This book was based on the content of my lecture delivered in March 1953 to the participants of the 12th Odessa Mathematical Olympiad for high school students. The Olympiad was organized and held at the Faculty of Physics and Mathematics of Odessa State University. I.I. Mechnikov. The mentioned lecture contained only § § 2, 5 and 8 as set out in this book, the remaining paragraphs of no less interest, of course, could not be included in one two-hour lecture. The content of the book is quite accessible for students of the ninth and tenth grades, since according to the methods used to solve problems, it does not go beyond the course of high school mathematics, although in essence these are problems of higher mathematics.