Using the mapping method for solving some methodical problems

There is an opinion among many teachers that programming olympiads are mostly of a sporting nature and do not interfere scientific research. At the same time it has become a tradition to hold summer conferences of the Russian tournament of cities, where participants are offered various research problems. In general, these problems breaks down into a number of subtasks, that require research in some specific areas. In this paper a series of programming problems is discussed, each of which requires a separate study. All problems are similar in content and can be solved using mapping method, either analytically or computationally. All tasks, at the time, were proposed at the Olympiads of the Republican level, Russian or International levels, which, in our opinion, connects the sports and scientific approaches.