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1 + 2abc\ge a^2 + b^2 + c^2, with a,b,c in [-1,1]

Source: Ukraine TST 2012 p12

April 30, 2020
inequalitiesSubsetsalgebra

Problem Statement

We shall call the triplet of numbers a,b,ca, b, c of the interval [−1,1][-1,1] qualitative if these numbers satisfy the inequality 1+2abc≥a2+b2+c21 + 2abc\ge a^2 + b^2 + c^2. Prove that when the triples a,b,ca, b, c, and x,y,zx, y, z are qualitative, then ax,by,czax, by, cz is also qualitative.
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