Belorusija 2012

Lemma. Let $0<x<y<z$ and $\lambda >1$. Then ${\sum }_{cyc}{x}^{\lambda }y<{\sum }_{cyc}x{y}^{\lambda }$.

Proof. Consider the function $f(y)={\sum }_{cyc}(x^{\lambda }y-x{y}^{\lambda })$. It takes the value $0$ for $y=x$ and $y=z$. We have also $f^{\prime \prime }(y)=\lambda (\lambda -1)y^{\lambda }(z-x)>0$, hence $f$ is convex. It follows that $f(y)<0$ for $y\in (x;z)$. The lemma is proved.

Now use the lemma for $x=a^{12}$, $y=b^{12}$, $z=c^{12}$, and $\lambda =20/12$ thus obtaining the inequality needed.