I.F Sharygin Olympiad

Geometry Level 4

In the space, five points are marked. It is known that these points are the centers of five spheres, four of which are pairwise externally tangent, and all these four are internally tangent to the fifth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fifth (the largest) sphere. If the ratio of the greatest and the smallest radii of the spheres is

\(\large\ \frac { a + \sqrt { b } }{ c } \) where \(a, b, c\) are positive integers, with \(b\) square-free.

Find \(a+b+c\)

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