Problem
ALG-B1-M11-P013 Inequality
#13
★★★☆☆ Level 3 of 5
Let \(x,y,z>0\). Prove \(\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}\ge\frac{x+y+z}{2}\).
Add the denominators.
By Cauchy, the left-hand side is at least \(\frac{(x+y+z)^2}{(x+y)+(y+z)+(z+x)}=\frac{x+y+z}{2}\).
Strategy: recognize Engel form.