Consider $n$ distinct points in the plane. For each segment formed by a pair of these points, the midpoint of the segment is colored blue. How many blue points must there be, at a minimum?

If two midpoints coincide, they count as only one point.

## Solutions
The answer is $2 n-3$.

First, we can achieve this bound: we align the points on the integers of the x-axis $((0,0),(1,0), \ldots,(n-1,0))$. Then the blue points will be exactly the points with integer or half-integer coordinates between $(0,0)$ and $(n-1,0)$ (thus $(0.5,0),(1,0),(1.5,0), \ldots)$, which makes $2 n-3$ points.

To show that this bound is optimal, we introduce an x-axis, ensuring that each point has a different x-coordinate (this is possible since there are infinitely many directions for the axis, but only a finite number of points). We consider the leftmost and rightmost points. The leftmost point allows us to form $n-1$ midpoints with the remaining points (the x-coordinates of these $n-1$ midpoints are then distinct since all points have different x-coordinates). The rightmost point, on the other hand, also allows us to form $n-1$ distinct points by the same argument, but we have counted the midpoint formed by our two extreme points twice, so there are at least $(n-1)+(n-1)-1=2 n-3$ blue points.

## V. Group C

## Content of this section

1 First part: Arithmetic & Algebra . . . . . . . . . . . . . . . . . . . . . . 248

1 Polynomials 1 (Vincent) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

2 Order 1 (Félix) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

3 Polynomials 2 (Georges) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

4 Order 2 (Melvil & Maryam) . . . . . . . . . . . . . . . . . . . . . . . . . . 266

5 Inequalities (Thoé) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6 Problems of magnitude (Quentin) . . . . . . . . . . . . . . . . . . . . . . 284

7 Functional equations (Emilhan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8 TD Diophantine equations (Antoine) . . . . . . . . . . . . . . . . . . . . 296

2 Mid-course training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

3 Second part: Geometry & Combinatorics . . . . . . . . . . . . . . . . . . . 303

1 Double counting 1 (Raphael) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

2 Power of a point (Benoît & Martin) . . . . . . . . . . . . . . . . . . . . . 308

3 South Pole (Aline) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

4 Graphs (Jean) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

5 Radical axes (Gaëtan & Gaspard C.) . . . . . . . . . . . . . . . . . . . . 351

6 Variants/invariants (Aimeric) . . . . . . . . . . . . . . . . . . . . . . . . . 362

7 Double counting 2 (Maena & Émile) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

8 TD of Geometry (Pierre-Marie) . . . . . . . . . . . . . . . . . . . . . . . . 370

4 End-of-course training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

## 1 First part: Arithmetic & Algebra

## 1 Polynomials 1 (Vincent)

This course is an introduction to polynomials, which are very nice functions. It is largely based on the written course by Igor Kortchemski, available on the POFM website. More specifically, it covers sections 1, 2.1, 2.2, and 2.6 of this course.

## First definitions and operations

## Definition 1.

A function $P$ from $\mathbb{R}$ to $\mathbb{R}$ is called a polynomial with real coefficients if there exists an integer $n \geqslant 0$ and real numbers $a_{0}, a_{1}, \ldots, a_{n}$ such that

$$
P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}
$$

for all real $x$.

We will see later, in Corollary 5, that, provided we choose $n$ to be minimal, this representation is unique. Once this result is admitted, we can then define the following concepts.

## Definition 2.

When $a_{n} \neq 0$, we say that

- the number $n$ is the degree of $P$; it is commonly denoted as deg $(P)$;
- the numbers $a_{0}, a_{1}, \ldots, a_{n}$ are the coefficients of $P$; the number $a_{k}$ is called the coefficient of degree $k$ of $P ;{ }^{1}$
- the number $a_{n}$ is the leading coefficient of $P$;
- the number $a_{0}$ is the constant coefficient of $P$; it is thus called because, when $P$ is constant, the only value it takes is $a_{0}$.

When the numbers $a_{0}, a_{1}, \ldots, a_{n}$ are all zero, we say that $P$ is the zero polynomial, and we define (somewhat arbitrarily, but this definition will prove practical) its degree as being equal to $-\infty$.

Finally, it may happen that we impose other constraints on the coefficients $a_{i}$. For example, we may be interested in polynomials $P$ for which these coefficients are non-negative or zero; strictly positive; rational; integers; or simply complex numbers, not necessarily real. ${ }^{2}$

In general, when $\mathcal{E}$ is a set of numbers, we denote by $\mathcal{E}[X]$ the set of polynomials whose coefficients belong to $\mathcal{E}$. Thus, $\mathbb{R}[X]$ denotes the set of polynomials with real coefficients, and $\mathbb{Q}[X]$ the set of polynomials with rational coefficients.

In the following, we will use the notations $P$ and $P(X)$, or even $a_{n} X^{n}+a_{n-1} X^{n-1}+$ $\cdots+a_{1} X+a_{0}$ interchangeably to denote this polynomial. However, we strongly advise against using the notation $P(x)$ to denote