International Mathematical Olympiad 1960 Problem 5

Consider the cube $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ (with face $ABCD$ directly above face $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$).

(a) Find the locus of the midpoints of segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B^{\prime}D^{\prime}$.

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