Problem
ALG-B2-M03-P006 Three quadratic denominators
#6
★★★★☆ Level 4 of 5
Let \(x,y,z>0\). Prove \[\frac{x^2}{x^2+xy+y^2}+\frac{y^2}{y^2+yz+z^2}+\frac{z^2}{z^2+zx+x^2}\ge\frac12.\]
Hint 1. Again use Engel form.
Hint 2. Bound the denominator sum from above by \(2(x+y+z)^2\).
By Cauchy, the left side is at least \[\frac{(x+y+z)^2}{2(x^2+y^2+z^2)+xy+yz+zx}.\] Since \(2(x^2+y^2+z^2)+xy+yz+zx\le2(x+y+z)^2\), the left side is at least \(\frac12\).
Cauchy-Schwarz module training problem. Method tags: cauchy, bounds.