KöMaL 2019 B5037

A polyhedron $P$ is divided into the smaller polyhedra $P_1, \ldots, P_k$ and also divided into the smaller polyhedra $Q_1, \ldots, Q_k$ such that the polyhedra $P_i$ and $Q_i$ are congruent for all $i=1, \ldots, k$. Show that it is possible to select some points in the interior of $P$ such that there are the same number of points (at least one) in the interior of polyhedra $P_i$ and $Q_i$ for all $i=1, \ldots, k$. (The position of each polyhedron is fixed in the space.)