Problem
ALG-B1-M07-P023 Sum and product of factors
#23
★★★★☆ Level 4 of 5
Let \(a,b,c>0\) and \(abc=1\). Prove \[(a+b+1)(b+c+1)(c+a+1)\ge27.\]
Show that each factor is at least \(3\) using AM-GM.
By AM-GM, \(a+b+1\ge3\sqrt[3]{ab}\), \(b+c+1\ge3\sqrt[3]{bc}\), \(c+a+1\ge3\sqrt[3]{ca}\). Multiplying, the left-hand side is at least \(27\sqrt[3]{a^2b^2c^2}=27\), since \(abc=1\). Equality holds when \(a=b=c=1\).
Here one must choose the right three numbers inside each factor.