Original Czech edition published by Masaryk University, Brno, 1996
2000, XI, 344 p.
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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.
Content Level »Lower undergraduate
Keywords »algebra - diophantine equation - number theory - prime number
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.