RO-NMO 2017 Problem 121

a) Let be a continuous function $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} .$ Show that there exists a natural number $n_0$ and a sequence of positive real numbers $\left( x_n \right)_{n>n_0}$ that satisfy the following relation. $$n\int_0^{x_n} f(t)dt=1, n_0<\forall n\in\mathbb{N}$$ b) Prove that the sequence $\left( nx_n \right)_{n> n_0}$ is convergent and find its limit.