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The practice of mathematics teaching and its scientific, methodical and didactic support in the system of in-service education of modern teachers generates a topical problem of modernization of operational and technological and reflexive functionality of competence-oriented learning of solving high complexity problems, among which the tasks of mathematical olympiads as an indicator of the quality of the established professional competence stand out.In the competency-based and methodical context of working with mathematically gifted students and preparing them for mathematical competitions, the transformation and genesis of the problem material, which is discussed with teachers on in-service training courses, are consistently considered from the perspective of forming productive convolved didactic structures with regard to the features of flexibility, differentiation of levels, algorithmic and structural recognizability, essential for creating convoluted associations.Implementation of the convergence for theoretical approaches to these methodical problems is hampered, for example, by the internal contradictions caused by the subject-object status of teachers undergoing professional development.Our researches and scientific and practical findings, including those aimed at overcoming such contradictions, consolidate the comprehensive use of balanced dynamic synergetic mechanisms based on the emergent effect (as opposed to more traditional mechanisms of dynamic transitions such as “educational activity ⟶ quasi-professional activity ⟶ educational-professional activity ⟶ professional activity”) in the practice of teacher professional development. Such interpretation fundamentally changes the significance and functions of the case method (a form of situational learning), depriving it of the features of an intermediate organizational form in the interpretation of other studies.In the course of the research the methods of systematic scientific and methodological analysis, synthesis, generalization of theoretical positions, modelling and practical conclusions are used.The article highlights and clarifies the structure and interaction of the components of professional competencies of the teacher, the specifics of the approach to designing and developing effective specialized competency-based cases, aimed at stimulating work with mathematically gifted students. The article pays attention to some differences between developing the methodical competencies of a future teacher of mathematics and improving the competencies of a practising teacher. The article presents a model example of a tested situational training devoted to an important class of olympiad-type geometry problems, accumulating a significant layer of mathematical skills of both teachers and students.&#x0D;Keywords: professional teacher development, scientific and methodical support, case method, basic competencies of teachers, productive didactic structures, mathematics teaching methodology, mathematically gifted students, olympiad-type problems in geometry.

PECULIARITIES OF MODELLING OF SPECIALIZED METHODICAL CASES IN THE CONTEXT OF PROFESSIONAL DEVELOPMENT OF MATHEMATICS TEACHERS