Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,

$$
f\left(x^{666}+y\right)=f\left(x^{2023}+2 y\right)+f\left(x^{42}\right)
$$
This exercise is designed to be scary. In reality, it is less difficult than it looks. The general philosophy is to simplify the equation as much as possible. We can look for $x, y$ such that the values of $x^{666}+y$ and $x^{2023}+2 y$ are simple, but this does not seem easy at all. Better, we will try to equalize them, by taking for example $y=x^{666}-x^{2023}$. After this substitution, we end up with $f\left(x^{42}\right)=0$, which tells us that $f$ is zero on the positive reals.

To conclude, we can for example set $y=0$:

$$
f\left(x^{666}\right)+f\left(x^{2023}\right)=f\left(x^{42}\right)
$$

which by nullity on the positives gives $f\left(x^{2023}\right)=0$ hence $f$ is identically zero. This function is a solution, and it is the only one.

The following exercises have not been covered in class. We only provide the main idea of the solution.