The famous problems formulated by David Hilbert at the Paris International Mathematical Congress in 1900 had a decisive influence on the development of mathematics in the 20th century. One of the goals of this brochure is to show that many well-known and quite complex mathematical problems arise in a completely natural way, so that even a high school student can understand the reasons for the appearance of these problems and their formulation. The text of the brochure is an adaptation of the recording of a lecture given by the author on October 23, 1999 at Small Faculty of Mechanics and Mathematics for schoolchildren in grades 9-11.