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. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ b) Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.
Problem
Source: Romanian National Olympiad 2017, grade xii, p. 1
Tags: function, real analysis, Definite integral, Sequences, limits, calculus, integration