Problem
ALG-B2-M02-P011 Four terms
#11
★★★★★ Level 5 of 5
If \(x,y,z>0\) and \(x^3yz=32\), prove \(3x+y+z\ge10\).
Hint 1. Apply AM-GM to \(x,x,x,y,z\).
Hint 2. The product of the five chosen numbers is \(32\).
By AM-GM, \(\frac{x+x+x+y+z}{5}\ge\sqrt[5]{x^3yz}=\sqrt[5]{32}=2\). Hence \(3x+y+z\ge10\).
AM-GM module training problem. Method tags: weighted-am-gm, normalisation.