(Benelux 2010) Find all polynomials $P \in \mathbb{R}[X]$ such that for all real numbers $a, b, c$ we have:

$$
p(a+b-2 c)+p(b+c-2 a)+p(c+a-2 b)=3 p(a-b)+3 p(b-c)+3 p(c-a)
$$
By injecting $a=b=c=0$, we find $P(0)=0$. By taking $b=c=0$, we obtain $P(-2a)=3P(-a)+P(a)$, and this for all $a$. We assume $P$ is of degree $n$. By examining the leading coefficients, we get $2^{n}=(-1)^{n}+3$, so $n$ is 1 or 2, and $P$ is of the form $a X^{2}+b X$. We verify conversely that these polynomials work.