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Publisher:

Dover Publications

Publication Date:

2013

Number of Pages:

173

Format:

Paperback

Price:

14.95

ISBN:

9780484695730

Category:

Problem Book

[Reviewed by , on ]

Russell Jay Hendel

12/3/2013

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 principl**e: 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.

Notation | |||||||

The Problems | |||||||

The Hints | |||||||

The Solutions | |||||||

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References |

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