### 1960 IMO 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 point of $B’D’$;

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