RU-ALLRUSSIAN 1978 Problem 4

If P, Q are points in space the point [PQ] is the point on the line PQ on the opposite side of Q to P and the same distance from Q. $\mathrm{K}_{0}$ is a set of points in space. Given $\mathrm{K}_{\mathrm{n}}$ we derive $\mathrm{K}_{\mathrm{n} + 1}$ by adjoining all the points [PQ] with P and Q in $\mathrm{K}_{\mathrm{n}}$ .

(1) $\mathrm{K}_{0}$ contains just two points A and B, a distance 1 apart, what is the smallest n for which $\mathrm{K}_{\mathrm{n}}$ contains a point whose distance from A is at least 1000?

(2) $\mathrm{K}_{0}$ consists of three points, each pair a distance 1 apart, find the area of the smallest convex polygon containing $\mathrm{K}_{\mathrm{n}}$ .

(3) $\mathrm{K}_{0}$ consists of four points, forming a regular tetrahedron with volume 1. Let $\mathrm{H}_{\mathrm{n}}$ be the smallest convex polyhedron containing $\mathrm{K}_{\mathrm{n}}$ . How many faces does $\mathrm{H}_{1}$ have? What is the volume of $\mathrm{H}_{\mathrm{n}}$ ?