IMO 1977 LL GDR17

Origin: GDR

Problem

A ball K of radius r is touched from the outside by mutually equal balls of radius R. Two of these balls are tangent to each other. Moreover, for two balls K1 and K2 tangent to K and tangent to each other there exist two other balls tangent to K1, K2 and also to K. How many balls are tangent to K? For a given r determine R.

Solution

Centers of the balls that are tangent to K are vertices of a regular poly- hedron with triangular faces, with edge length 2R and radius of circum- scribed sphere r + R. Therefore the number n of these balls is 4, 6, or 20. It is straightforward to obtain that: (i) If n = 4, then r + R = 2R( \sqrt 6/4), whence R = r(2 + \sqrt 6). (ii) If n = 6, then r + R = 2R( \sqrt 2/2), whence R = r(1 + \sqrt 2). (iii) If n = 20, then r + R = 2R

5 + \sqrt 5/8, whence R = r  5 −2 \sqrt 5+ (3 − \sqrt 5)/2 .