Problem
ALG-B1-M07-P020 Fractions with x + 1
#20
★★★★☆ Level 4 of 5
Let \(x,y,z>0\) and \(x+y+z=6\). Prove \[\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}\ge4.\]
Again apply Cauchy in Engel form and add the denominators.
By Cauchy, the left-hand side is at least \(\frac{(x+y+z)^2}{(x+1)+(y+1)+(z+1)}=\frac{36}{9}=4\). Equality holds when \(x=y=z=2\).
The problem looks different, but the method is the same: a sum of squares over positive denominators.