In how many ways can we tile a horizontal strip $2 \times n$ with bricks $1 \times 2$ or $2 \times 1$
When looking at the small cases, we quickly notice that the answer is the $n$-th term of the Fibonacci sequence defined by $\mathcal{F}_{0}=\mathcal{F}_{1}=1$, and for $n \geq 2, \mathcal{F}_{n}=$ $\mathcal{F}_{n-1}+\mathcal{F}_{n-2}$.

We prove this by induction, making sure to initialize at ranks 1 and 2.

For the inductive step, we assume that the result holds at ranks $n-1$ and $n$, and we notice that at rank $n+1$, the last brick is either vertical, in which case there are $\mathcal{F}_{n}$ possibilities, or horizontal, in which case there are $\mathcal{F}_{n-1}$. These two possibilities are disjoint and cover all cases, so there are $\mathcal{F}_{n+1}=\mathcal{F}_{n}+\mathcal{F}_{n-1}$ ways to tile the strip of side $n+1$.

## 3 Saturday 19 morning: Antoine Séré

The entire course can be found in the Montpellier 2014 Olympiad stage polycopied, page 57 for invariants and page 63 for the extremum principle.

## 4 Group B: Arithmetic

## 1 Thursday 17 afternoon: Victor Vermès

This course is an introduction to arithmetic. It reviews the concepts of divisibility, GCD, and prime numbers. The theorem of Bézout, the lemma of Gauss, and the prime factorization are demonstrated and applied to some exercises. This course corresponds to parts 1 and 2 of the beginner arithmetic polycopied by Jean-Louis $\mathrm{Tu}$, and here is the solution to the exercises.

## 2 Friday 18 afternoon: Raphaël Ducatez

This course focused on congruences. It can be found in part 3 of the polycopied by Jean-Louis Tu.

## 3 Saturday 19 afternoon: Victor Vermès

This course complements the two previous arithmetic courses by emphasizing the usefulness of factorization and introducing infinite descent, applying it to exercises. This course corresponds to part 4 of the beginner arithmetic polycopied by Jean-Louis Tu, and here is the solution.

## 5 Group C: Combinatorics

## 1 Thursday 17 morning: Félix Breton