10

Part of 1982 IMO Longlists

Problems(1)

The Spheres, radiis and areas

Source: IMO LongList 1982 - P10

r_1, \ldots , r_n

be the radii of

n

S_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

\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.

geometry3D geometryspheregeometric transformationhomothetycombinatorial geometrygeometry unsolved