Question 20
Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition
$$
P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c)
$$
for all real numbers $a, b, c$ with $a + b + c = 0$.
$$
P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c)
$$
for all real numbers $a, b, c$ with $a + b + c = 0$.