IMO 1974 LL USS44

Origin: USS

Problem

We are given n mass points of equal mass in space. We define a sequence of points O1, O2, O3, . . . as follows: O1 is an arbitrary point (within the unit distance of at least one of the n points); O2 is the center of gravity of all the n given points that are inside the unit sphere centered at O1; O3 is the center of gravity of all of the n given points that are inside the unit sphere centered at O2; etc. Prove that starting from some m, all points Om, Om+1, Om+2, . . . coincide.