RO-NMO 2008 Problem 121

Let $a>0$ and $f: [0,\infty) \to [0,a]$ be a continuous function on $(0,\infty)$ and having Darboux property on $[0,\infty)$ . Prove that if $f(0)=0$ and for all nonnegative $x$ we have $$xf(x) \geq \int^x_0 f(t) dt ,$$ then $f$ admits primitives on $[0,\infty)$ .