Reviews (2)

A great bookfor young students.
I bought this book to help me learn how to solve problems. However, when it arrived, I realised it was destined as a book for 12 to 14 year old students. Still, I gave it a try ( I am 19 years old). The problems are well stated, easy to do, and methodologicaly sound. I found the problems too easy, but my little brother ( 9 years old ) had trouble. It's great for some young students who would like to learn the basics of problem solving.

The Russians do Math Right
In sharp contrast to standard US math education, which
is generally a death march from algebra to calculus, this
book suggests a wonderful new way to organize the ideas
of elementary mathematics. The organizational principle
here is around fundamental ideas that underlie
every mathematical proof ever conceived: parity, the
pigeonhole principle, induction, counting (combinatorics),
etc. Each section starts off with easy problems that anyone
can get, and leads you through to more and more challenging
illustrations of that section's principle; the last problems
of each section are often quite sophisticated and rewarding.
Do the problems in this book, and you can't help but just
be smarter for it.

Reviews (4)

Difficult, but fun, problems.
The reviewer below seems rather confused. This book has problems which COULD be solved in a mechanical way, but almost all the solutions given are simple and creative. Futhermore, there are often two solutions given. I find the problems hard, but maybe I am just not very smart. How could this book alter your perception of truth? By giving you knowledge?

Mechanical versus Wise
Understanding is to understand and from that derive sequences of logical reasoning. While this kind of problem solving books merely stiffens one's (well, at least my) creativity and affects my perception of truth. Not worth reading if you are this kind of people.

Reviews (1)

Reviews (1)

Learning on the mathematical Olympiad problems
The present book is of obligatory reading for all those students and teachers interested in mathematical olympiads, as well as for the lovers of problems of ingenious solution.

The International Mathematics Olympiadbegan in 1959. It is an annual competition among the most brilliant highschool students of the world. The book contains questions and solutions ofthe seven international olympiads among 1978 (twentieth olympiad) and 1985(twenty-sixth olympiad) and it can be considered like a continuation ofSamuel L. Greitzer's book that contains the first 19 olympiads.

Contraryto the official solutions that the leaders of each delegation receiveduring an olympiad, those included in the book are very detailed. In manycases the problems include two alternating solutions. The theorems that arenecessary are demonstrated. A reading of these solutions gives a quite nearpanorama of what a student should know to participate with success in thistype of events. Obviously it is very useful also for a trainer ofstudents.

The book also includes 40 additional problems of the maintopics considered in these competitions:

Reviews (1)

Reviews (7)

This book is wonderful for what it attempts
[...]

This book does an incredibly admirable job of providing explanation of, and various methods of solving problems including, but not limited to, the irrationality of sqrt(2), the Fibbonacci Sequence and its relation to the Golden Ratio, Imaginary and Surreal numbers, and so on.

Vakil is an engaging and sensitive writer who engages the reader who has no knowledge of thse problems, to the reader who has a fundamental understanding of the topics. Ravi Vakil was my professor of multivariate calculus, and I think his marvelous personality comes through in this superb introduction to some of mathematics' finest puzzles.

Excellent --- for middle school to very early high school
I understand that Ravi Vakil has a great record in the IMO and Putnam competitions.When I read the reviews and the back cover on a web site which explicitly stated:"Reveals some of [Vakil's] most powerful problem solving strategies," -- Well, of course I bought it immediately.Unfortunately most of the things in the book are quite elementary as far as problem solving goes.It is meant as an exposure for people who haven't seen these things so far(particularly talented jr high /middle school and early high school).Of course you will find some very interesting facts in here (and the rare mistakes-- Euler's Formula being called DeMoivre's Theorem) and will leave many young students in awe as Mr. Vakil shows off some beautiful mathematics.It is a good book, but I gave it only 3 stars because I expected much more( much more "powerful" stuff as the backcover said).If your a very experienced problem solver you will find many of the things in this book trivial(you will find some gems though), but if your a beginner with some talent, then you will love this book.

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Excellent Book on Problem Solving
This is an awesome book on problem solving.While it is true that you should have a couple of seperate references to comprehend fully the mathematics the author uses, I'm sure it's because he assumes a working knowledge on much of the material (contrary to the preface that states that it is a transition from regular high school material---Well, this it is not!).Many exercises and Olympiad questions covering the material presented in each chapter are given.Hints and solutions to the exercises(but not Olympiad questions--whose answer you can find on many sites) are given.There are alot of interesting topics covered (numbers, combinatorics, algebra, induction, pigeonhole, etc) in ways that I hadn't seen them presented.This is an excellent buy for those interested in solving tough and interesting mathematics problems.

