The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004 (Problem Books in Mathematics) [Hardcover]

Duan Djuki (Author), Vladimir Jankovi (Author), Ivan Mati (Author), Nikola Petrovi (Author)

Book Description

Publication Date: February 23, 2006 | ISBN-10: 0387242996 | ISBN-13: 978-0387242996 | Edition: 1

This is the ultimate collection of challenging high-school-level mathematics problems. It is the result of a two year long collaboration to rescue these problems from old and scattered manuscripts, and produce the definitive source of IMO practice problems in book form for the first time. This book attempts to gather all the problems and solutions appearing on the IMO and contains a grand total of 1900 problems. It is an invaluable resource for high-school students preparing for mathematics competitions, and for anyone who loves math.

Editorial Reviews

Review

From the reviews:

"The International Mathematical Olympiad, or IMO is the premier international competition for talented high school mathematics students. … This book collects statements and solutions of all of the problems ever set in the IMO, together with many problems proposed for the contest. … serves as a vast repository of problems at the Olympiad level, useful both to students … and to faculty looking for hard elementary problems. No library will want to be without a copy, nor will anyone involved in mathematics competitions … ." (Fernando Q. Gouvêa, MathDL, March,2006)

From the Back Cover

The International Mathematical Olympiad (IMO) has within its almost 50-year-old history become the most popular and prestigious competition for high-school students interested in mathematics. Only six students from each participating country are given the honor of participating in this competition every year. The IMO represents not only a great opportunity to tackle interesting and challenging mathematics problems, it also offers a way for high school students to measure up with students from the rest of the world.

The IMO has sparked off a burst of creativity among enthusiasts in creating new and interesting mathematics problems. In an extremely stiff competition, only six problems are chosen each year to appear on the IMO. The total number of problems proposed for the IMOs up to this point is staggering and, as a whole, this collection of problems represents a valuable resource for all high school students preparing for the IMO.

Until now it has been almost impossible to obtain a complete collection of the problems proposed at the IMO in book form. "The IMO Compendium" is the result of a two year long collaboration between four former IMO participants from Yugoslavia, now Serbia and Montenegro, to rescue these problems from old and scattered manuscripts, and produce the ultimate source of IMO practice problems. This book attempts to gather all the problems and solutions appearing on the IMO, as well as the so-called "short-lists", a total of 864 problems. In addition, the book contains 1036 problems from various "long-lists" over the years, for a grand total of 1900 problems.

In short, "The IMO Compendium" is the ultimate collection of challenging high-school-level mathematics problems. It will be an invaluable resource, not only for high-school students preparing for mathematics competitions, but for anyone who loves and appreciates math.

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Product Details

• Hardcover: 760 pages
• Publisher: Springer; 1 edition (February 23, 2006)
• Language: English
• ISBN-10: 0387242996
• ISBN-13: 978-0387242996
• Product Dimensions: 9.5 x 6.4 x 1.5 inches
• Shipping Weight: 2.5 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #1,099,004 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

11 of 11 people found the following review helpful:

4.0 out of 5 stars Best IMO compendium but authors' English is not sufficient, August 28, 2008

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ILAN VARDI (Paris, France) - See all my reviews
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This review is from: The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004 (Problem Books in Mathematics) (Hardcover)

This is a great collection of IMO problems, but I am disappointed by the authors' command of English and sloppiness, especially since they have rewritten the problem statements to save space, often leading to unclear or even incorrect statements. For example, problem 4, IMO 1965 is incorrectly stated as: "Find four real numbers x1,x2,x3,x4 such that the sum of any of the numbers and the product of other three is equal to 2." The statement is not as clear as the original one found on the IMO website, the last four words can be replaced with the two words "equals 2" to save space, and the article "the" is missing (a typical native Slavic language speaker grammar error in English). Much more serious is that this formulation has the trivial answer 1,1,1,1 while the original question was to find all solutions, not just a single one. An error in problem statement is the worst mistake possible in a problem book, even worse than a solution error, and it is even worse here as it is for an original IMO problem, not the long or shortlisted one where this could have been understandable due to the large number of problems and the lesser attention given to them over the years. I just got the book yesterday, so have not checked it out thoroughly, but I expect to find many more inaccuracies and mistakes. For me this is not such a problem and I even find it motivating to look for errors, but it can be a serious problem for students using the book to learn about doing IMO problems.

