MathDB
Turkey TST 1991 - P6

Source:

March 13, 2011
geometry3D geometrytetrahedronpyramidgeometry proposed

Problem Statement

Let UU be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices O,A,B,CO,A,B,C. Let VV be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that V≤(U−∣OA∣−∣BC∣)(U−∣OB∣−∣AC∣)(U−∣OC∣−∣AB∣)(27⋅3)V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}.
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