(*) We call $I$ the set of points in the plane such that their abscissa and ordinate are irrational numbers, and $R$ the set of points whose both coordinates are rational. How many points of $R$ at most can lie on a circle of irrational radius whose center belongs to $I$?
(Solved by Yakob Kahane)

If we have two points on a circle, then the center of said circle lies on the perpendicular bisector of the two points. If the said two points have rational coordinates, their perpendicular bisector has an equation with rational coefficients.

We assume henceforth that there exist $A, B, C$ three points with rational coordinates on the same circle $\Gamma$ whose center is not at rational coordinates. We denote $L_{1}, L_{2}$ the equations of the perpendicular bisectors of $[A B]$ and $[A C]$ with rational coefficients:

\[
\begin{aligned}
& L_{1}: a_{1} x+b_{1} y=c_{1} \\
& L_{2}: a_{2} x+b_{2} y=c_{2}
\end{aligned}
\]

Their point of intersection is the solution to a system of two equations with two unknowns and rational coefficients; the calculation of the solution (i.e., the point of intersection) remains within $\mathbb{Q}$; but this point $O$ of intersection is the center of the circumcircle of $A B C$ and thus of $\Gamma$, which contradicts the hypothesis.

It remains to find two rational points on a circle with a center of irrational coordinates. The reader will verify that a circle with center $O(\sqrt{2} ; 1-\sqrt{2})$ passes through $A(0 ; 0)$ and $B(1 ; 1)$, which concludes the proof.

## X. Memorable Quotes

- Matthieu L "The problem is not that there is a problem"
- "It's a bit like being a fakir"
- Étienne M "Arithmetic is simpler when it's easy"
- Lucie W "Do sharp angles exist?"
- Victor "You can use it for drafts... like this course"
- Antoine: "If you are the Queen of England, I am the Prime Minister. Are you the Queen of England?" Olivine: "Yes!"
- Matthieu L: "An airplane is like an elevator, just a bit more complicated"
- Noémie: "François, how many ears do you have?"
- "Does François have a cut ear?"
- Victor: "Clara? Is she one of the two Chinese girls downstairs?"
- Clara B: "There are two players who are not playing"
- Antoine: "God created the world and prime factorization"
- Savinien: "Let $a_{1}, a_{2}, \ldots, a_{4}$ be real numbers"
- Thomas: "We need to check that it works with any value of 2016"
- Mathias: $" \sqrt{4} \equiv 3[5] "$ (convinced of what he is saying)
- Clara: "I would be willing to live my whole life with Lucie but not with a child"

Lucie: "I would live with a child, but not with Clara"

- Clément: "I don't know the Knights of the Round Table, except for D'Artagnan"
- Clara B: "My circle has a radius of 0 and I am supposed to place 4 distinct points on it"
- Rémi: "The blue circle has seven points of intersection with the red line"

Pierre-Alexandre: "Which one? The one with the loops?"

- Lucie W: "Who is motivated to drink Toblerone?"
- Lucie W: "A photo without us is of no interest"
- Savinien, in a copy: "We choose the endpoint of the side of the other side of the side of the triangle which is the $3^{rd}$ vertex of the triangle"
- Vincent J: "Bézout is the one who invented the bézounours"
- Vincent J: "Are we, in our group, Swiss? No? So we are not a group because we don't have a neutral element"
- Vincent J (during his lecture): "What is a positive braid, apart from a braid that has participated in the Tour de France?"
- Sébastien: "He shows us how to prove that"
- Tanguy: "It's trivialong"

Clément: "Ah, yeah, it's a bit like a triathlon"

[^0]:    1. For the attention of the geometric reader: here, unless otherwise stated, a geometric transformation is considered as a function $t$ associating a geometric figure with another figure and satisfying a certain number of properties. However, the fact that the ambient spaces of departure and arrival are the same up to isomorphism (and therefore that $t$ can be considered by its action within a given figure) is not taken into account.
    2. We do not specify here the distinction between direct and indirect transformations, so we only consider direct transformations that, if they preserve angles, also preserve oriented angles.
[^1]:    8. Strictly speaking, on a trilateral.