Problem
ALG-B1-M10-P020 Three fractions
#20
★★★★☆ Level 4 of 5
Let \(x,y,z>0\). Prove \[\frac{x^2}{2x+y}+\frac{y^2}{2y+z}+\frac{z^2}{2z+x}\ge\frac{x+y+z}{3}.\]
Add the denominators and apply Cauchy.
By Cauchy, the left-hand side is at least \(\frac{(x+y+z)^2}{3(x+y+z)}=\frac{x+y+z}{3}\).
Cauchy is hidden in nonstandard denominators.