Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy:

$$
f(f(f(n)))+f(f(n))+f(n)=3 n
$$

## - Correction -
First, we need to note that $f$ is injective. Indeed, if we assume $f(n)=f(m)$, then

$$
3 n=f(f(f(n)))+f(f(n))+f(n)=f(f(f(m)))+f(f(m))+f(m)=3 m
$$

so $n=m$.

Let's start with $f(0): f(f(f(0)))+f(f(0))+f(0)=0$ and since $f$ takes values in $\mathbb{N}$, $f(0)=0$.

Now, let's try to determine $f(1)$. Since $f$ is injective, $f(1) \neq 0$, and therefore $f(f(1)) \neq 0$, and thus $f(f(f(1))) \neq 0$. Since $f$ takes integer values, this means that these three quantities are at least 1. And since their sum is 3, this means that $f(1)=1$.

We see the induction emerging: let $n \geq 1$, we assume that for all $k<n, f(k)=k$. Then by the injectivity of $f, f(n) \geq n$, the same for $f(f(n))$ and $f(f(f(n)))$. Since their sum is $3 n, f(n)=n$.

Conversely, the function $f(n)=n$ is indeed a solution to the equation.

## 3 Third Period

## 1 Essential: Geometry

Program:

- Vector calculus, barycenters: Thales' theorem, Pythagorean theorem, Al Kashi's formula.
- Circles: Inscribed angles, cyclic points, power of a point with respect to a circle.
- Transformations: translations, homotheties, rotations, similarities, reflections.
- Triangle geometry: Ceva's theorem, Menelaus' theorem, isobarycenter, orthocenter, circumcircle, incircle, excircles, Euler circle, Euler line, points of tangency of the incircle, sine rule, area of a triangle.