Question 95
Let $n \ge 3$ be a positive integer. Find all non-constant real polynomials $f_1(x), f_2(x), \dots, f_n(x)$ such that
$$
f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x)), \quad 1 \le k \le n
$$
for every real $x$ (here $f_{n+1}(x) \equiv f_1(x)$ and $f_{n+2}(x) \equiv f_2(x)$).
$$
f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x)), \quad 1 \le k \le n
$$
for every real $x$ (here $f_{n+1}(x) \equiv f_1(x)$ and $f_{n+2}(x) \equiv f_2(x)$).