Mathematical Olympiad Challenges [Englisch] [Taschenbuch]

It should be noted that being an exceptional problem solver does not necessarily make one a good mathematician, but it helps. This is certainly true of the second author who is also a renowned mathematician in the field of knot theory and three dimensional topology.

As mentioned the two authors have a sterling record in the arena of problem solving and in coaching would be problem solvers. I am more familiar with Razvan Gelca who led the University of Michigan team to a top five finish in the highly competitive and extremely challenging Putnam exam. This exam is administered yearly and is open to all college students in North America; usually around 430 universities and colleges send teams to compete in the Putnam. The exam has been offered since the thirties and finishing at the top carries a great deal of prestige. Razvan's superior abilities led to the spectacular success of the Michigan team which was no mean feat.

My own experience with the book has been one of revelation with each passing page. I used the book to teach the problem solving course at the University of Michigan, Ann Arbor, and it helped me immensely. The book possesses a variety of topics in elementary mathematics, ranging from algebra to geometry to trigonometry to number theory. Each chapter is divided into sections and each section has a theme. In keeping with the theme, the authors mention some useful formulae and/or facts that may be used in that section. This is followed by a demonstration of some dazzling problem solving techniques applied to a couple of problems. This is then followed by a list of challenging problems of varying levels of difficulty, all related to the theme of the section. There are roughly 18 such sections and many, many problems to think about. The rest of the book, which is the bulk of it, is dedicated to providing elegant solutions to every problem posed in the first part. Occasionally a problem merits more than one solution and sometimes the way is pointed to some interesting mathematics. The authors also acknowledge the source of many of the problems in the book which is a good indicator of the pedigree of the problem. Almost every solution is a gem and each problem demands its own style of solution. As noted earlier, problem solving is a skill and the authors try and succeed in conveying that idea in the problems and solutions they present.

Here is a sample problem from the book; if you can't do it and want to know how, check out the book:

"Show that any cube can be divided into 'n' cubes for any integer 'n' bigger than 54."

In summary if you are interested in figuring out puzzles, if you are a problem solver of elementary mathematical problems, or if you are just plain curious how a large fraction of mathematicians got hooked on mathematics, I would highly recommend you give this book a try. You may learn something and may even enjoy yourself in the process.

The book is centered around a number of tools, tricks, techniques, as you want to name them, which are then used to solve a number of problems depending on them.

The problems are carefully arranged in increasing order of difficulty (maybe that's the reason I was able to solve some of the first entries!), so that the reader is not immediately discouraged by problems too hard to solve.

The actual selection of the problems clearly reflects the taste of the authors. Not a bad taste, I would say, if one notices that Titu Andreescu was the man behind the brilliant success of the USA olimpic team in 1994 (all whose members got then the maximum possible number of points).

I was one of the coaches of an olimpic team myself and I know how fast one goes through a list, I would say through *any* list of problems with these gifted guys. From this point of view, the book is an essential instrument for all who contemplate being involved in problem solving training.

But the book is a good teaching tool for high school teachers who wish to challenge their best students with more interesting problems. As another potential pool of customers, I would say that those old sea wolfs as myself, who get bored from time to time of the technicalities of professional maths will find this book a nice companion.

Yes, I like this book and I warmly recommend it to all lovers of problem solving.