46. A regular $n$-gon $M$ is rotated around its center by an angle of $\pi / n$ to obtain the $n$-gon $M^{\prime}$. Into what minimum number of convex polygons can the figure $M \cup M^{\prime}$ be cut?
92.46. Answer. $n+1$. The example is obvious. We will prove that it is impossible to cut it into a smaller number of polygons. For this, consider all the external vertices of the figure $M \cup M^{\prime}$ and, numbering the polygons into which it is cut, we will consider that these vertices are painted in the corresponding colors. It is left to the reader as an exercise to prove the following lemma and derive the statement of the problem.

Lemma. On a circle, $n$ points are marked, which are colored in several colors such that any two adjacent points are of different colors, and if $A, B$ are points of the same color, and $C$ and $D$ are points of another color, then the segments $A B$ and $C D$ do not intersect. Then the number of colors is no less than $[(n+3) / 2]$.