Romanian Mathematical Olympiad

a) Denote by $T$ and $R$ the intersections of the straight lines $A{C}^{\prime }$ and $A^{\prime }M$, respectively $A{C}^{\prime }$ and $A^{\prime }N$. In the triangle $A^{\prime }{C}^{\prime }T$, $C^{\prime }M$ is an angle bisector and an altitude, so $A^{\prime }M=MT$. In the triangle $A^{\prime }AR$, $AN$ is an angle bisector and an altitude, so $A^{\prime }N=NR$. The segment $[MO]$ joins the midpoints of two sides of the triangle $A^{\prime }TB$, hence $MO\parallel TB$, and the segment $NO$ joins the midpoints of two sides of triangle $A^{\prime }RB$, hence $NO\parallel RB$. The requirement follows from the fact that the planes $(TRB)$ and $(A{C}^{\prime }B)$ are the same.



b) The distance between the two planes is equal to the distance from $O$ to the plane $(A{C}^{\prime }B)$, and this last distance is half the distance from $A^{\prime }$ to the same plane. As the distance from $A^{\prime }$ to the plane is $\frac{1}{2}{A}^{\prime }D=\frac{1}{2}\sqrt{2}$, the required distance is $\frac{1}{4}\sqrt{2}$.