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x, y, and z are all positive real numbers. x+y+z = 1

prove: x^2/(y+2z) + y^2/(z+2x) + z^2/(z+2y) is greater or equal to 1/3

This has to be proved using the Cauchy inequality.

Any help is appreciated!!

May 24, 2019

$$\displaystyle 1=[x+y+z]=[(x/k_{1})k_{1}+(y/k_{2})k_{2}+(z/k_{3})k_{3}]$$
Apply Cauchy-Schwarz to that and then deduce appropriate forms for $$\displaystyle k_{1},k_{2}\text{ and }k_{3}.$$