Problem
ALG-B2-M02-P006 Shifted product
#6
★★★☆☆ Level 3 of 5
Let \(a,b,c>0\) and \(a+b+c=6\). Prove \((1+a)(1+b)(1+c)\le27\).
Hint 1. Look at the sum of \(1+a,1+b,1+c\).
Hint 2. The product is maximized when these three numbers are equal.
The sum of \(1+a,1+b,1+c\) is \(9\). By AM-GM, their product is at most \(\left(\frac93\right)^3=27\). Equality occurs at \(a=b=c=2\).
AM-GM module training problem. Method tags: am-gm, equality-case.