examday

/TMUA

For real numbers $a$ and $b$, let

What is the greatest positive number $C$ with the following property?

For every pair $(a,b)$ satisfying $a^2+4b^2<C$, the two polynomials have five distinct real roots which, in increasing order, are a root of $P$, a root of $Q$, a root of $P$, a root of $Q$ and a root of $P$.