CZ-SK-MO 1996 Problem 2

Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$ . Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).