Reviews (9)

This book has most topics that you see at an Olympiad
This book has a lot of topics that you see at a mathematical competition, like IMO. It has Number Theory, Geometry, Games, Combinatorics and many strategies to solve a difficult and exciting problem. I recommend this book for you.

A book for everyone who loves mathematics
I disagree with another reviewer(Math Messiah) who thinks this is a book only for intelligent people like himslf. The problems are wide ranging and numerous and even dumb people (me, for instance) can find fun and excitement in many of them. I also have an advice for the Math Messiah who is on brink of some major discoveries: hold on to that brink, and don't drop.

Reviews (5)

Problem Solving at its best!
Finally, a book that develops problem solving techniques in a structured way for the Putnam. However, the techniques can be generally applicable to solve problems in many different areas.
The power of using "first principles" to solve problems that at first sight seem almost impossible is brought out clearly. (Example, show that any sequence of consecutive integers is divisible by n!. Where does even start to solve such problems? The book shows you how.)

You can read the book from the beginning to end, but a better way is to read it at random. Read the first chapter, though. It does a marvellous job of enumerating the different types of techniques.

Enjoy this book. You'll be amazed at how simple ideas can lead to difficult problems.

A stimulating book...
I dive inside the book. The organization by strategies is fine and ease the reading, the problems are challenging enough. If you are interessed in mathematics, student or not, buy it. You will not regret it.

How can I have not given this 5 stars yet?
I would like to suggest that the audience for this book is somewhat broader than just the "all-star" math students. I will indeed use this book to help students preparing to take the Putnam exam this fall - it is the best book I have seen for this purpose.

But in just the first two chapters, Larson presents strategies such as working backwards, modifying the problem, mathematical induction, and the pigeonhole principle, in a way which all math majors can benefit from. A graduating senior reported that once he started participating in math contests, his performance in ALL of his math classes improved. Problem solving unifies mathematical understanding. This student took the Putnam last fall (which, if you've got this far without knowing, is a six-hour national undergraduate mathematics competition taken by 3000 students in the US and Canada each year, approximately half of whom score zero) just to see what he could do, to gauge his improvement in mathematics. There isn't much more of a compliment a student could give me, intentional or otherwise.

That can't all be attributed to this book, but it is that good. The presentation is unique; the organization - by strategy, rather than by year or whatever you see in other problem books - is illuminating by itself, and has improved my pedagogy; and he just makes hard problems look easy.

Reviews (2)

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 (2)

The best reference for Proofs
This is an advanced book, with a lot of information on every page.I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about everymajor branch of mathematics.

It's highly recommended for anyone who is*serious* about mathematical proofs.Although the book is packed withmaterial, it's a small book, so it's one of the first I choose to take withme when I travel.

Reviews (3)

Very usefull book.
I teach students, a mathematical olympic team, and this book will help them to get the training they need in order to have more medalls and achievements in their carreers. It has everything they need to do their exams and practice.

Excellent introduction to problem solving
The last time I competed in a mathematical competition was in elementary school.So when I recently picked up and tried to solve some old Olympiad and Putnam questions I was dumbfounded, to say the least. Then I discovered "The Mathematical Olympiad Handbook." This book is an excellent intro to problem solving.It is an immense help in dealing with the frustration of problems that will not yield.No solutions are given-- only outlines to the solutions(which you should only read part of if you get stuck, as the outlines will easily lead to a solution).This structure was an excellent choice as it gives novice problem solvers a chance to solve problems on their own with a little help if they need it.Highly recommend!

[people new to problem solving should also check out "The Art and Craft of Problem Solving" by Paul Zeitz]

Reviews (1)

Eloquent imagery, but the characters aren't quite believable
The language in this book is incredible-- the metaphors perfectly capture the essence of what they describe, and the melodic rise and fall of the sentences mirrors the tragic (but in the end uplifting) plot into which thereader is drawn.As I read the book, I felt I was becoming an activeparticipant in it, taking part in the story and not just observing it, andit was so hard to put the book down when I didn't know how things wouldturn out!

The elegant simplicity of the solutions presented to theseemingly complex problems that face the characters is a biting comment onour own society's predilection for grinding out the answer to everything,when in reality what we need when confronting humanity's great issues is asense of perspective.

However, I found the characters somewhat flat. Why, for example, does the set N choose to define himself in such narrowterms?Has he no dreams, no fantasies, no weaknesses, nothing but arecursive definition?Many of the characters had similar problems; I haddifficulty relating to the motives behind some of their self-simplifyingactions.

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Reviews (7)

lots and lots of fun problems
I'm only starting the fourth chapter, "Some Problems from Arithmetic" or something like that, but I already love it. I'm not clever enough to solve more than few of the problems, but I find reading the solutions to the problems that I thought looked interesting to be very entertaining. Some of the solutions are so pretty and clever that I make my parents sit on my bed while I show them the best ones on my chalkboard (I'm just about start 11th grade). I highly recommend this book to anyone who appreciates the joy of numbers.

