The IMO compendium
Problem Books in Mathematics
The International Mathematical Olympiad (IMO) is nearing its fiftieth an-
niversary and has already created a very rich legacy and firmly established
itself as the most prestigious mathematical competition in which a high-school
student could aspire to participate. Apart from the opportunity to tackle in-
teresting and very challenging mathematical problems, the IMO represents
a great opportunity for high-school students to see how they measure up
against students from the rest of the world. Perhaps even more importantly,
it is an opportunity to make friends and socialize with students who have
similar interests, possibly even to become acquainted with their future col-
leagues on this first leg of their journey into the world of professional and
scientific mathematics. Above all, however pleasing or disappointing the final
score may be, preparing for an IMO and participating in one is an adventure
that will undoubtedly linger in one’s memory for the rest of one’s life. It is
to the high-school-aged aspiring mathematician and IMO participant that we
devote this entire book.
The goal of this book is to include all problems ever shortlisted for the
IMOs in a single volume. Up to this point, only scattered manuscripts traded
among different teams have been available, and a number of manuscripts were
lost for many years or unavailable to many.
In this book, all manuscripts have been collected into a single compendium
of mathematics problems of the kind that usually appear on the IMOs. There-
fore, we believe that this book will be the definitive and authoritative source
for high-school students preparing for the IMO, and we suspect that it will be
of particular benefit in countries lacking adequate preparation literature. A
high-school student could spend an enjoyable year going through the numer-
ous problems and novel ideas presented in the solutions and emerge ready to
tackle even the most difficult problems on an IMO. In addition, the skill ac-
quired in the process of successfully attacking difficult mathematics problems
will prove to be invaluable in a serious and prosperous career in mathematics.
However, we must caution our aspiring IMO participant on the use of this
book. Any book of problems, no matter how large, quickly depletes itself if
niversary and has already created a very rich legacy and firmly established
itself as the most prestigious mathematical competition in which a high-school
student could aspire to participate. Apart from the opportunity to tackle in-
teresting and very challenging mathematical problems, the IMO represents
a great opportunity for high-school students to see how they measure up
against students from the rest of the world. Perhaps even more importantly,
it is an opportunity to make friends and socialize with students who have
similar interests, possibly even to become acquainted with their future col-
leagues on this first leg of their journey into the world of professional and
scientific mathematics. Above all, however pleasing or disappointing the final
score may be, preparing for an IMO and participating in one is an adventure
that will undoubtedly linger in one’s memory for the rest of one’s life. It is
to the high-school-aged aspiring mathematician and IMO participant that we
devote this entire book.
The goal of this book is to include all problems ever shortlisted for the
IMOs in a single volume. Up to this point, only scattered manuscripts traded
among different teams have been available, and a number of manuscripts were
lost for many years or unavailable to many.
In this book, all manuscripts have been collected into a single compendium
of mathematics problems of the kind that usually appear on the IMOs. There-
fore, we believe that this book will be the definitive and authoritative source
for high-school students preparing for the IMO, and we suspect that it will be
of particular benefit in countries lacking adequate preparation literature. A
high-school student could spend an enjoyable year going through the numer-
ous problems and novel ideas presented in the solutions and emerge ready to
tackle even the most difficult problems on an IMO. In addition, the skill ac-
quired in the process of successfully attacking difficult mathematics problems
will prove to be invaluable in a serious and prosperous career in mathematics.
However, we must caution our aspiring IMO participant on the use of this
book. Any book of problems, no matter how large, quickly depletes itself if
the reader merely glances at a problem and then five minutes later, having
determined that the problem seems unsolvable, glances at the solution.
The authors therefore propose the following plan for working through the
book. Each problem is to be attempted at least half an hour before the reader
looks at the solution. The reader is strongly encouraged to keep trying to solve
the problem without looking at the solution as long as he or she is coming up
with fresh ideas and possibilities for solving the problem. Only after all venues
seem to have been exhausted is the reader to look at the solution, and then
only in order to study it in close detail, carefully noting any previously unseen
ideas or methods used. To condense the subject matter of this already very
large book, most solutions have been streamlined, omitting obvious derivations
and algebraic manipulations. Thus, reading the solutions requires a certain
mathematical maturity, and in any case, the solutions, especially in geometry,
are intended to be followed through with pencil and paper, the reader filling
in all the omitted details. We highly recommend that the reader mark such
unsolved problems and return to them in a few months to see whether they
can be solved this time without looking at the solutions. We believe this to
be the most efficient and systematic way (as with any book of problems) to
raise one’s level of skill and mathematical maturity.
We now leave our reader with final words of encouragement to persist in
this journey even when the difficulties seem insurmountable and a sincere wish
to the reader for all mathematical success one can hope to aspire to.
determined that the problem seems unsolvable, glances at the solution.
The authors therefore propose the following plan for working through the
book. Each problem is to be attempted at least half an hour before the reader
looks at the solution. The reader is strongly encouraged to keep trying to solve
the problem without looking at the solution as long as he or she is coming up
with fresh ideas and possibilities for solving the problem. Only after all venues
seem to have been exhausted is the reader to look at the solution, and then
only in order to study it in close detail, carefully noting any previously unseen
ideas or methods used. To condense the subject matter of this already very
large book, most solutions have been streamlined, omitting obvious derivations
and algebraic manipulations. Thus, reading the solutions requires a certain
mathematical maturity, and in any case, the solutions, especially in geometry,
are intended to be followed through with pencil and paper, the reader filling
in all the omitted details. We highly recommend that the reader mark such
unsolved problems and return to them in a few months to see whether they
can be solved this time without looking at the solutions. We believe this to
be the most efficient and systematic way (as with any book of problems) to
raise one’s level of skill and mathematical maturity.
We now leave our reader with final words of encouragement to persist in
this journey even when the difficulties seem insurmountable and a sincere wish
to the reader for all mathematical success one can hope to aspire to.
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