Justify that the triangles $B C M$ and $A B M$ have the same area without calculating it.

Justify that the following statement is correct: There are infinitely many planes that do not contain a point whose three coordinates are equal.

Justify that the probability that three different motifs appear on the badges has the value $\frac { ( n - 1 ) \cdot ( n - 2 ) } { n ^ { 2 } }$.

Prove that the quadrilateral ABFE is a trapezoid with two sides of equal length.

Draw the pyramid EFGHS${}_{15}$ in Figure 1. The lateral face $\mathrm { EFS } _ { 15 }$ and the base EFGH of this pyramid form an angle. Justify without further calculation that the measure of this angle is less than $45 ^ { \circ }$; use the following information for this purpose: For the midpoint $M$ of the square EFGH and the point $N$ with $\vec { N } = \frac { 1 } { 2 } \cdot ( \vec { E } + \vec { F } )$, we have $\overline { M S _ { 15 } } < \overline { M N }$.

The shadow region of the entire pyramid on the ground consists in the model of two congruent quadrilaterals. Draw this shadow region in Figure 3 and specify the special form of the mentioned quadrilaterals.

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\mathbf{C}$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$.
a) Show that $f\left([0,1]\cap\mathbf{Z}\left[\frac{1}{2}\right]\right)\subset\tau$.
b) Show that $f([0,1])\subset\tau$.

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, $\tau_0 = \widehat{0\,(-1)\,(-\mathrm{i})}$, $\tau_1 = \widehat{0\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\tau$.
Conversely, let $z\in\tau$.
a) Show that we can define two sequences $(z_n)_{n\geq 0}$ and $(r_n)_{n\geq 1}$ in the following way:

• $z_0 = z$ and, if $n\geq 1$:
• if $z_{n-1}\in\tau_0$ then $r_n = 0$ and $z_n = (\phi_0)^{-1}(z_{n-1})$
• otherwise $r_n = 1$ and $z_n = (\phi_1)^{-1}(z_{n-1})$.

Prove that, for every integer $n\in\mathbb{N}$, $z_n$ belongs to $\tau$.
b) Prove that $f\left(\sum_{n=1}^{\infty}\frac{r_n}{2^n}\right) = z$ (one may express $z$ in terms of $z_n$ and the $\phi_{r_i}$).
c) Write a function that takes as argument a complex number $z$ (which we will assume is in $\tau$) and a real number $\epsilon$ and which returns an approximate value to within $\epsilon$ of a preimage of $z$.

The map $f\in\mathcal{E}$ satisfies $Tf = f$, $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, and $f(x) = -\overline{f(1-x)}$ for all $x\in[0,1]$.
a) Prove that $f$ is not injective (one may use the relation $f(1-x) = -\overline{f(x)}$).
b) More generally show that there exists no continuous bijection from $[0,1]$ onto $\tau$ (one may use an argument of arc-connectedness).

We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$.
a) For $(i,j)\in\{0,1\}^2$, determine the complex expression of $\phi_i\circ\phi_j$, recognize it, specify its fixed point and the image of $\tau$. Make a drawing.
b) Let $r_1, r_2, \ldots, r_p$ be elements of $\{0,1\}$. Prove that $\phi = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_p}$ has a unique fixed point which we will not necessarily try to express simply.
c) Exhibit, using the map $f$, a fixed point of $\phi$.
d) Show that the set $X$ of complex numbers $z$ which are fixed points of the composition of a finite number of maps $\phi_0$ and $\phi_1$ is dense in $\tau$.

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Let $n \in \mathbb{N}^*$. Show that the polynomial function $T_n$ has exactly $n$ distinct zeros all belonging to $]-1,1[$. For $j \in \{1, 2, \ldots, n\}$, we denote by $x_{n,j}$ the $j$-th zero of $T_n$ in increasing order. Give the value of $x_{n,j}$.

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Let $n \in \mathbb{N}^*$ and $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$. Show that: $$\frac{T_n'(x)}{T_n(x)} = \sum_{j=1}^{n} \frac{1}{x - x_{n,j}}$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
Let $n \in \mathbb{N}^*$, $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$ and $P \in E_{n-1}$.
a) Show that: $$P(x) = \sum_{j=1}^{n} \frac{P(x_{n,j})}{T_n'(x_{n,j})} \frac{T_n(x)}{x - x_{n,j}}$$
b) Deduce that: $$P(x) = \frac{2^{n-1}}{n} \sum_{j=1}^{n} (-1)^{n-j} \sqrt{1 - x_{n,j}^2}\, P(x_{n,j}) \frac{T_n(x)}{x - x_{n,j}}$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Show that, for all $x \in [x_{n,1}, x_{n,n}]$, we have $$\sqrt{1 - x^2} \geq \frac{1}{n}$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$

Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$

Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)

Show that the set of non-zero similarities is a subgroup of $GL(E)$ under composition of applications.

Let $f$ be an antisymmetric endomorphism of $E$. Show that: $\forall x \in E, \langle x, f(x) \rangle = 0$.

Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.

Let $f$ be an antisymmetric endomorphism of $E$. Let $g$ be an antisymmetric endomorphism of $E$, such that $fg = -gf$. Show that: $\forall x \in E, \langle f(x), g(x) \rangle = 0$.

Let $f$ be an antisymmetric endomorphism of $E$. What is $f^{2} = f \circ f$ if $f$ is an orthogonal automorphism and antisymmetric of $E$?

Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$. We fix $x \in E \backslash \{0\}$. By considering $\Phi : f \mapsto f(x)$, linear application from $V$ to $E$, show that $\operatorname{dim}(V) \leqslant n$. Thus $1 \leqslant d_{n} \leqslant n$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $h^{-1} \circ h'$ has $n$ distinct eigenvalues. Show that there exists a basis of $E$ orthogonal for both $q$ and $q'$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $x \in E$ such that $q(x) = 0$ and such that $x \neq 0$.
We propose to demonstrate that there exists a plane $\Pi \subset E$ containing $x$ such that $(\Pi, q_{/\Pi})$ is an artinian plane (where $q_{/\Pi}$ denotes the restriction of the application $q$ to the plane $\Pi$).
a) Demonstrate that there exists $z \in E$ such that $\varphi(x,z) = 1$.
b) We set $y = z - \frac{q(z)}{2}x$. Compute $q(y)$.
c) Conclude.