RU-SPB 2018 Problem 102

Color every vertex of $2008$ -gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$ -gon as interesting. Convex $2009$ -gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$ -gon. Prove, that if we remove marked vertex, then we get interesting $2008$ -gon too.