RO-NMO 2011 Problem 103

Let be three positive real numbers $a,b,c.$ Show that the function $f:\mathbb{R}\longrightarrow\mathbb{R} ,$ $$f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} ,$$ is nondecresing on the interval $\left[ 0,\infty \right)$ and nonincreasing on the interval $\left( -\infty ,0 \right] .$