Problem
ALG-B2-M08-P011 Product 1 and sum
#11
★★★★☆ Level 4 of 5
Let \(a,b,c>0\), \(abc=1\). Prove \(a+b+c\ge3\).
Hint. Use AM-GM, or the representation \(a=x/y\), \(b=y/z\), \(c=z/x\).
By AM-GM, \(a+b+c\ge3\sqrt[3]{abc}=3\). Equality holds at \(a=b=c=1\).
Teaching goal: recognize the appropriate substitution and check the domain.