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Roots of polynomial composition

Source: Russian TST 2018, Day 7 P1 (Groups A & B)

March 30, 2023
polynomialrootsalgebra

Problem Statement

Let f(x)=x2+2018x+1f(x) = x^2 + 2018x + 1. Let f1(x)=f(x)f_1(x)=f(x) and fk(x)=f(fk−1(x))f_k(x)=f(f_{k-1}(x)) for all k⩾2k\geqslant 2. Prove that for any positive integer nn{}, the equation fn(x)=0f_n(x)=0 has at least two distinct real roots.
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