Problem
ALG-B2-M03-P005 Cyclic denominators
#5
★★★☆☆ Level 3 of 5
Prove for \(x,y,z>0\): \[\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}\ge\frac{x+y+z}{2}.\]
Hint 1. Apply Engel form directly.
Hint 2. The denominator sum is \(2(x+y+z)\).
By Cauchy, \[\sum\frac{x^2}{x+y}\ge\frac{(x+y+z)^2}{(x+y)+(y+z)+(z+x)}=\frac{x+y+z}{2}.\]
Cauchy-Schwarz module training problem. Method tags: cauchy, cyclic-sum.