MathNet 2019-part-04 — 184 problems (EN–VI) Question 104

A quadratic polynomial $P(x)$ with two distinct real roots is given. For all real numbers $a$ and $b$ with $|a|,|b| \geq 2017$ we have $P\left(a^{2}+b^{2}\right) \geq P(2 a b)$. Prove that at least one of the roots of $P$ is negative.