Tasks with mathematical content in Olympiads and competitions in informatics

Mathematical concepts and algorithms are present in the basic and advanced informatics courses of primary and secondary schools, in the tasks of the Unified State Exam and tasks of Olympiads and competitions in programming. the content lines of mathematics and informatics overlap in many ways, mainly in the sections of discrete mathematics: mathematical logic, graph theory, combinatorics, number systems, modular arithmetic, etc. the article provides examples of mathematical tasks that are useful when studying various sections of the informatics course. the use of semi-invariants and invariants in solving mathematical tasks and verifying programs is considered. Examples of tasks on non-traditional number systems that are used in creating hardware and software for personal computers are given. the issues of remainder arithmetic related to modern information encryption standards are discussed. Most of the tasks included in the article are created for well-prepared and motivated students. they were proposed by the authors at various Olympiads and competitions in informatics for schoolchildren in the Yaroslavl region.