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f_n(x) has 2^n different roots

Source: Bulgaria MO 2011

May 30, 2011
algebrapolynomialinductionalgebra proposed

Problem Statement

Let f1(x)f_1(x) be a polynomial of degree 22 with the leading coefficient positive and fn+1(x)=f1(fn(x))f_{n+1}(x) =f_1(f_n(x)) for n≥1.n\ge 1. Prove that if the equation f2(x)=0f_2(x)=0 has four different non-positive real roots, then for arbitrary nn then fn(x)f_n(x) has 2n2^n different real roots.
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