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In the paper problems which organizers of Olympiads are faced considered. The approach to solve this problems suggested. This approach is based on activity theory and includes using rather simple constructive problems as a first step to more complicated theoretical ones. The experience of implementing this approach within the framework of the Olympiad in Discrete Mathematics and Theoretical Informatics is described. The focus is set on computer manipulators -interactive dynamic models of mathematical and informatical object.

Abstract. The paper presents an experience of work with gifted schoolchildren in the field of informatics within the International Contest "Bebras". The idea of tasks that develops algorithmic or computational thinking is presented. The on-line Russian site of the contest and its coverage of schoolchildren of Russia and Belarus is demonstrated.

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The development of information technologies makes it possible to record a large amount of data in the learning process, both on the results of performing educational operations and on the psychophysical characteristics of students. In this regard, a number of ethical, pedagogical and methodological problems arise. Among them are the disproportionate control actions to the volume and properties of information retrieved during monitoring and problems associated with ensuring the information security of the student. In contrast to natural monitoring, monitoring the management of teaching has as an object of training a person, who can be considered, from the point of view of management, as a highly organized information system. The article is devoted to the study of the problem of non-invasive monitoring, involving the redirection of most of the monitoring information to the student himself. As a definition of non-invasive monitoring, the following is proposed: non-invasive monitoring is a type of pedagogical feedback that is used exclusively for correcting the student’s actions and is prohibited from using to control the effectiveness and transfer the results outside class. The article analyzes 15 different teaching paradigms, which are explicitly or implicitly adhered to by teachers, organizing the educational process, from the point of view of compliance with the tasks of non-invasive monitoring. The methodological aspects of the implementation of non-invasive monitoring from the point of view of computer support of the learning process are also analyzed. It is shown how non-invasive monitoring can be supported by software that provides active interaction of the student with the subject environment and gives the teacher more freedom in planning and achieving strategic goals of managing the educational process.