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Eventually centre of masses coincide [ILL 1974]

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January 3, 2011
geometry3D geometryspheregeometry proposed

Problem Statement

We are given nn mass points of equal mass in space. We define a sequence of points O1,O2,O3,…O_1,O_2,O_3,\ldots as follows: O1O_1 is an arbitrary point (within the unit distance of at least one of the nn points); O2O_2 is the centre of gravity of all the nn given points that are inside the unit sphere centred at O1O_1;O3O_3 is the centre of gravity of all of the nn given points that are inside the unit sphere centred at O2O_2; etc. Prove that starting from some mm, all points Om,Om+1,Om+2,…O_m,O_{m+1},O_{m+2},\ldots coincide.
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