IMO 1967 LL GBR19

Origin: GBR

Problem

The n points P1, P2, . . . , Pn are placed inside or on the bound- ary of a disk of radius 1 in such a way that the minimum distance dn between any two of these points has its largest possible value Dn. Calcu- late Dn for n = 2 to 7 and justify your answer.

Solution

Suppose n \leq6. Let us decompose the disk by its radii into n congruent regions, so that one of the points Pj lies on the boundaries of two of these regions. Then one of these regions contains two of the n given points. Since the diameter of each of these regions is 2 sin \pi n, we have dn \leq2 sin \pi n. This

value is attained if Pi are the vertices of a regular n-gon inscribed in the boundary circle. Hence Dn = 2 sin \pi n. For n = 7 we have D7 \leqD6 = 1. This value is attained if six of the seven points form a regular hexagon inscribed in the boundary circle and the seventh is at the center. Hence D7 = 1.