Problem

Source: Indonesia National Math Olympiad 2021 Problem 5 (INAMO 2021/5)

Tags: algebra, polynomial, inequalities



Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.