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This book is intended as a text for a problem-solving course at the first or second-year university level, as a text for enrichment classes for talented high-school students, or for mathematics competition training. It can also be used as a source of supplementary material for any course dealing with algebraic equations or inequalities, or to supplement a standard elementary number theory course. There are already many excellent books on the market that can be used for a problem-solving course. However, some are merely collections of prob lems from a variety of fields and lack cohesion. Others present problems according to topic, but provide little or no theoretical background. Most problem books have a limited number of rather challenging problems. While these problems tend to be quite beautiful, they can appear forbidding and discouraging to a beginning student, even with well-motivated and carefully written solutions. As a consequence, students may decide that problem solving is only for the few high performers in their class, and abandon this important part of their mathematical, and indeed overall, education.
List of contents
1 Algebraic Identities and Equations.- 1 Formulas for Powers.- 2 Finite Sums.- 3 Polynomials.- 4 Symmetric Polynomials.- 5 Systems of Equations.- 6 Irrational Equations.- 7 Some Applications of Complex Numbers.- 2 Algebraic Inequalities.- 1 Definitions and Properties.- 2 Basic Methods.- 3 The Use of Algebraic Formulas.- 4 The Method of Squares.- 5 The Discriminant and Cauchy's Inequality.- 6 The Induction Principle.- 7 Chebyshev's Inequality.- 8 Inequalities Between Means.- 9 Appendix on Irrational Numbers.- 3 Number Theory.- 1 Basic Concepts.- 2 Prime Numbers.- 3 Congruences.- 4 Congruences in One Variable.- 5 Diophantine Equations.- 6 Solvability of Diopha,ntine Equations.- 7 Integer Part and Fractional Part.- 8 Base Representations.- 9 Dirichlet's Principle.- 10 Polynomials.- 4 Hints and Answers.- 1 Hints and Answers to Chapter 1.- 2 Hints and Answers to Chapter 2.- 3 Hints and Answers to Chapter 3.
Report
"In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here." (Olaf Ninnemann, zbMATH 1055.00006, 2021)
From the reviews:
MATHEMATICAL INTELLIGENCER
"I have nothing but praise for this book, and I can't imagine a mathamtician who wouldn't want to own it."
MAA ONLINE
"This book contains over 300 examples and then over 700 exercises for the reader. The authors have even included a fourth chapter with hints and answers to the exercises. So, even if the solution isn't given in the back, at least the reader will have some hint about which direction to start looking. The book provides a nice introduction into many of the problems-solvers' 'tricks of the trade.' There is a lot presented here, so the reader (especially an undergraduate) may want to take the following approach: read just a few pages at a time, work out some of the exercises, then take some time to digest the material before going on."
