RO-NMO 2007 Problem 122

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function.

a) Show that for any integer $n\geq 1$ , there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x) dx=\frac{1}{n}\int_{0}^{1}f(x) dx$ holds for all $k=0,1,\ldots,n-1$ .

b) For each $n$ , consider the $a_{i}$ above (that depend on $n$ ) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$ . Show that the sequence $(b_{n})$ is convergent and compute it's limit.