2012 ToT Fall Senior A p6 centroid of points of icosahedron
Source:
March 22, 2020
icosahedron3D geometryCentroidgeometryspherecombinatorial geometry
Problem Statement
(a) A point is marked inside a sphere. Three perpendicular lines drawn through intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.
(b) An icosahedron with the centre is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from to the vertices of the icosahedron mark points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new points on the sphere. Let and be the centres of mass of old and new points respectively. Prove that .