THIS book is a translation of the second Polish edition, published in 1960, in which various improvements were made.The problems set at the contests require only a knowledge of school mathematics (i.e. elementary algebra, geometry and trigonometry) but are on the whole more difficult than the usual school exercises. Their degree of difficulty, however, is not uniform, for it is considered desirable that not only the most gifted pupils but also those of average ability should —with a certain effort— manage to solve some of the problems and gain a number of points at the contests.This book contains the problems set at the first five Olympiads. It has been prepared in order to provide secondary school pupils with suitable topics to be worked out on their own, individually or collectively. It aims at extending their knowledge of mathe­ matics and training them in mathematical thinking. Accordingly, the solutions of most of the problems have been given in an extend­ ed form, the readers’ attention being drawn to various details of the reasoning. In addition, several problems have been provided with commentaries containing generalizations or further development of the topics in question, including various supplementary data of elementary mathematics outside the scope of the school syllabus.