Determine all integers $x_{1}, x_{2}, \cdots, x_{9}, x_{10}$ such that

$$
0<x_{1}<x_{2}<\cdots<x_{9}<x_{10} \text { and } x_{9} x_{10} \leqslant 2\left(x_{1}+x_{2}+\cdots+x_{9}\right)
$$
We immediately check that $x_{1}=1, x_{2}=2, \ldots, x_{10}=$ 10 is a solution. Let's show that there is no other.

First method. Let $a=x_{9}$ and $b=x_{10}$ for short. We have

$$
a(a+1) \leqslant a b \leqslant 2((a-8)+\cdots+(a-1)+a)=18 a-72
$$

thus $72 \leqslant 18 a-a(a+1)=a(17-a)$. This proves that $a \leqslant 16$. Moreover, since $0 < x_{9}$, we deduce that $x_{10}=x_{9}+1$ and that

$$
x_{1}+x_{2}+\cdots+x_{9}=1+2+\cdots+x_{9} .
$$

Consequently, all integers between 1 and $x_{9}$ appear in the list $x_{1}, \ldots, x_{9}$, which implies that $x_{k}=k$ for all $k=1,2, \ldots, 10$.

## 4 Initial Evaluation

The first day of the internship is dedicated to dividing the interns into four balanced level groups, based on an "initial evaluation" questionnaire presented below:

## GEOMETRY

Have you ever used the inscribed angle theorem? YES? NO?

Do you know what similar triangles are? YES? NO?

Do you know what a homothety is? YES? NO?

Have you ever used Ceva's theorem? YES? NO?

Do you know the law of sines? YES? NO?

If yes, write it down:

Solve the following problem:

Let $A B C$ be a triangle, $P \in] A B[, Q \in] B C[$ and $R \in] C A[$. The circumcircles of $A P R$ and $B P Q$ intersect at $P$ and $X$. Show that $X$ is on the circumcircle of $C Q R$.

## COMBINATORICS

Have you ever used the pigeonhole principle? YES? NO?

Do you know what a binomial coefficient is? YES? NO?

If yes, what is: $\binom{n}{k}+\binom{n}{k+1}$?

Do you know what a proof by induction is? YES? NO?

If yes, solve the following problem:

We draw $n$ lines in the plane, no two of which are parallel and no three of which are concurrent. Show that these $n$ lines divide the plane into $\frac{n^{2}+n+2}{2}$ regions.

## ALGEBRA

Do you know how to perform polynomial division? YES? NO?

If yes, write the division of $X^{3}+X+1$ by $X-2$.

Do you know the arithmetic-geometric mean inequality? YES? NO?

If yes, write it down:

Find all functions $f$ such that for all real $x$ and $y$, we have: $f(x+y-f(x))=y$

## ARITHMETIC

## II. ANIMATH CUP AND INITIAL EVALUATION

Do you know what the GCD of two integers is? YES? NO?

If yes, what is the GCD of 21 and 91?

Have you ever used Bézout's theorem? YES? NO?

Do you know how to manipulate congruences? YES? NO?

If yes, what is: $2014^{2014}$ modulo 7?

Do you know the Chinese remainder theorem? YES? NO?

If yes, write it down:

If you have time left, here are two problems to solve (directly on the sheet):

1) In the city of Baghdad, the good Caliph has just invented a new game: there are twenty pebbles, and each of the two players, in turn, can take one, two, or five of these pebbles. The winner is the one who takes the last pebble(s). The good Caliph loves this game: as he is the Caliph, he plays first and always wins. How does he do it?

2) Let $A B C$ be a triangle with an angle of $60^{\circ}$ at $A$, $I$ the center of its inscribed circle, $F$ the point on $[A B]$ such that $(I F)$ is parallel to $(A C)$, and $P$ the point on $[B C]$ such that $B C=3 B P$. Show that the angle $\widehat{B F P}=\frac{1}{2} \widehat{A B C}$.

II. ANIMATH CUP AND INITIAL EVALUATION

## III. First Period

## 1 Group A: Basic Strategies

## 1 Tuesday 19 morning: François Lo Jacomo

Basic strategies traditionally refer to strategies used in many areas of mathematics. This course will focus on two of them: the pigeonhole principle and invariants. We will end, if time permits, with some tiling exercises.

## Pigeonhole Principle

The pigeonhole principle seems elementary, but it has only been used in mathematics since the 19th century:

If you place at least $n+1$ objects into $n$ drawers, at least one drawer will contain at least two objects. More generally, if you place at least $k n+1$ objects into these same $n$ drawers, at least one drawer will contain at least $k+1$ objects. For example: knowing that a human has fewer than 250,000 hairs and that there are 2,500,000 Parisians, at least two Parisians have the same number of hairs. We can even specify that there are at least 10 Parisians with the same number of hairs. If there were 2,500,001 Parisians each with fewer than 250,000 hairs, we could assert that at least 11 of them have the same number of hairs.