KURSCHAK 1967

$A$ is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of $A$ also belongs to $A$ .

A convex $n$ -gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$ .

For a vertex $X$ of a quadrilateral, let $h(X)$ be the sum of the distances from $X$ to the two sides not containing $X$ . Show that if a convex quadrilateral $ABCD$ satisfies $h(A) = h(B) = h(C) = h(D)$ , then it must be a parallelogram.