Prove that if than . ( )
Who thinks these horrible things up (and why do I spend hours brooding over them)?
I'll assume that a, b and c are all positive (the problem's hard enough as it is, and I don't want to be bothered with negative quantities).
Notice first that (and similarly for b and c), so the inequality can be written
. . . . . . . .(1)
If bc+ca+ab=1 then
. . . . . . . .(2)
(with similar inequalities for b and c).
Also, . . .(3)
(just multiply out both sides to verify that).
By the arithmetic-geometric mean inequality,
. . . . . . . .(4)
(that's the only place where I assume that a, b and c are positive).
It follows that
. . . . . . . .by (2)
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .by (3)
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .by (4).
By (1), that was what we wanted to prove.