RO-NMO 2006 Problem 83

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$ . The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$ . Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.

a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$ .

b) Find the minimal value of the angle between the lines $MN$ and $BC'$ .