Problem
ALG-B2-M09-P012 Shifted product with abc
#12
★★★★☆ Level 4 of 5
Let \(a,b,c>0\), \(abc=1\). Prove \[(1+a)(1+b)(1+c)\ge8.\]
Hint. Expand the product and use \(a+b+c\ge3\), \(ab+bc+ca\ge3\).
Expanding, \((1+a)(1+b)(1+c)=1+(a+b+c)+(ab+bc+ca)+abc\). Since \(abc=1\), \(a+b+c\ge3\), and \(ab+bc+ca\ge3\), the product is at least \(1+3+3+1=8\).
Mixes fixed product and symmetric sums.