CZ-SK-MO 1954 Problem 4

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$ ). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$ . (a) Let $M=B'Y\cap D'X$ . Find the locus of all $M$ s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.