KURSCHAK 1964

$ABC$ is an equilateral triangle. $D$ and $D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .

At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls $G$ and $G'$ and two boys $B$ and $B'$ , such that each of $B$ and $G$ danced, $B'$ and $G'$ danced, but $B$ and $G'$ did not dance, and $B'$ and $G$ did not dance.

Show that for any positive reals $w, x, y, z$ we have $$\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}$$