If $a,b,c>0$ and $abc=1$, prove $$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1$$
I replaced x=1/a y=1/b z=1/c and x/(2x+1)+y/(2y+1)+z/(2z+1) <=1 but can't seem to get further than that. How should I use AM-GM inequality here?
2026-03-29 17:11:49.1774804309
Proving $\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1$ for positive $a$, $b$, $c$ with $abc=1$with AM-GM
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