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3-variable inequality with squares

Source: Italy MO 2023 P5

May 7, 2023
algebrainequalities proposed

Problem Statement

Let a,b,ca, b, c be reals satisfying a2+b2+c2=6a^2+b^2+c^2=6. Find the maximal values of the expressions a) (a−b)2+(b−c)2+(c−a)2(a-b)^2+(b-c)^2+(c-a)^2; b) (a−b)2⋅(b−c)2⋅(c−a)2(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2. In both cases, describe all triples for which equality holds.
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