KURSCHAK 1962

Show that the number of ordered pairs $(a, b)$ of positive integers with lowest common multiple $n$ is the same as the number of positive divisors of $n^2$ .

Show that given any $n+1$ diagonals of a convex $n$ -gon, one can always find two which have no common point.

$P$ is any point of the tetrahedron $ABCD$ except $D$ . Show that at least one of the three distances $DA$ , $DB$ , $DC$ exceeds at least one of the distances $PA$ , $PB$ and $PC$ .