This paper deals with a set of geometrical problems for mathematical problem-solving at different difficulty levels. All of these are presented as national flags and one has to investigate invariant area proportions when changing the locus of any corner of the flag. This dynamic element suggests the usage of dynamical geometry environments. Considering applications in mathematics classes, it allows training students' ability to find a proof above all in the phases of production of a conjecture and exploration of the conjecture. First, the paper introduces results of selected studies on reasoning and proof in mathematics classes. Then the formulation of the problems and their sample solutions are given. Finally, the paper reports on experiences teaching some of the posed problems.