RO-NMO 2003 Problem 123

Let be a continuous function $f:\mathbb{R}\longrightarrow\mathbb{R}$ that has the property that $$xf(x)\ge \int_0^x f(t)dt ,$$ for all real numbers $x.$ Prove thata) the mapping $x\mapsto \frac{1}{x}\int_0^x f(t) dt$ is nondecreasing on the restrictions $\mathbb{R}_{<0 }$ and $\mathbb{R}_{>0 } .$ b) if $\int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt ,$ for any real number $x,$ then $f$ is constant.

Mihai Piticari