A Good Variety of Mind-Expanding Problems
This book consists of "320 unconventional problems in algebra, arithmetic, elementary number theory and trigonometry." The problems are mathematically accessible to students at the high school level or higher, as they call more upon analytical thinking than upon advanced mathematical techniques. There is a range of difficulties, with harder problems marked with stars in the book. (The hardest problems are marked with double stars.) The problems are divided into twelve sections: "Introductory Problems," "Alterations of Digits in Integers," "The Divisibility of Integers," "Some Problems from Arithmetic," "Equations Having Integer Solutions," "Evaluating Sums and Products," "Miscellaneous Problems from Algebra," "The Algebra of Polynomials," "Complex Numbers," "Some Problems of Number Theory," "Some Distinctive Inequalities," and "Difference Sequences and Sums." Much of the book is devoted to providing hints and solutions, which are both thorough and clear. This is a great resource for preparing for competition, for developing your analytical thinking, or just for having fun (that is, if you are the sort of person who finds solving math problems fun).

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Competition problems does not mean hard, only good
Whether it be known as the Soviet Union or Russia, there is a history of mathematical excellence in that country that few others can rival. The Leningrad Mathematical Olympiad is the oldest mathematics competition in Russia, having been founded in 1934 and being the impetus for other such competitions in the country.
There are several levels of the competition, starting at grade five and going through grade 10. This book presents the problems and solutions for those competitions for the years 1987 through 1991. At each level there are two sets of problems, one for the initial round and the second for the elimination round.Not only is this collection an excellent source of challenge type problems for use in middle and high school classes, competitions and clubs, they also provide insight into the level of mathematics instruction in Russia. While some of the problems are of course taxing, they are well stated and many rely more on insight rather than the application of heavy mathematical skill.
The phrase "competition problems"should mean well-stated and capable of direct solution rather than extremely difficult. These problems satisfy those requirements, and if you are looking for problems to challenge middle or high school students, this is a very good place to search.

Disclaimer: The author of this review is a member of the editorial board of The Contests in Mathematics Series.

Reviews (2)

Oh, no, this is not for young adults but for problem-lovers
This is a dream book for 3 kinds of people:

Anyone who is (has been) seriously practice(d) the mathematics contest. Any teacher/professer who tried(or is tring) to train students. Anyone who loves the intriguing mathematical problems and know the deep joy of solving problems.

This book contains more then 20 national MO problem-sets in 1998/1999 from around the world, as well as some missalaneous regional contest problems. For 1998's there are full solutions for each problem, while for 1999's there are only problems. It is obviously the next issue will be 1999/2000, with solutions for 1999 and problems for 2000. I hope this series will go on and on.

I must say that this book is never for the young adults but for those who love solving and appreciating 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.

Reviews (2)

Good reference for the budding problem solver.
I did Putnam in 2002 and solved 6 full problems. The future Putnam competitor should make it a point to consult past Putnam problems to successfully make the switch from olympiad problems to the Putnam. Not that it is a big change, but to do well in Putnam you have to know at least some basic trick in calculus, analysis and such. This is a good starting point in making that transition.

On a whole another level, even research mathematicians (problem solver type) can benefit from the 'tricks' and nuances of the 'toy' problems (e.g Putnam problems) in honing their problem solving instinct.

Reviews (1)

A well selected and organized problem set in combinatorics.
This is an excellent training source in combinatorics for mathematics olympiads and contests.

Problems in this book are divided into 2 sets, with each set containing 51 problems. Problems in the first set are introductory ones (hence, relatively easy), and that in the second set are advanced problems (And harder, much harder indeed in the last few problems). Problems are very well selected and carefully organized according to their difficulty and technic involved. They span from the very easy first some enumerative problems to the really hard IMO-level, even notorious National-MO problems. Moreover, problems are up-to-date hence it avoids well-known cliche problems. The book is published in the end of 2002, and in this book some of the 102 problems are selected from the IMO that year!

Reviews (4)

Infinitely Many Stars (Best of the Best)
Congratulations to the authors!!!
If you would like to have a mathematical problem book different
from the others, I highly recommend you to have this excellent book.
The book is not only one problem book, but also a rich resource
of very interesting notes about the origins of many very amazing
mathematical problems, theorems, and facts.
The book's style is very live, and I am a lover of the book.

Mathematical Miniatures
Mathematical Miniatures is a wonderful book, with a new approach to an old favorite topic: problem solving.

The book is comprised of 50 chapters, each one telling a mathematical story. The majority of theproblems come from math Olympiads and the most clever solutions and applications are always accompanied by the name of the problem solver. One cannot help but be in awe at amazing insights in solving tremendously complicated problems, performed by high school students under the constrictive time frame of the Olympiad tests.

The topics of the problems are reflective of the Math Olympiads topics, and will particularly appeal to the pure mathematician. But the beauty of the ingenious, albeit elementary mathematics will grab any passionate mathematician's attention. The sequence of 50 chapters - 50 mathematical topics is broken by nine "coffee breaks", and preceded by a warm up set. Each of the breaks proposes to the reader a handful of problems - some easy, some quite hard - all of them teasing and inviting. This is to say to the reader: don't just read about how others have cleverly solved problems, try your hand at some! I have to confess to falling for this trap several times and going through the house mumbling to myself "hmm... I used to get this in 5 minutes when I was in high school". Luckily all these problems come with solutions as well.

Overall, a delightful book, recommended not only for one's mathematical reading pleasure, but also for math clubs in either high school or college.

For the professional mathematician
A bit too over my head.Really a book for the professional, not the amateur.I love math, as explained in a college textbook, but this book was a bit too Poincaire-brief and condensed for me.I need a more verbose, baby-steps book.

So that's why I'm giving it two stars.Of course, once you are a master, then less is more.