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The Green Book of Mathematical Problems

Publisher: 
Dover Publications
Number of Pages: 
173
Price: 
14.95
ISBN: 
9780484695730

This book contains one hundred challenging mathematics problems drawn from undergraduate-level math competitions. The book is simply organized: problems, hints, and solutions. Of course, the book should prove useful to students preparing for any type of math competition, such as the William Lowell Putnam Mathematical Competition. But the book is also useful as a problem supplement in standard undergraduate courses. There are a variety of interesting techniques illustrated, some of which are enumerated below. Instructors can either assign these for homework problems, or else use these problems for lecture illustrations of techniques and topics. Throughout the review, a number (or numbers) indicates the corresponding problem(s) in the book.

The following basic topics are illustrated:

·         Groups: 91

·         Ordinary differential equations: 98

·         Matrices and determinants: 45, 57, 67, 94, 100

·         Pure Geometry: 37, 48, 60, 62, 88, 99

·         Discrete Mathematics (Diophantine equations, Recursions): 7–9, 39, 58, 82, 89

·         Number Theory (Primes): 55, 80

Additionally the following standard calculus topics are illustrated:

·         Formulae and inequality for finite sums: 11, 12, 27, 29, 43, 77, 79, 97

·         Infinite and Power series: 16, 20, 31, 35, 40, 95

·         Integrals: 6, 21, 26, 28, 42, 53, 59, 63, 64, 76, 86

·         Sequences: 14

The following special theorems are illustrated:

·         Rouché’s theorem: 4

·         Cauchy’s inequality: 15

·         Jordan’s inequality: 81

·         The Abel limit theorem: 76

·         DeMoivre’s theorem: 32

·         Hurwitz’s theorem: 62

·         The Bolzano-Weierstrass theorem: 98

·         Helly’s theorem: 88

·         The extended mean value theorem: 92

·         Dirichlet’s box principle: 47

Certain “exotic” topics, not necessarily covered in standard undergraduate courses, such as functional equations (75) and problems in sets of numbers (34, 49, 52) are also illustrated.

Finally, certain solutions illustrate solving a problem by breaking up the problem and enumerating cases (2, 100).

In conclusion, I think the book has adequate resources justifying it being on any instructor’s bookshelf.


Russell Jay Hendel (RHendel@Towson.Edu) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

Date Received: 
Friday, June 21, 2013
Reviewable: 
Yes
Include In BLL Rating: 
No
Kenneth Hardy and Kenneth S. Williams
Publication Date: 
2013
Format: 
Paperback
Audience: 
Category: 
Problem Book
Russell Jay Hendel
12/3/2013

 

Notation
The Problems
The Hints
The Solutions
Abbreviations
References

 

Publish Book: 
Modify Date: 
Friday, June 21, 2013

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