IMO Shortlist 1982 problem 18
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Avg: 0,0 Let
be a point of three-dimensional space and let
be mutually perpendicular straight lines passing through
. Let
denote the sphere with center
and radius
, and for every point
of
, let
denote the sphere with center
and radius
. We denote by
the intersection of
with the straight lines
, respectively, where we put
if
meets
at two distinct points and
otherwise (
). What is the set of centers of gravity of the (possibly degenerate) triangles
as
runs through the points of
?
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![l_1, l_2, l_3](/media/m/5/8/9/589bd14418838013f437e552b9cdd560.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
![P_1, P_2, P_3](/media/m/0/9/b/09b69a1e26a9b8213563a1a2f1d307c1.png)
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
![l_1, l_2, l_3](/media/m/5/8/9/589bd14418838013f437e552b9cdd560.png)
![P_i \neq O](/media/m/e/a/4/ea4f316ece95f62da0d948dfa4e5f43b.png)
![l_i](/media/m/b/2/6/b2647bded77d01395618ce39635fe411.png)
![S_M](/media/m/e/f/6/ef66189d71bb23040b4d5d15e37f498d.png)
![P_i = O](/media/m/c/8/7/c87a1442d45d41b4fbcf77362b0581f5.png)
![i = 1, 2, 3](/media/m/a/7/b/a7bbda360cece7bb6d0d6a1f43153f17.png)
![P_1P_2P_3](/media/m/3/5/5/3556b2178ef8de9f97d48162d503e6c2.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1982