IMO Shortlist 1982 problem 18


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April 2, 2012
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Let O be a point of three-dimensional space and let l_1, l_2, l_3 be mutually perpendicular straight lines passing through O. Let S denote the sphere with center O and radius R, and for every point M of S, let S_M denote the sphere with center M and radius R. We denote by P_1, P_2, P_3 the intersection of S_M with the straight lines l_1, l_2, l_3, respectively, where we put P_i \neq O if l_i meets S_M at two distinct points and P_i = O otherwise (i = 1, 2, 3). What is the set of centers of gravity of the (possibly degenerate) triangles P_1P_2P_3 as M runs through the points of S?
Source: Međunarodna matematička olimpijada, shortlist 1982