Problem
ALG-B2-M02-P010 Three shifts
#10
★★★★★ Level 5 of 5
Let \(a,b,c>0\) and \(abc=1\). Prove \((1+a)(1+b)(1+c)\ge8\).
Hint 1. Estimate each factor separately.
Hint 2. Use \(1+a\ge2\sqrt a\).
By AM-GM, \(1+a\ge2\sqrt a\), \(1+b\ge2\sqrt b\), \(1+c\ge2\sqrt c\). Multiplying gives \((1+a)(1+b)(1+c)\ge8\sqrt{abc}=8\). Equality occurs at \(a=b=c=1\).
AM-GM module training problem. Method tags: am-gm, equality-case, fixed-product.