RO-NMO 2003 Problem 122

Let be an odd natural number $n\ge 3.$ Find all continuous functions $f:[0,1]\longrightarrow\mathbb{R}$ that satisfy the following equalities. $$\int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n, \forall k\in\{ 1,2,\ldots ,n-1\}$$ Titu Andreescu