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.