(IMO 2004, exercise 2) Find all polynomials $P$ with real coefficients that satisfy, for all real $a, b, c$ such that $a b + b c + c a = 0$:

$$
P(a-b) + P(b-c) + P(c-a) = 2 P(a+b+c)
$$
First, by setting $b=c=0$, we observe that $P$ is an even function, and thus contains only terms of even degree. Evaluating at zero, we find that the constant term must be zero. Next, we try $a=6 x, b=3 x$, and $c=-2 x$. This gives $P(3 x)+P(5 x)+P(-8 x)=2 P(7 x)$. Let $n$ be the degree of $P$. By comparing the leading coefficients, we get $3^{n}+5^{n}+(-8)^{n}=2 \cdot 7^{n}$. This is impossible for $n \geq 5$. Therefore, $P$ must be of the form $a X^{4}+b X^{2}$. We verify conversely that these polynomials satisfy the given conditions.