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TOT 154 1987 Autumn J5 A^2 + B^2 + C^2 \le 2(AB + BC + CA)

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April 19, 2020
inequalitiesalgebra

Problem Statement

We are given three non-negative numbers A,BA , B and CC about which it is known that A4+B4+C4≤2(A2B2+B2C2+C2A2)A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2) (a) Prove that each of A,BA, B and CC is not greater than the sum of the others. (b) Prove that A2+B2+C2≤2(AB+BC+CA)A^2 + B^2 + C^2 \le 2(AB + BC + CA) . (c) Does the original inequality follow from the one in (b)?
(V.A. Senderov , Moscow)
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