Find all $(k, z) \in \mathbb{N}^{2}$ such that $2^{k}+1=z^{2}$.
Let $(k, z)$ be a solution. We factorize: $2^{k}=(z-1)(z+1)$, so $z-1$ and $z+1$ are powers of 2. Let $a<b$ such that $z-1=2^{a}$ and $z+1=2^{b}$. The difference between these two equations gives: $2^{b-a}\left(2^{a}-1\right)=2$. Therefore, $2^{b-a}=1$ and $2^{a}=1$ (by the uniqueness of the prime factorization). Therefore, $b=a+1$ and $a=0$. Thus, $z=3$, and we verify conversely that this is indeed a solution.

## 5 Tuesday 23 morning: Colin Davalo

This course has not yet been integrated.

## 6 Tuesday 23 afternoon: Florent Noisette

This course has not yet been integrated.

## 3 Group C: Geometry and Functional Equations

## 1 Sunday 21 morning: Cécile Gachet

Study Principle In this course, we focus on different geometric transformations that map points to points, lines to lines, and circles to circles. The study plan is as follows:

1. We consider a quantity that we would like to see preserved by our transformation (lengths, ratios of lengths, geometric angles, oriented angles);
2. We define the transformations that preserve this quantity and study their structure:

- Center of a transformation: does such a transformation have a fixed point? If so, how to construct it?
- "Degrees of freedom": can we determine a transformation from the given data of a certain number of points and their images? Conversely, for which $n$ can we fix $A_{1}, \ldots, A_{n}$ and $B_{1}, \ldots, B_{n}$ such that there exists an appropriate transformation that maps $A_{1}$ to $B_{1}, \ldots, A_{n}$ to $B_{n}$? (and what additional constraints must be imposed on these points to increase the degree of freedom $n$?)
- Composition of two transformations of the same type
- Decomposition of a transformation as a product of "elementary" factors.

3. If we have two types of transformations (those that preserve the schtroumpfs and those that preserve the bagadas, for example), we study the relative position of the set of transformations that preserve the schtroumpfs and the set of transformations that preserve the bagadas, and how our composition law behaves with respect to them (see the diagram below).

Additional points covered/not covered We emphasized the difference between a direct transformation and an indirect transformation (starting observation: axial symmetry is an isometry, it preserves angles but multiplies all oriented angle values by -1), this question playing an important role in the study of the number of degrees of freedom of a transformation. We studied the notion of interior similarity. We provided the construction of the Miquel point, from which we could deduce the theorem of the concurrence of the four circles at the Miquel point.

The general study of axial symmetries (especially their composition) was not carried out. The existence of a center for every similarity was assumed.

## Summary Diagram

Exercises covered In addition to the study of the various transformations that we tried to conduct as several small exercises, the following exercises were posed: