Lucile wrote on the board $s$ 2023-tuples of integers. She then allows herself operations of the following form: she chooses two 2023-tuples $\mathbf{u}=\left(u_{1}, u_{2}, \ldots, u_{2023}\right)$ and $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{2023}\right)$, not necessarily distinct, from those she has already written, and then she also writes on the board the two 2023-tuples $\mathbf{u}+\mathbf{v}=\left(u_{1}+v_{1}, \ldots, u_{2023}+v_{2023}\right)$ and $\max (\mathbf{u}, \mathbf{v})=\left(\max \left(u_{1}, v_{1}\right), \ldots, \max \left(u_{2023}, v_{2023}\right)\right)$.
What are the values of $s$ for which, if Lucile judiciously chooses the $s$ 2023-tuples she initially wrote, and by repeating the above operations, she can write any 2023-tuple of integers?
First, if $s=1$, Lucile has written a single vector $\mathbf{u}$. Any vector $\mathbf{x}$ she writes subsequently will have coordinates of the same sign as $\mathbf{u}$, and she will therefore not be able to write all possible vectors.
Next, if $s=2$, let $\mathbf{u}$ and $\mathbf{v}$ be the two vectors Lucile has initially written. If two coordinates $u_{i}$ and $v_{i}$ have the same sign, any vector $\mathbf{x}$ Lucile writes subsequently will also have a coordinate of the same sign as $u_{i}$ and $v_{i}$. Consequently, for all integers $i \leqslant 2023$, the coordinates $u_{i}$ and $v_{i}$ must have opposite signs. By swapping the vectors $\mathbf{u}$ and $\mathbf{v}$ as well as the coordinates in positions 1 to 3, we can assume that $u_{1}$ and $u_{2}$ are strictly positive, and that $u_{1} v_{2} \leqslant u_{2} v_{1}$.
We then show by induction that any vector $\mathbf{x}$ Lucile writes subsequently will also satisfy the inequality $u_{1} x_{2} \leqslant u_{2} x_{1}$. Indeed, from two vectors $\mathbf{x}$ and $\mathbf{y}$ that already satisfy this inequality, Lucile can form two vectors:
$\triangleright \mathbf{z}=\mathbf{x}+\mathbf{y}$, for which $u_{1} z_{2}=u_{1} x_{2}+u_{1} y_{2} \leqslant u_{2} x_{1}+u_{2} y_{1}=u_{2} z_{1}$;
$\triangleright \mathbf{z}=\max (\mathbf{x}, \mathbf{y})$, for which $u_{1} z_{2}=\max \left(u_{1} x_{2}, u_{1} y_{2}\right) \leqslant \max \left(u_{2} x_{1}, u_{2} y_{1}\right)=u_{2} z_{1}$.
In particular, Lucile will never be able to write vectors whose first two coordinates are 0 and 1.
Finally, if $s \geqslant 3$, Lucile can write the vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ such that $u_{i}=-1, v_{i}=i$, and $w_{i}=-i^{2}$, and add $s-3$ useless vectors. From these vectors, Lucile can then successively create the following vectors, where the integer $k$ is a parameter that we freely vary between 1 and 2023:
$\triangleright \mathbf{a}^{(k)}=\left(k^{2}-1\right) \mathbf{u}+2 k \mathbf{v}+\mathbf{w}$, for which $a_{i}^{(k)}=1-k^{2}+2 k i-i^{2}=1-(i-k)^{2}$;
$\triangleright \mathbf{b}^{(k)}=\mathbf{a}^{(k)}+\mathbf{u}$, for which $b_{i}^{(k)}=-(i-k)^{2}$;
$\triangleright \mathbf{0}=\max \left\{\mathbf{b}^{(k)}: 1 \leqslant k \leqslant 2023\right\}$, for which $\mathbf{0}_{i}=0$;
$\triangleright \mathbf{c}^{(k)}=\max \left\{\mathbf{a}^{(k)}, \mathbf{0}\right\}$, for which $c_{i}^{(k)}=1$ if $i=k$, and $c_{i}^{(k)}=0$ otherwise.
From this point, Lucile can write any vector $\mathbf{x}$, as she can express it as the sum

$$
\mathbf{x}=|\mathbf{x}| \mathbf{u}+\sum_{k=1}^{2023}\left(|\mathbf{x}|+x_{k}\right) \mathbf{c}^{(k)}
$$

where we have set $|\mathbf{x}|=1+\max \left\{\left|x_{k}\right|: 1 \leqslant k \leqslant 2023\right\}$.
In conclusion, the values of $s$ sought are the integers $s \geqslant 3$.
Comment from the graders: This very difficult problem was little addressed and even less successfully solved. Many students managed to find simple initial configurations for $s=2 \times 2023$ or $s=2023+1$, but these did not help to demonstrate that $s=3$ was sufficient. Even if obtaining a complete solution to this problem was a daunting task, it is nonetheless important to repeat to students the following advice:
$\triangleright$ as they did, search for simple initial configurations for values of $s$ that could be the minimum values sought;
$\triangleright$ read the problem statement carefully: several students believed that Lucile was only interested in natural numbers, which nothing in the statement suggested;
$\triangleright$ consider the case of $k$-tuples when $k$ is small, for example $k=1, k=2$, or $k=3$; only three students actively searched for proof elements in this direction, although it is obviously essential to try to understand the small cases before understanding the large ones;
$\triangleright$ observe whether the operations studied preserve certain properties: here, remembering that the maximum and the sum of two convex functions is convex was a possible approach to proving that $s \geqslant 3$, and that any possible initial configuration for $s=3$ must include as simple concave sequences as possible, such as the sequence $\mathbf{w}$ given in the solution.