IMO 1966 - Explicitly dated longlist selection (53-57)

(USS, 1966) Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac16 S$.

(USS, 1966) Find the last two digits of a sum of eighth powers of $100$ consecutive integers.

(USS, 1966) Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ,70^\circ]$.

(USS, 1966) Let $ABCD$ be a tetrahedron such that $AB\perp CD$, $AC\perp BD$, and $AD\perp BC$. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.

(USS, 1966) Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.