Reviews (5)

This book is amazing!
I remember the first time I touched this book, i fell so in love with it that it was very hard for me to remember how many other things I had to do during my day. It really illustrated how every problem you solve (or at least try really hard) can be an entire lesson you can use later on.
It is very well organized, even the problems in each section are set in a way that each one helps with the previous one in case a more creative solution doesn't show up...
I love this book, and I really recommend it for any student studying for any math contest around the world. It really helped me, and I'm sure it will do the exact same thing to anyone with the desire to spend countless hours solving beautiful math problems. Good luck, God bless you all :)
Pura vida.

Fantastic Book!
This is a marvelous book for lovers of mathematical problems.Scattered about are wonderful problems in Geometry, Trig, Algebra and Analysis, Invariants, and Number Theory.A truly delightful read that will have you working on some problems for hours.Each section introduces the reader to the concept or technique needed to solve the problems in each section.The problem sets start off with a few "warm-up" problems that quickly build up to some that require keen (some brilliant!) insight.A true gem among most problem books since this book is not merely a book of problems, but also contains clear presentations and introductions to various concepts in mathematics.The solutions are a true delight, the ingenuity and beauty of mathematical problem solving is captured exquisitely in this fabulous book.Highly Recommended.A++

Review for Mathematical Olympiad Challenges.
The book, Mathematical Olympiad Challenges", is a delightful book on problem solving written by two of the leaders of the craft. Mathematical problem solving is a skill that can be honed like any other and this book is an ideal tool for the job. Problem solving usually involves elementary mathematics; this does not mean "easy mathematics". An elementary mathematical problem is one that is easily stated and can be understood by anyone who has had basic training in the subject (up to calculus). The solution, though, may be quite hard and may require a great deal of ingenuity and thought.

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."

Reviews (14)

Essential for budding (and experienced) problem-solvers
I join the ranks of previous reviewers here who honestly feel that having read this book in high school would have almost certainly changed my life.I, too, did very well in high school math competitions, but the maturity I am gleaning from this gem may have vaulted me into a different league.

It contains hundreds of problems from various levels of competition, from AIME problems all the way through some of the toughest Putnam problems (which, if you know anything about the Putnam, are about as hard as competition problems come).But the biggest help are the vital insights and exciting ways of looking at these problems.Don't take my word for it--many past IMO contestants have suggested this book too.

Particularly helpful is the way the author divides the book into sections based on often-used concepts and techniques.For example, you will see applications of the pigeonhole principle from the most basic (e.g. "In a drawer with socks of 2 colors, show that after picking any 3 socks, we must have a pair of same-colored socks.") through some rather difficult ones (1994 Putnam A4, an Erdos problem, and more).

The same goes for a multitude of others--the invariants section includes both the classic chocolate bar-cutting problem and Conway's rather difficult checker problem.Then, not only does he solve the latter beautifully, but incorporates nontrivial questions that ensure the reader has completely understood the solution (e.g., "Could we have replaced lambda with an arbitrary integer?Why not?").

You don't have to be a math competition buff to gain from this book, however.If you're simply interested in mathematical puzzles and problems, and are looking to expand your repertoire, this book will help you.Anyone with a good dose of intelligence and motivation will benefit.

For an additional problem book, check out Mathematical Olympiad Challenges by Andreescu and Gelca.For purely Putnam treatment, there are several volumes written by Kedlaya.And if you're a CS student, looking for honing those CS math skills to be razor sharp, you should definitely look into Concrete Mathematics by Graham, Knuth, and Patashnik.

Happy solving.

The Book I wish I had in High School
When I was in high school, I placed second in the Alabama State Mathematics Contest and won many others. However, I might could have been competitive with the IMO style problems had I had this book and would be much better off today had I seen this book earlier.

This book is for the exceptionally brilliant and the mentally tough.It is absolutely necessary to approach this book in a different way from a standard math textbook. You MUST attempt the examples BEFORE looking at the example solutions, NO MATTER HOW DIFFICULT OR FRUSTRATING. You may be bamboozled by the problems, but even trying to understand the problems before looking at the solutions and thinking about how a solution might proceed will pay huge dividends in the long run.

For example, in the first chapter Zeitz presents an example asking the reader to prove that the product of four consecutive integers cannot be a perfect square. The solution involves some clever algebraic trickery not visible to the inexperienced, but persistence and getting your hands dirty is key.

If you persist in spite of the considerable difficulty, you will find that you get better very, very quickly. You will also notice that it isn't just contest problems it helps you solve. I have found that I have solved my homework sets in the Berkeley graduate engineering program much more easily since working these problems. You will start to see creative and clever solutions where they exist in everything problem oriented.

PATIENCE PATIENCE PATIENCE!

Brilliant
As a high school student that is essentially bored with the regular, ho-hum classes that my school offers, this book is perfect.It gives a problem-solving foundation for math enthusiasts desiring to compete nationally in contests like the AMC, AIME, and USAMO.The problems are excellent and cover a wide range of difficulty (past ASHMEs, USAMOs, and, finally, IMOs); and the solutions are well-written, logical, and intelligible.In short, if you are looking to "get better" at problem solving, this is the book for you.