3. Let $F(x)$ and $G(x)$ be polynomials of degree 2021. It is known that for all real $x$, $F(F(x)) = G(G(x))$ and there exists a real number $k, k \neq 0$, such that for all real $x$, $F(k F(F(x))) = G(k G(G(x)))$. Find the degree of the polynomial $F(x) - G(x)$.
Solution. Since $F(F(x))=G(G(x))$, then $F(k G(G(x)))=G(k G(G(x)))$. This equality holds for all $x$, i.e., for more than 2022 values of the variable (Since the polynomial $G(x)$ takes an infinite number of values). Therefore, the polynomials coincide, their difference has degree 0.

Answer. The degree of $F(x)-G(x)$ is 0, the polynomials coincide.