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Problem 4

Source: Last round of the Bulgarian Mathematical Oympiad 2009

June 18, 2009
algebrapolynomialalgebra proposed

Problem Statement

Let n≥3 n\ge 3 be a natural number. Find all nonconstant polynomials with real coeficcietns f1(x),f2(x),…,fn(x) 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 x (with fn+1(x)≡f1(x) f_{n +1}\left(x\right)\equiv f_{1}\left(x\right) and fn+2(x)≡f2(x) f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)).
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