IMC 2006 · Problem 1

Let $V$ be a convex polygon with $n$ vertices.

(a) Prove that if $n$ is divisible by $3$ then $V$ can be triangulated (i.e. dissected into non-overlapping triangles whose vertices are vertices of $V$) so that each vertex of $V$ is the vertex of an odd number of triangles.

(b) Prove that if $n$ is not divisible by $3$ then $V$ can be triangulated so that there are exactly two vertices that are the vertices of an even number of the triangles.