Update, November 17, 2008. I have found more errors in IMO contest statements 1981.4, 1987.4, 1971.3, 1979.1, 1978.3, 1978.5, 1985.4, 1989.1, 1991.2. This includes errors in English (some of which make the statement longer than the original) as well as changing the problem statement for 1987.4 and 1989.1. The case 1989.1 is particularly serious. The original problem asked to partition the numbers 1,...,1989 into 117 sets each with equal sum, while the book asks for 17 sets instead. It seems that the book formulation avoids the subtleties of this problem by considering a less complicated formulation, the general case seems much harder when the number of sets in the partition is greater than the number of elements in each set (in particular I give a solution when this is not the case). Here is an outline of my analysis:

Consider the general problem of partitioning {1,...,n} into d sets of equal size and equal element sum, so n = c*d. Clearly, the problem is the same for the set {0,...,n-1}, and write these numbers as a cxd array whose jth row is 117j + i, i=0,...,d-1. Each permutation of {0,...,d-1} induces a permutation on a row. One then considers permutations x |-> x + a mod d (cyclical shift by a) and x |-> d-1-x (reverse a row). If c is even, then one changes the array by reversing c/2 rows and leaving c/2 rows unchanged. If c is odd and d <= c, then one changes the array by cyclically shifting d rows (row j shifted by j), and then, since c-d is even, one can reverse (c-d)/2 rows and leave (c-d)/2 rows unchanged. It is clear that the columns of the resulting array give a solution. This gives a solution for n = c*d with c even or n = c*d odd and d <=c. In particular, this solves the book version with d = 17, c = 117.

For n = c*d, odd and d > c, I could not find a simple analysis in this case. For example if n = 15, d = 5, I found an ad hoc solution using Sudoku style trial and error, but I don't see a pattern:

{1,10,13}, {2,8,14},{3,9,12},{4,5,15},{6,7,11}.

However, I did find a general solution when d = r*s and r <= s <= c. This solves the IMO problem since r = 9 < s = 13 < c = 17. Once again, consider the cxd array with jth row jd + i, i = 0,...,d-1. By the first part above, since r <= s, one can partition {0,...,d-1} into r sets B1,...,Br each of size s and all with equal sum. There is a permutation sigma of {0,...,d-1} in which each Bk is shifted cyclically by 1, note that sigma has order s. One repeatedly applies this permutation to rows 1,...,s of the array (that is, apply sigma^j to row j, j =0,...,s-1) and on the remaining c-s rows, half will be left the same and the other half reversed. The columns of the resulting array give the required partition of {0,...,n-1}.

8 of 8 people found the following review helpful:

5.0 out of 5 stars Rare collection of IMO problems/solutions in one volume, October 30, 2006

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science-buff (Fremont, CA, USA) - See all my reviews

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This review is from: The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004 (Problem Books in Mathematics) (Hardcover)

46-year compendium of all the IMO shortlist and longlist problems. This is a rare book as none of the other IMO problem books so far covers both long and short-list problems.

The book has a complete listing of all long and short list problems but only provides solutions to shortlist problems as the book is already thick enough as it is. Would be great if the authors could expand the book or have a sequel to cover the other equally interesting longlist problems.

As expected, this is a very expensive Springer-Verlag book. Highly recommended though.

4 of 4 people found the following review helpful:

5.0 out of 5 stars Outstanding collection of problems, November 27, 2006

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Y. Xiao - See all my reviews
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This review is from: The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004 (Problem Books in Mathematics) (Hardcover)

This book contains roughly 1900 problems in 45 years of IMO history. Most of the problems are accompanied with a solution. This is an excellent resource for anyone preparing for a high calibre math contest such as olympiads or the William Lowell Putnam Mathematical Competition.

This book also has an extremely comprehensive collection of definitions and theorems which span from analysis to algebra to combinatorics in the first part of the book, which is a nice summary of tools useable on math contests and in math courses.