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111 Problems in Algebra and Number Theory

Adrian Andreescu and Vinjai Vale
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Allen Stenger
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This is an interesting and compact course in inequalities, polynomials, and number theory, taught through a large number of worked examples. In addition there are 111 problems to work yourself (that also include solutions and some hints). The problems are slanted very much toward the math olympiads (although this is not explicit): Nearly all solutions depend on making some clever observation, after which the solution is fairly short.

The problems are thus at the advanced high-school or early undergraduate level. The Algebra of the title means college algebra and is permeated by inequalities, even though it nominally deals in additional topics such as polynomials (especially factorization of polynomials and Gauss’s Lemma) and symmetric functions. It covers all the standard inequalities such as Cauchy–Schwarz, Hölder, and Arithmetic Mean–Geometric Mean, along with numerous clever specializations of them. It does not cover convexity or Jensen’s inequality, which is understandable since it does not assume calculus. The Number Theory portion is more traditional, covering divisibility, factoring, Euler’s Theorem, and some Diophantine equations.

The publisher, XYZ Press, is the publishing arm of Awesome Math, a training company founded n 2006 by well-known Olympiad coach Titu Andreescu. They put out a large number of similar problem books that are distributed through the AMS Bookstore.

Bottom line: a well-done book and a reasonable way to get introduced to these topics, although the true audience seems to be math competition contestants.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.