The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics

by D.O. Shklarsky, N.N. Chentzov, Isaak M. Yaglom

This book contains 320 unconventional problems in algebra, arithmetic, elementary number theory, and trigonometry. Most of the problems first appeared in competitive examinations sponsored by the School Mathematical Society of the Moscow State University and the Mathematical Olympiads held in Moscow. Although most of the problems presuppose only high school mathematics, they are not easy; some are of uncommon difficulty and will challenge the ingenuity of any research mathematician. Nevertheless, many are well within the reach of motivated high school students and even advanced seventh and eighth graders.
The problems are grouped into twelve separate sections. Among these the divisibility of integers, equations having integer solutions, evaluating sums and products, miscellaneous algebraic problems, the algebra of polynomials, complex numbers, problems of number theory, distinctive inequalities, difference sequences and sums, and more.
Complete solutions to all problems are given; in many cases, alternate solutions are detailed from different points of view. Solutions to more advanced problems are given in considerable detail. Moreover, when advanced concepts are employed, they are discussed in the section preceding the problems. Useful in a variety of ways in high school and college curriculums, this challenging volume will be of particular interest to teachers dealing with gifted and advanced classes.

Genres: Mathematics, Science

480 pages, Paperback

First published September 28, 1993

Reviews

[Jose Moa · Rating 4 out of 5]

This is a collection of 320 problems proposed in the USSR mathematical olympiads ,i think till the 50s,many of them very difficult to solve and some extremely difficult,that could challenge proffesional mathematicians.
The neccesary background is high school mathematics and are more focussed in numeric or algebraic tricks that in deep concepts,yet are some subjects as the little Fermat theorem,Euler theorem and Wilson theorem concerning elementary number theory that are explained in the book.
The book dont have pure geometrical problems ,diferential,integral calculus,algebraic structure sets or topology.

All problems have explained solutions and for those that dont want a complete solution it has a chapter of hints

The problems are classified with more or less fortune in the following chapters:
Introductory Problems
Alterations of digits in integers
The divisibility of integers
Some problems from arithmetic
Equations having integer solutions
Evaluating sums and produts
Miscellaneus problems from algebra
The algebra of polinomials
Complex numbers
Some problems in number theory
some distinctive inequalities
Difference sequences and sums

I think is a good book for those high school students that want to participate in mathematical competitions and for anybody that have a certain mathematical background or interest, because the book have problems containing elegant , curious or striking results

[Murilo Andrade · Rating 3 out of 5]

This is a very interesting book. As many USSR books, it is very heavy on computations rather than modern ideas. It offers 320 problems from many areas of mathematics, all with (very) elementary solutions. There is almost no prerequisite to read it, and although it usually does not show the "best" (i.e., most simple, elegant or clean) solution, it generally shows the low risk path of heavy calculations. The last (american) edition is still plenty of typos and small errors, but they are in general easily identifiable. Note that it is also a big book, mainly because the solutions are very long, in favor of being elementary.

The problems of the book are divided by field into a dozen chapters, and within chapters, problems are roughly grouped by similarity and ordered by difficulty. I found particularly useful the first 3 chapters, as well as the Algebra one.

[William Schram · Rating 5 out of 5]

Very good. Includes 320 problems divided into different categories along with thorough solutions. Includes recreational problems and some real head-scratchers. Some merely require logic, while other problems require some grounding in trigonometry and algebra. Most of the book is devoted to the solutions to the problems, so you aren't left hanging at the end of the day.

I mostly got this book to recover some of my lost math, and a lot of these problems will surely get you more of a grounding in math.

[Frog and or Toad · Rating 4 out of 5]

I will probably never read this entire book, but thumbing through it in the bookstore and working through some of the problems in my head was highly enthralling. Recommended if you like math.