BALKAN-SL 2012 A3

Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition \begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*} for all real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=0$