MathDB
The Spheres, radiis and areas

Source: IMO LongList 1982 - P10

March 16, 2011
geometry3D geometryspheregeometric transformationhomothetycombinatorial geometrygeometry unsolved

Problem Statement

Let r1,…,rnr_1, \ldots , r_n be the radii of nn spheres. Call S1,S2,…,SnS_1, S_2, \ldots , S_n the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that S1r12+S2r22+⋯+Snrn2=4π.\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.
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