Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.
Source: Russian TST 2018, Day 7 P1 (Groups A & B)
Tags: polynomial, roots, algebra
Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.