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Maximum value of the sum [Iran Second Round 1990]

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December 1, 2010
geometry3D geometryspheregeometry proposed

Problem Statement

(a) Consider the set of all triangles ABCABC which are inscribed in a circle with radius R.R. When is AB2+BC2+CA2AB^2+BC^2+CA^2 maximum? Find this maximum.
(b) Consider the set of all tetragonals ABCDABCD which are inscribed in a sphere with radius R.R. When is the sum of squares of the six edges of ABCDABCD maximum? Find this maximum, and in this case prove that all of the edges are equal.
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