The purpose of this article is to determine the role and place of school geometry in the subject olympiad system. For this, the authors turn to the experience of Russia in organizing and conducting geometric olympiads for schoolchildren, exploring the specifics of the olympiads named after named I.F. Sharygin, named S.A. Anischenko, named A.P. Savina, Moscow and Iran olympiads. The objectives and themes of full-time, extramural, oral geometric olympiads are defined. It is revealed that the topics of topology, projective, affine, combinatorial sections of geometry constitute the content of olympiad geometry. The study showed that the tasks of the olympiad work on geometry checked mathematical skills to perform actions with geometric figures, coordinates and vectors; build and explore simple mathematical models; apply acquired knowledge and skills in practical activities. The conclusions are made about the need to include tasks of geometric content in the block of olympiad tasks for the development of spatial thinking of schoolchildren.

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The main content of the article is devoted to the geometric problems of mathematical school olympiads. The study revealed the types of operating with spatial images in the process of solving problems, the stages of forming spatial representations of students in the study of geometry and objectives of the course of visual geometry. It was concluded that the formation of spatial, topological, spatial, projective representations goes through successive stages, developing the geometric skills of schoolchildren. Olympiad tasks, designed on the basis of school programs and textbooks on geometry, make it possible to check the formation of the geometric skills of schoolchildren. The article may be of interest to mathematics teachers, students and schoolchildren who are interested in methods of solving olympiad problems in geometry.