IMO 1960 Problem 5

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$ ).

a) Find the locus of the midpoints of the segments $XY$ , where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$ ;

b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$ .