MathNet 2008-part-02 — 165 problems (EN–VI) Question 41

Let $P(x) = x^2 + a x + b$ be a real polynomial with $a \lt 2$. Suppose $P(P(x)) = 0$ has four distinct real roots and the sum of some two of them is $\le -1$. Prove that $P(x+y) \ge P(x) + P(y)$ for all non-negative real numbers $x, y$.