Question 164
A regular $2018$-gon is inscribed in a circle. The numbers $1,2,\ldots,2018$ are placed at the vertices of the $2018$-gon, one number at each vertex, such that the sum of every two consecutive numbers (of that configuration) is equal to the sum of the two numbers diametrically opposite to them. Find the number of all such configurations. (Configurations obtained from each other by a rotation around the center of the circle are considered the same.)