Concepts and Problems for Mathematical Competitors

This book presents the mathematical methods that students need when preparing for competitions, whether for the International Mathematical Olympiad (IMO) for high school students or the Putnam competition for undergraduate students. 
The book features six parts, each subdivided into several chapters.  The six major sections address counting methods, number theory, inequalities and the theory of equations, metrical geometry, analysis, and number representations and logic. Each chapter provides detailed solutions to a series of problems as well as approximately 30 more problems; the text thus provides about 1,000 problems to be solved. In addition to its value to students preparing for competitions, this volume is a useful resource for those seeking a thorough and practical review of mathematical methods.
For high school, undergraduate, and graduate students in mathematics

Ramón Martin Rodríguez-Dagnino is a Professor in the Electrical and Computer Engineering Dept. at the Tecnologico de Monterrey (ITESM) in Monterrey, Mexico: Ramón M. Rodríguez-Dagnino teaches there as well. Alexander Sarana teaches at Zhytomyr University, Ukraine.

Part I. Counting Methods
1. Mathematical Induction
2. Counting in Two Ways
3. Correspondence or Equivalent Representations
4. Combinatorics
5. Invariants
6. Parity
7. Extremal Principle
8. Dirichlet's Principle
9. Graphs
Part II. Theory of Numbers
10. Divisibility and Remainders, Euclid's Algorithm
11. Equations with Integers
12. Rational and Irrational Numbers
Part III. Inequalities and Theory of Equations
13. Methods of Proving Inequalities
14. The Average Values, Cauchy's Inequality
15. Non-Standard Equations and Systems of Equations
16. Applying Inequalities when Solving Equations and Systems of Equations
17. Application of Properties of Functions
18. Problems Containing Whole and Fractional Part of Numbers
19. Functional Numbers
Part IV. Metrical Geometry
20. Placement of Figures on a Plane, Coating, Cutting and Coloring Figures
21. Gaming Problems
22. Planimetric Problems
23. Transformation of the Plane, Geometric Constructions
24. Vector Methods
25. Geometric Inequalities and Extremes
26. Stereometry Problems
Part V. Analysis
27. Sequences
28. Limit of a Sequence and of a Function
29. Applications of Derivative and Integral
30. Parametric Problems
31. Jensen's Inequality
Part VI. Number Representations and Logic
32. Numbers with Some Given Properties
33. Logical Problems
Glossary
Bibliography