Problem
ALG-B1-M07-P014 Three factors
#14
★★★☆☆ Level 3 of 5
Let \(x,y,z>0\) and \(xyz=1\). Prove that \((1+x)(1+y)(1+z)\ge8\).
Estimate each factor \(1+x\) by \(2\sqrt{x}\).
By AM-GM, \(1+x\ge2\sqrt{x}\), \(1+y\ge2\sqrt{y}\), \(1+z\ge2\sqrt{z}\). Multiplying, \((1+x)(1+y)(1+z)\ge8\sqrt{xyz}=8\). Equality holds when \(x=y=z=1\).
The problem trains multiplying several sharp estimates.