Problem-Solving Through Problems

Problem-Solving Through Problems is a practical anthology of some of the best elementary problems in different branches of mathematics. This classroom tested material will teach its readers important principles and broad strategies for coping with the experience of solving problems. It has been found very helpful for students preparing for the Putnam exam.

Résumé

From the reviews:

"This is a very welcome addition. The main message of the book is that the only way to learn to solve problems is to solve problems! I found this book very helpful. I am quite sure the book will be in constant use and I have no hesitation in recommending it." (The Mathematical Gazette)

Contenu

1. Heuristics.- 1.1. Search for a Pattern.- 1.2. Draw a Figure.- 1.3. Formulate an Equivalent Problem.- 1.4. Modify the Problem.- 1.5. Choose Effective Notation.- 1.6. Exploit Symmetry.- 1.7. Divide into Cases.- 1.8. Work Backward.- 1.9. Argue by Contradiction.- 1.10. Pursue Parity.- 1.11. Consider Extreme Cases.- 1.12. Generalize.- 2. Two Important Principles: Induction and Pigeonhole.- 2.1. Induction: Build on P(k).- 2.2. Induction: Set Up P(k + 1).- 2.3. Strong Induction.- 2.4. Induction and Generalization.- 2.5. Recursion.- 2.6. Pigeonhole Principle.- 3. Arithmetic.- 3.1. Greatest Common Divisor.- 3.2. Modular Arithmetic.- 3.3. Unique Factorization.- 3.4. Positional Notation.- 3.5. Arithmetic of Complex Numbers.- 4. Algebra.- 4.1. Algebraic Identities.- 4.2. Unique Factorization of Polynomials.- 4.3. The Identity Theorem.- 4.4. Abstract Algebra.- 5. Summation of Series.- 5.1. Binomial Coefficients.- 5.2. Geometric Series.- 5.3. Telescoping Series.- 5.4. Power Series.- 6. Intermediate Real Analysis.- 6.1. Continuous Functions.- 6.2. The Intermediate-Value Theorem.- 6.3. The Derivative.- 6.4. The Extreme-Value Theorem.- 6.5. Rolle's Theorem.- 6.6. The Mean Value Theorem.- 6.7. L'Hôpital's Rule.- 6.8. The Integral.- 6.9. The Fundamental Theorem.- 7. Inequalities.- 7.1. Basic Inequality Properties.- 7.2. Arithmetic-Mean-Geometric-Mean Inequality.- 7.3. Cauchy-Schwarz Inequality.- 7.4. Functional Considerations.- 7.5. Inequalities by Series.- 7.6. The Squeeze Principle.- 8. Geometry.- 8.1. Classical Plane Geometry.- 8.2. Analytic Geometry.- 8.3. Vector Geometry.- 8.4. Complex Numbers in Geometry.- Glossary of Symbols and Definitions.- Sources.