Does there exist a regular polygon in which exactly half of the diagonals are parallel to the sides?
A diagonal is parallel to a side if there is an equal number of sides between the ends of the side and the ends of the diagonal. Therefore, a diagonal is parallel to some side if one of the polygons cut off by it has an even number of sides. Thus, if the number of sides of a regular polygon is odd, then each of its diagonals is parallel to some side. If, however, the number of sides is even (equal to $2k$), then from each vertex there are $2k-3$ diagonals, of which only every second one, counting from the side, is parallel to some side, that is, the number of diagonals parallel to the sides is less than half.

## Answer

None.