KURSCHAK 1957

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$ . Find the locus of $P$ as $D$ varies.

A factory produces several types of mug, each with two colors, chosen from a set of six. Every color occurs in at least three different types of mug. Show that we can find three mugs which together contain all six colors.