I found this question recently looking around the internet, apparently it was on the IMO shortlist many years back. I haven't been able to solve it, and I am looking for hints and/or full solutions. Apparently it can be done very elegantly.
Consider N non overlapping spheres of equal radius placed in 3D space. Let $S$ be the set of points on the surface of these spheres which are not visible from any other sphere. Show that the total area of $S$ is equal to the surface area of one sphere.
Thoughts so far: the problem is trivial in one dimension, and likely equivalent in difficulty in 2D and 3D (it looks like it still holds in 2D, hence I imagine it works for any number of dimensions). It is also obviously true for 2 spheres, and can be verified with some effort for 3. I've had many ideas but none of them have yet led me anywhere I thought promising.
from Hot Weekly Questions - Mathematics Stack Exchange
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