Question bank VMO 1967

Study the variation and draw the graph of the function
$$y=\frac13|x^3-x^2-2x|-|x+1|.$$

The two banks of a river are parallel horizontal straight lines. The current flows parallel to the banks from left to right with speed $u$. A ferry of constant speed $v$ relative to the water starts at $O$ on the lower bank, taken as the axis $Ox$, to carry a vehicle to the opposite bank.
(a) If the actual path of the ferry makes an angle $\alpha$ with $Ox$, what is its actual velocity $\vec V$? Discuss.
(b) What angle $\alpha$ minimizes the crossing time?

A circle $(L)$ with center $O$ is inscribed in a rhombus $ABCD$. A variable tangent to $(L)$ meets the lines $AB,AD,BC,CD$ at $M,N,P,Q$, respectively.
(a) Conjecture a relation between the lengths $BM$ and $DN$, and prove it. What other noteworthy relations occur in the figure? Discover as many as possible.
(b) Four circles, all passing through $O$ and centered respectively at $A,B,C,D$, intersect the four segments $AB,BC,CD,DA$ at eight points. Examine the special properties of the convex octagon having these points as vertices, and prove them. Apply this to constructing a regular octagon.
(c) Formulate a solid-geometry problem by rotating the figure of the preceding problem about the axis $AC$.