Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality : $${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$
Source: China Northern MO 2009 p5 CNMO
Tags: algebra, inequalities
Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality : $${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$