2001 IMO Shortlist Problems/A6

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Problem

Prove that for all positive real numbers $a,b,c$,

$\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \geq 1.$

Solution

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Resources

The leader of the Bulgarian team had come up with a generalization of this with putting k in place of 8 and replacing 1 in the RHS by 3/sq root(1+k)