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Found problems: 2

1967 IMO Longlists, 37
Tags: three variable inequality , algebra , inequality , muirhead , inequalities
Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

1967 IMO Shortlist, 3
Tags: algebra , inequalities , inequality , three variable inequality , muirhead
Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]