RO-NMO 2011 Problem 122

Let be a continuous function $f:[0,1]\longrightarrow\left( 0,\infty \right)$ having the property that, for any natural number $n\ge 2,$ there exist $n-1$ real numbers $0<t_1<t_2<\cdots <t_{n-1}<1,$ such that $$\int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt.$$ Calculate $\lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} .$