For introductory undergraduate courses in mathematics and
problem-solving, students preparing for such academic contests as
the William Lowell Putnam Mathematical Competition, and advanced high
school students studying for the American Mathematical Olympiad.
This book presents the principles and specific problem-solving methods
that can be used to solve a variety of mathematical problems. The
book provides clear examples of various problem-solving methods accompanied
by numerous exercises and their solutions.
Introduces and explains specific problem-solving methods
(with examples) and then gives a set of exercises and complete
solutions for each method.
Each chapter includes an additional set of problems to
challenge the reader.
By studying the principles and applying them to the exercises,
the reader will gain problem-solving ability as well as general mathematical
Eventually, the reader should be able to produce results
that have the whole air of intuition.
Organized according to specific problem-solving techniques
in separate chapters. These techniques include:
Induction (chapter 4)
Pigeonhole principle (chapter 9)
Chapters and exercises are arranged in order of increasing
difficulty. Presents a wide variety of problemssome old favorites
and some new gems.
Problem sets illustrate significant mathematical ideas
and have elegant but not tedious solutions.
Some chapters also include a moderate amount of theory
in order to provide context.
Includes hundreds of worked-out examples.
2. Direct and Indirect Reasoning.
5. Specialization and Generalization.
8. Various Moduli.
9. Pigeonhole Principle.
10. Two-Way Counting.
11. Inclusion-Exclusion Principle.
12. Algebra of Polynomials.
13. Recurrence Relations and Generating Functions.
14. Maxima and Minima.
15. Means, Inequalities, and Convexity.
16. Mean Value Theorems.
17. Summation by Parts.
19. Deus Ex Machina.
20. More Problems.