RU-ALLRUSSIAN 1986 Problem 8

A regular hexagon has side 1000. Each side is divided into 1000 equal parts. Let S be the set of the vertices and all the subdividing points. All possible lines parallel to the sides and with endpoints in S are drawn, so that the hexagon is divided into equilateral triangles with side 1.

Let X be the set of all vertices of these triangles. We now paint any three unpainted members of X which form an equilateral triangle (of any size). We then repeat until every member of X except one is painted. Show that the unpainted vertex is not a vertex of the original hexagon.