Challenging Mathematical Problems with Elementary Solutions
This book is the first of a two-volume translation and adaptation of
a well-known Russian problem book entitled Non-Elementary Problems
in an Elementary Exposition. * The first part of the original, Problems on
Combinatorial Analysis and Probability Theory, appears as Volume I, and
the second part, Problems from Various Branches of Mathematics, as
Volume II. The authors, Akiva and Isaak Yaglom, are twin brothers,
prominent both as mathematicians and as expositors, whose many excel-
lent books have been exercising considerable influence on mathematics
education in the Soviet Union.
This adaptation is designed for mathematics enthusiasts in the upper
grades of high school and the early years of college, for mathematics
instructors or teachers and for students in teachers' colleges, and for all
lovers of the discipline; it can also be used in problem seminars and
mathematics clubs. Some of the problems in the book were originally
discussed in sections of the School Mathematics Circle (for secondary
school students) at Moscow State University; others were given at
Moscow Mathematical Olympiads, the mass problem-solving contests
held annually for mathematically gifted secondary school students.
The chief aim of the book is to acquaint the reader with a variety
of new mathematical facts, ideas, and methods. The form of a problem
book has been chosen to stimulate active, creative work on the materials
presented.
The first volume contains 100 problems and detailed solutions to
them. Although the problems differ greatly in formulation and method
of solution, they all deal with a single branch of mathematics: combina-
torial analysis. While little or no work on this subject is done in American
high schools, no knowledge of mathematics beyond what is imparted
in a good high school course is required for this book. The authors have
tried to outline the elementary methods of combinatorial analysis with
some completeness, however. Occasionally, when needed, additional
explanation is given before the statement of a problem.
a well-known Russian problem book entitled Non-Elementary Problems
in an Elementary Exposition. * The first part of the original, Problems on
Combinatorial Analysis and Probability Theory, appears as Volume I, and
the second part, Problems from Various Branches of Mathematics, as
Volume II. The authors, Akiva and Isaak Yaglom, are twin brothers,
prominent both as mathematicians and as expositors, whose many excel-
lent books have been exercising considerable influence on mathematics
education in the Soviet Union.
This adaptation is designed for mathematics enthusiasts in the upper
grades of high school and the early years of college, for mathematics
instructors or teachers and for students in teachers' colleges, and for all
lovers of the discipline; it can also be used in problem seminars and
mathematics clubs. Some of the problems in the book were originally
discussed in sections of the School Mathematics Circle (for secondary
school students) at Moscow State University; others were given at
Moscow Mathematical Olympiads, the mass problem-solving contests
held annually for mathematically gifted secondary school students.
The chief aim of the book is to acquaint the reader with a variety
of new mathematical facts, ideas, and methods. The form of a problem
book has been chosen to stimulate active, creative work on the materials
presented.
The first volume contains 100 problems and detailed solutions to
them. Although the problems differ greatly in formulation and method
of solution, they all deal with a single branch of mathematics: combina-
torial analysis. While little or no work on this subject is done in American
high schools, no knowledge of mathematics beyond what is imparted
in a good high school course is required for this book. The authors have
tried to outline the elementary methods of combinatorial analysis with
some completeness, however. Occasionally, when needed, additional
explanation is given before the statement of a problem.
Thus the majority of the problems in this book and in its companion
volume represent questions in higher (Unon-elementary") mathematics
that can be solved with elementary mathematics. Most of the problems
in this volume are not too difficult and resemble problems encountered
in high school. The last three sections, however, contain some very
difficult problems. Before going on to the problems, the reader should
consult the uSuggestions for Using the Book."
The book was translated by Professor James McCawley, Jr., of the
University of Chicago and edited and revised by Professor Basil Gordon
of the University of California at Los Angeles.
Problem 85 was sent by the Russian authors for inclusion in the
American edition, and appears here for the first time. A number of
revisions have been made by the editor:
I. In order to make volume I self-contained, some problems were
transferred to volume II. To replace these, problems 1,3,12, and
100 were added. Problem 12, in which the principle of inclusion
and exclusion is presented, is intended to unify the treatment of
several subsequent problems.
2. Some of the problems have been restated in order to illustrate the
same ideas with smaller numbers.
3. The introductory remarks to section I, 2, 6, and 8 have been
rewritten so as to explain certain points with which American
readers might not be familiar.
4. Adaptation of this book for American use has involved these
customary changes: References to Russian money, sports, and
so forth have been converted to their American equivalents; some
changes in notation have been made, such as the introduction of
the notation of set theory where appropriate; some comments
dealing with personalities have been deleted; and Russian biblio-
graphical references have been replaced by references to books
in English, whenever possible.
The editor wishes to thank Professor E. G. Straus for his helpful
suggestions made during the revision of the book. The Survey wishes to
express its particular gratitude to Professor Gordon for the valuable
improvements he has introduced.
volume represent questions in higher (Unon-elementary") mathematics
that can be solved with elementary mathematics. Most of the problems
in this volume are not too difficult and resemble problems encountered
in high school. The last three sections, however, contain some very
difficult problems. Before going on to the problems, the reader should
consult the uSuggestions for Using the Book."
The book was translated by Professor James McCawley, Jr., of the
University of Chicago and edited and revised by Professor Basil Gordon
of the University of California at Los Angeles.
Problem 85 was sent by the Russian authors for inclusion in the
American edition, and appears here for the first time. A number of
revisions have been made by the editor:
I. In order to make volume I self-contained, some problems were
transferred to volume II. To replace these, problems 1,3,12, and
100 were added. Problem 12, in which the principle of inclusion
and exclusion is presented, is intended to unify the treatment of
several subsequent problems.
2. Some of the problems have been restated in order to illustrate the
same ideas with smaller numbers.
3. The introductory remarks to section I, 2, 6, and 8 have been
rewritten so as to explain certain points with which American
readers might not be familiar.
4. Adaptation of this book for American use has involved these
customary changes: References to Russian money, sports, and
so forth have been converted to their American equivalents; some
changes in notation have been made, such as the introduction of
the notation of set theory where appropriate; some comments
dealing with personalities have been deleted; and Russian biblio-
graphical references have been replaced by references to books
in English, whenever possible.
The editor wishes to thank Professor E. G. Straus for his helpful
suggestions made during the revision of the book. The Survey wishes to
express its particular gratitude to Professor Gordon for the valuable
improvements he has introduced.
Enregistrer un commentaire