solved by Matthieu Vogel

Let $u: \mathbb{N} \rightarrow \mathbb{N}^{*}$ be a sequence of non-zero natural numbers. We say that an index $i$ of this sequence is nice if $u_{1}+\cdots+u_{i}$ and $u_{1}+\cdots+u_{i+1}$ are coprime.

Since divisibility is transitive, it is necessary and sufficient to find $P: i \rightarrow \mathbb{N}^{*}$ such that $S_{k}\left(P_{i}\right) \mid S_{k}\left(P_{i+1}\right)$ for all $i, k$. We show that there exists a permutation of $\mathbb{N}^{*}$ that is nice and that for any nice permutation $P_{i}$, we can construct a nice permutation $P_{i+1}$ such that $S_{k}\left(P_{i}\right) \mid S_{k}\left(P_{i+1}\right)$ for all $i, k$.

First, let's find a permutation $a_{1}, a_{2}, \ldots$ of $\mathbb{N}^{*}$ that is nice. We define it by induction. $a_{1}=1$. If $i$ is even, we take $a_{i+1}=p$ a prime $\neq a_{1}, \ldots, a_{i}$ large enough so that $\operatorname{gcd}\left(a_{1}+\cdots+a_{i}, a_{i}+\cdots+a_{i}+a_{i+1}\right)=\operatorname{gcd}\left(a_{1}+\cdots+a_{i}, p\right)=1$, hence $i$ is a nice index. If $i \geqslant i$ is odd, we take $a_{i+1}=\min \left(\mathbb{N}^{*} \backslash\left\{a_{1}, \ldots, a_{i}\right\}\right)$, hence the sequence is a permutation of $\mathbb{N}^{*}$.

Secondly, let $a_{1}, a_{2}, \ldots$ be a nice sequence. Construct a sequence $b_{1}, b_{2}, \ldots$ that is nice such that $a_{1}+\cdots+a_{i} \mid b_{1}+\cdots+b_{i}$ for all $i$. We construct the sequence $b$ by induction. We take $b_{1}=a_{1}$. Then, to construct $b_{i}$, if $i$ is not nice in the sequence $a$, we take $b_{i}$ such that $b_{1}+\cdots+b_{i} \equiv 0\left(\bmod a_{1}+\cdots+a_{i}\right)$ and $b_{i} \neq b_{1}, \ldots, b_{i-1}$. If $i$ is a nice index in $a$, we will construct both $b_{i}$ and $b_{i+1}$. Let $A=a_{1}+\cdots+a_{i}$ and $A^{\prime}=a_{1}+\cdots+a_{i+1}$, $A$ and $A^{\prime}$ are coprime because $i$ is a nice index. We will show that if $x \neq b_{1}, b_{2}, \ldots a_{i-1}$, we can find $b_{i}, b_{i+1}$ such that all divisibility conditions are satisfied and $b_{i+1}=x$. It will then suffice to take $x=\min \left(\mathbb{N}^{*} \backslash\left\{b_{1}, \ldots, b_{i-1}\right)\right.$ an infinite number of times so that the sequence $b$ is a permutation of $\mathbb{N}^{*}$ and $x$ a large enough prime so that $i$ is a nice index in the sequence $b$ an infinite number of times so that the sequence $b$ is nice. If we fix $b_{i+1}=x$, it is sufficient to find $b_{i}$ such that $b_{1}+\cdots+b_{i} \equiv 0(\bmod A)$ and $b_{1}+\cdots+b_{i}+x \equiv 0\left(\bmod A^{\prime}\right)$. It exists by the Chinese Remainder Theorem and we take it large enough so that it is different from $a_{1}, \ldots, a_{i-1}, x$.

## VIII. Memorable Quotes

- Tristan: "I'm studying math, I haven't seen numbers in years."
- Rémi: "No, you can't be in two places at the same time. Physicists can, but not you."

Tristan: "That's rather positive, it means you're not a physicist."

- Aurélien: "If you choose fascist friends, it will show."

— Pierre-Marie: "I don't say stupid things, ask Aurélien."

- Zinedine: "Especially Paul, he's a human."

— Théo, talking about graphs: "This tree is ugly, we want to cut it down."

- Théo: "So there's something Tristan realizes. Normally he's already realized it. But well! It's Tristan."

- Pierre-Marie: "We have a pretty little equation that's wandering in the forest."

- Pierre-Marie: "Anyone who uses Zsigmondy should jump out the window." Later in the proof, "Well, we're going to have to use Zsigmondy."

- Rémi: "The people who got 7/7 can be counted on the fingers of a one-armed man."

- Anatole, 13 years old, to first-year students: "Good night, little ones!"

- Gaspard: "Tic tac toc without pif paf pof is much better than tic tac."

— "I have to write a letter to someone."

Martin: "Is her name Élise, just in case?"

- Rémi: "I'm a pretty primitive person."

- "When will we get the T-shirts?" Rémi: "Yes."

The T-shirts only arrived on the last day.

- Georges: "The anims are all young and dynamic, even François Lo Jacomo is young and dynamic!"

— Martin: "There's fa, there's do. What else is there as a note already?"

- Alexander: "Everything is valued." Pierre-Marie: "Nothing is serious."

— Victor: "A 4x3 square."

- Victor: "There are bald people with zero hair."

- Théo: "So yes, I write very badly... If you can't decipher me, it's your fault."

- At the departure of the anims:

"Goodbye!"

"I don't think they can hear us."

"I can shout!"

"Even if you shout, I don't count on you to bring them back."

- Emile: "For the class, I take a break at 4:30 PM." The classes end at 4:30 PM.

- Eva: "It would be great to have a compass in the eye."

- Théo: "You're confusing Euclidean division with Euclidean division."

- Emma: "I've never completed anything in my life."

- Matthieu Vogel: "Do you often go to Bodo's camp?"

Amélie: "Yes, every time."

Matthieu V.: "That's why I'm bad at math."

- Théo, during the anims meeting: "This morning, I gave a sleeping class to the group."

- Stéphane: "Stanislas is modern, but in the old style."

- Emma: "I see the world in color, and it hurts my head."

- Théo: "But yes, it's possible with the kiwi card."

— Raphaël: "There's a shooting going on downstairs."

Everyone looks towards the window.

Raphaël, disappointed: "Okay... you can go look for two minutes." Everyone rushes to the window.

[^0]: 1. We will abuse the terminology and refer to the remainder of the division by 7 as the value modulo 7.