On the new problems in stereometry

It were given original combinative problems with regular polyhedrons, which including well-known tetrahedrons and cubes, and also the lesser-known octahedrons, dodecahedrons and icosahedrons due to its complexity. The solution of the problem of the calculation of the edge of octahedron, dodecahedron and icosahedron by using the the side of inscribed and circumscribed cubes has been given. The radius of the circumsphered circle and midsphere around the dodecahedron and icosahedron has been calculated. Two triangular pyramids and two tetragonal ones, as well as the triangular pyramid with a cone have been arranged. In the second chapter of the paper, it were shown several non-trivial combinations of the bodies with a common vertex, in which the height of one body was a lateral edge of the other one and in which the volume of generalities bodies was found. Two triangular pyramids and two tetragonal ones, as well as the triangular pyramid with a cone have been arranged. Each problem was going with the detailed figure, and the solution to the problems with Platon body was including several supporting ones as well. This paper could be used by high school mathematics teachers not only as methodological support, but also as a clear example in preparation for the Olympiad tasks in math for students.