(solved by Alice Héliou). Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying:

$$
f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y))
$$

for all real numbers $x$ and $y$.
. By setting $x=y$, we obtain $f(0)=0$. Then with $x=-1$ and $y=0$, we have $f(1)=$ $-f(-1)$. With, on the one hand, $y=1$, and on the other hand, $y=-1$, we have for all $x$

$$
\begin{aligned}
& f\left(x^{2}-1\right)=(x-1)(f(x)+f(1)) \\
& f\left(x^{2}-1\right)=(x+1)(f(x)-f(1))
\end{aligned}
$$

From the equality $(x-1)(f(x)+f(1))=(x+1)(f(x)-f(1))$, we derive $f(x)=f(1) x$. Therefore, any function satisfying the equation is a linear function. Conversely, any function of the form $f(x)=k x$ satisfies this equation.