Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.