Problem 1. (3 points) A convex polyhedron with 100 edges is given. All vertices are cut by planes such that no two cutting planes intersect inside or on the boundary of the polyhedron. How many vertices and how many edges does the resulting polyhedron have?
Solution. On every edge of the given polyhedron, there lie two vertices of the obtained one, with three edges emanating from each vertex of the obtained polyhedron. Therefore, the obtained polyhedron has $2 \cdot 100 = 200$ vertices and $\frac{200 \cdot 3}{2} = 300$ edges.