Find all polynomials $P \in \mathbb{R}[X]$ such that for all real numbers $a, b, c$ satisfying $a b+b c+c a=0$, we have:

$$
P(a-b)+P(b-c)+P(c-a)=P(a+b+c)
$$

## 2 Solution
By setting $a=b=c=0$, we find $P(0)=0$. If $b=c=0, P(a)=P(-a)$ for all real $a$: $P$ is even, so all its monomials are even (it's a good little exercise to show this property).

By taking $b=2a$, and $c=-2a/3$, we have

$$
P(-a)+P(4a/3)+P(8a/3)=2P(7a/3)
$$

We look at the term of the highest degree $k, p_{k}X^{k}$. For $a \to +\infty$, this term "crushes" the others, and thus the dominant term in $a$ must be the same on both sides.

Therefore:

$$
\begin{aligned}
p_{k}(-1)^{k}+p_{k}(8/3)^{k}+p_{k}(-5/3)^{k} & =2p_{k}(7/3)^{k} \\
3^{k}+8^{k}+5^{k} & =2 \times 7^{k}
\end{aligned}
$$

(indeed, $k$ is even). And, $(8/7)^{k}>2$ for $n \geq 8$ and there is no equality for $k=6$. Therefore, $k \in \{0,2,4\}$. Conversely, $x^{2}$ and $x^{4}$ are solutions to the problem. We quickly deduce that the solutions are the polynomials

$$
P(X)=aX^{4}+bX^{2}
$$

with $a, b \in \mathbb{R}$.

## 4 Group D

## 1 Statement

Write the different problems on separate sheets. On each sheet, write your name, your group, and the problem number.

Rulers, set squares, and compasses are allowed. Protractors are prohibited. Calculators are prohibited, as are all electronic devices.