Problem
ALG-B2-M10-P012 Shifted product
#12
★★★★★ Level 5 of 5
Let \(a,b,c>0\), \(abc=1\). Prove \[(1+a)(1+b)(1+c)\ge8.\]
Hint. Expand the product and estimate \(a+b+c\) and \(ab+bc+ca\).
\((1+a)(1+b)(1+c)=1+(a+b+c)+(ab+bc+ca)+abc\). With \(abc=1\), we have \(a+b+c\ge3\) and \(ab+bc+ca\ge3\). Hence the product is at least \(1+3+3+1=8\).
Teaching goal: choose a combination of methods and always check the equality case.