Appendices 144 9.1. Venn Diagram 144 9.2. Angular Geometry 145 9.3. Right Triangles 146 9.4. Isosceles Triangles 148 9.5. Equilateral Triangles 148 9.6. Area of Figures 149 9.7. Patterns (Sequences) 150 9.8. Alternating Patterns (Sequences) 151 9.9. Summation Properties 152 9.10. Finite Summations 152 9.11. Laws of Exponents 153 9.12. Factoring Methods 154 9.13. Binomial Expansion 154 Bibliography 156 Index 158 Pascal's,Triangle;,Summations;,Factorial;,Recursive,Sequences;,Algebraic,Proofs;,Induction;,Integers;,Divisors;,Binomial,Expansion;,Triangular,Geometry;,Proportions;,Graphs;,Hamiltonian,Cycles;,Triangles;,Exponents Pascal's Triangle,Summations,Factorial,Recursive Sequences,Algebraic Proofs,Induction,Integers,Divisors,Binomial Expansion,Triangular Geometry,Proportions,Graphs,Hamiltonian Cycles,Triangles,Exponents This book aims to introduce high school students to all the necessary topics that frequently emerge in international Math Olympiad competitions. In addition to introducing the topics, the book will also provide several repetitive–type guided problems to help develop vital techniques in solving problems correctly and efficiently.