This book is an absolute must have for those of us who love challenging mathematical problems. The book claims in its preface to be a continuation of *Mathematical Contests 1997-1998: Olympiad Problems and Solutions from around the World*, published by the American Mathematics Competitions. I made more than one attempt to find a copy of that book when preparing this review and was unsuccessful. (I have now been informed that the books from 1995-1996, 1996-1997, and 1997-1998 can be ordered from the American Mathematics Competitions web page.) We are fortunate that the Mathematical Association of America has decided to begin a new series of books, the MAA Problem Books Series, and has begun that series with this wonderful collection of problems from around the world.

The concept for the book is very easy to describe. It includes eighteen National Mathematics Contests from 1998 (with solutions), and seven Regional Mathematics Contests also from 1998 (and also with solutions). It follows these with twenty-two National Mathematics Contests and eight Regional Contests from 1999. Additionally, there is a section which explains some of the notation used in the problems and a section which explains some of the terminology. Finally, there is a very nice index which lists all of the problems by type (Algebra, Combinatorics, Geometry, etc.). Presumably a followup volume will include the solutions from the 1999 contests and questions from the 2000 contests, and, hopefully, this will be an ongoing series. (Editor's note: It is indeed; see below.)

This is a wonderful book for puzzle lovers (and of course for students or teams preparing for this sort of contest). There is a rich collection of solved problems (sometimes with more than one solution given) which provides superb practice and a nice (and fun) way of learning problem solving techniques. The more recent sets of problems (those without solutions) could serve as good practice exams for teams and could be great starting points for discussions of solution techniques. The authors of the book state in the preface that "this collection is intended as practice for the serious student who wishes to improve his or her performance on the USAMO. ... The problems themselves should provide much enjoyment for all those fascinated by solving challenging mathematics questions."

It is pointed out in the preface of the book that the problems are far from being of uniform difficulty. This is certainly true. There are some problems which are very simple and some which are rather hard, and there is a wide variety of problems so there is something for everybody. I would like to share a bit of the flavor of the book by including a few of the problems it contains (for the solutions you will have to get a copy of the book). The first two problems listed below are from the 1998 Ireland National Contest. The third and fourth problems are from the 1998 Russian National Contest.

First Problem (an easy warm up): Show that a disc of radius 2 can be covered by seven (possibly overlapping) discs of radius 1.

Second Problem: Show that no integer of the form xyxy in base 10 can be the cube of an integer. Also find the smallest base b > 1 in which there is a cube of the form xyxy.

Third Problem: The roots of two quadratic polynomials are negative integers, and they have one root in common. Can the values of the polynomials at some positive integer be 19 and 98?

Fourth Problem: I choose a number from 1 to 144, inclusive. You may pick a subset of {1, 2, ..., 144} and ask me whether my number is in the subset. An answer of "yes" will cost you 2 dollars, an answer of "no" only 1 dollar. What is the smallest amount of money you will need to be sure to find my number?

This book belongs in the hands of everyone interested in competing in a National Mathematics Contest and would find a comfortable home in the mathematics collections of any undergraduate or high school library.

Carl D. Mueller ([email protected]) is Associate Professor of Mathematics at Georgia Southwestern State University in Americus, GA.