falantrng wrote: The degree of the polynomial $P(x)$ is $2017.$ Prove that the number of distinct real roots of the equation $P(P(x)) = 0$ is not less than the number of distinct real roots of the equation $P(x) = 0.$ Distinct roots of $P(x)$ are $r_1,r_2,...,r_k$ And since $P(x)$ has odd degree, each equation $P(x)=r_i$ has at least one real root $s_i$ such that $P(P(s_i))=0$ (and all $s_i$ are distinct)