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Prove that $S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{&#039;}$

Source: Moldova TST 2001

August 6, 2023

3D geometry

Problem Statement

Let

A_i

and

A_i^{'}

(i=1,2,3,4)

be diametrically opposite vertexes of a rectangular cuboid and

M{}

a point inside it. Prove that

S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{'}

, where

S{}

is the total surface area of the rectangular cuboid.