Problem
ALG-B1-M07-P017 Sum with neighboring denominators
#17
★★★★☆ Level 4 of 5
Let \(x,y,z>0\). Prove \[\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge\frac{x+y+z}{2}.\]
Apply Cauchy in Engel form to the whole sum.
By Cauchy, the left-hand side is at least \(\frac{(x+y+z)^2}{(y+z)+(z+x)+(x+y)}=\frac{(x+y+z)^2}{2(x+y+z)}=\frac{x+y+z}{2}\). Equality holds when \(x=y=z\).
This is a classic pattern: square of the numerator over the sum of the other two variables.