BG-MO 1965 Problem 4

In the space there are given crossed lines $s$ and $t$ such that $\angle(s,t)=60^\circ$ and a segment $AB$ perpendicular to them. On $AB$ it is chosen a point $C$ for which $AC:CB=2:1$ and the points $M$ and $N$ are moving on the lines $s$ and $t$ in such a way that $AM=2BN$ . The angle between vectors $\overrightarrow{AM}$ and $\overrightarrow{BM}$ is $60^\circ$ . Prove that:

(a) the segment $MN$ is perpendicular to $t$ ; (b) the plane $\alpha$ , perpendicular to $AB$ in point $C$ , intersects the plane $CMN$ on fixed line $\ell$ with given direction in respect to $s$ ; (c) all planes passing by $ell$ and perpendicular to $AB$ intersect the lines $s$ and $t$ respectively at points $M$ and $N$ for which $AM=2BN$ and $MN\perp t$ .