BG-MO 2009 Problem 4

Let $n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$ , for which $$f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), 1\le k\le n,$$ for every real $x$ (with $f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$ ).