IMO Shortlist 1966 problem 6


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2. travnja 2012.
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Let m be a convex polygon in a plane, l its perimeter and S its area. Let M\left( R\right) be the locus of all points in the space whose distance to m is \leq R, and V\left(R\right) is the volume of the solid M\left( R\right) .


a.) Prove that V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.

Hereby, we say that the distance of a point C to a figure m is \leq R if there exists a point D of the figure m such that the distance CD is \leq R. (This point D may lie on the boundary of the figure m and inside the figure.)

additional question:

b.) Find the area of the planar R-neighborhood of a convex or non-convex polygon m.

c.) Find the volume of the R-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron.

Note by Darij: I guess that the ''R-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is \leq R.
Izvor: Međunarodna matematička olimpijada, shortlist 1966