Nazar K. Agakhanov, Candidate of Sciences in Mathematical Physics, Associate Professor, Moscow Institute of Physics and Technology, President of the Central Content and Methodology Committee of the All-Russia Mathematical Olympiad for School Students, member of the International Mathematical Olympiad (IMO) Board. 
Address: 9 Institutsky ln, 141701 Dolgoprudny, Moscow Oblast, Russian Federation. E-mail: nazar_ag@mail.ru (corresponding author) 

Olga G. Marchukova, Candidate of Sciences in Pedagogy, Senior Lecturer, Department of Education and Psychology, Tyumen Oblast State Institute of Regional Education Development. 
Address: 56 Sovetskaya Str., 625000 Tyumen, Russian Federation. E-mail: vera-nadegda@bk.ru 

Oleg K. Podlipskii, Candidate of Sciences in Mathematical Physics, Associate Professor, Moscow Institute of Physics and Technology, Vice-President of the Central Content and Methodology Committee of the All-Russia Mathematical Olympiad for School Students. 
Address: 9 Institutsky Ln, 141701 Dolgoprudny, Moscow Oblast, Russian Federation. E-mail: ok@phystech.edu 

The evolution of math Olympiad training for school students in Russia since the second half of the 20th century is analyzed in this article in the context of sociocultural transformations and changes in the post-industrial society’s didactic knowledge. A retrospective analysis of a large body of empirical data (organization charts and content of the All-Union and All-Russia Mathematical Olympiads for school students in 1974–2021 and the International Mathematical Olympiads in 1994–2020) reveals trends in the development of the mathematical Olympiad movement, in particular changes in the network infrastructure of mathematical Olympiads, objectives and content of Olympiad problems, Olympiad material design practices, and value-and-meaning orientations of math Olympiads. Accordingly, new approaches are proposed to prepare school students for math competitions. 
Analysis of changes allows substantiating the insufficiency of the “cognitive-reproductive” method of school student training widely applied both in Russia and beyond, which is based on a thematic principle of selecting problems by content and demonstrating examples of their solving to students. This method does not conform to the objectives of contemporary math competitions. 
Today, it is important to find and recognize the potential for mathematical creativity in solving problems within multidisciplinary professional spheres. The goal of modern Olympiad training is not only to teach school students a system of problem-solving procedures but also to promote their ability to identify the semantic structure of problems in order to pick adequate solving strategies. The assumption that Olympiad math problems can be classified by the logic of their solutions is supported by a relevant taxonomy of Olympiad math problem solving techniques.