IMO 1967 LL POL37

Prove that for arbitrary positive numbers the following in-

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IMO 1967 LL POL37

Origin: POL

Problem

Prove that for arbitrary positive numbers the following in- equality holds: a + 1 b + 1 c \leqa8 + b8 + c8 a3b3c3 .

Solution

Using the A–G mean inequality we obtain 8a2b3c3 \leq2a8 + 3b8 + 3c8, 8a3b2c3 \leq3a8 + 2b8 + 3c8, 8a3b3c2 \leq3a8 + 3b8 + 2c8. By adding these inequalities and dividing by 3a3b3c3 we obtain the desired one.