A factory used a 3D printer to produce the prototype of a part. The prototype has the shape of a convex polyhedron, obtained by the juxtaposition of two distinct solids, one with the shape of a regular hexagonal prism and the other with the shape of a straight hexagonal pyramid frustum. The larger base of the pyramid frustum coincides with one of the bases of the prism.
After printing the prototype, it was sent to the customization sector for painting its surface. The criterion defined for painting considers that congruent faces must be painted with the same color, and non-congruent faces must have different colors. What is the quantity of colors used to paint the prototype?
(A) 9
(B) 8
(C) 6
(D) 4
(E) 3

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 $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.
Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We denote by $C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of functions $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ such that the two partial derivatives $\frac { \partial f } { \partial \theta _ { 1 } } , \frac { \partial f } { \partial \theta _ { 2 } }$ exist at every point of $\mathbb { R } ^ { 2 }$ and both define continuous functions on $\mathbb { R } ^ { 2 }$.
We set $\forall t \in \left[ 0 , + \infty \left[ , \theta ( t ) = \left( t \sqrt { \lambda _ { 1 } } + \varphi _ { 1 } , t \sqrt { \lambda _ { 2 } } + \varphi _ { 2 } \right) \right. \right.$.
The Ergodic Theorem states: Let $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Then, $$\lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } f \circ \theta ( t ) d t = ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \tag{4}$$
Let $j , l \in \mathbb { Z }$. Prove the Ergodic Theorem in the special case of the function $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto f \left( \theta _ { 1 } , \theta _ { 2 } \right) = e ^ { i \theta _ { 1 } j } e ^ { i \theta _ { 2 } l }$. (In the case where $( j , l ) \neq ( 0,0 )$ one may verify that $j \sqrt { \lambda _ { 1 } } + l \sqrt { \lambda _ { 2 } }$ is non-zero and then one may calculate each side of (4) separately in this special case).

In the rest of the second part, $f$ is an element of $\mathcal{C}_{0}$. For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Show that $\lim_{j \rightarrow +\infty} \max_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| = 0$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ For all $(j, k) \in \mathcal{I}$, $(i, \ell) \in \mathcal{I}$, calculate $c_{j,k}(\theta_{i,\ell})$.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Explicitly determine the set $\mathcal{D}_{P_C}$ when the polynomial $P$ is defined by $P(x,y) = x^3$.

Let $m \in \mathbb{N}$. We denote by $\mathcal{H}_m$ the vector subspace of harmonic polynomials of degree less than or equal to $m$. Determine the dimension of $\mathcal{H}_m$.

Explicitly determine a basis of $\mathcal{H}_3$.

We work on $\mathbb{R}^n$ for a natural integer $n \geqslant 3$. We admit that the Dirichlet problem on the unit ball of $\mathbb{R}^n$, associated with a continuous function defined on the unit sphere $S_n(0,1)$, admits a unique solution. Let $m \in \mathbb{N}^*$.
Determine the dimension of $\mathcal{H}_m$ as a function of $m$ and $n$.

Throughout this question, $\mathcal{S}$ is a simplex in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{S}}$. We want to show that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant 2\left\lfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{a(\mathcal{S})+1}\right)^n \right\rfloor + 1$$ We then set $a = a(\mathcal{S})$, and $k = \rfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a}{a+1}\right)^n \lfloor$.
10a. Express, for $\beta \in \mathbb{R}^*$ and $x \in \mathbb{R}^n$, $\operatorname{Vol}(\beta \mathcal{S})$ and $\operatorname{Vol}(\mathcal{S} - x)$. Show that for $\lambda \in [0,1[$ sufficiently close to $1$, $\operatorname{Vol}\left(\frac{\lambda a}{a+1}\mathcal{S}\right) > k$.
10b. For $\lambda$ as in the previous question, let $v_0, \ldots, v_k$ be the $k+1$ distinct points in $\frac{\lambda a}{a+1}\mathcal{S}$ satisfying $v_i - v_j \in \mathbb{Z}^n$ for all $i, j$, whose existence is guaranteed by Theorem 1. Show that the points $v_i - v_j$ are in $\lambda \mathcal{S}$. Deduce that the $v_i - v_j$ are in $\mathring{\mathcal{S}}$.
10c. Show that there exists an index $j \in \{0, \ldots, k\}$ such that the $(2k+1)$ points $0, \pm(v_i - v_j)$, for $i \in \{0, \ldots, k\} \setminus \{j\}$ are distinct. Deduce the statement of question 10, then that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{2}\right)^n$$

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.

We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$.
Study the limits of $F$ at 0 and at $+\infty$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Calculate $\delta\left(H_0\right)$ and, for $k \in \llbracket 1, n \rrbracket$, express $\delta\left(H_k\right)$ in terms of $H_{k-1}$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Give the coordinates of the polynomial $X^3 + 2X^2 + 5X + 7$ in the basis $(H_0, H_1, H_2, H_3)$ of $\mathbb{R}_3[X]$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $k \in \mathbb{Z}$. Calculate $H_n(k)$. Distinguish three cases: $k \in \llbracket 0, n-1 \rrbracket$, $k \geqslant n$, and $k < 0$. For the latter case, set $k = -p$.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. Show that $J : \mathbb { R } \rightarrow \mathbb { R }$ is continuous, and calculate $J ( 0 )$ and $J \left( \frac { \pi } { 2 } \right)$.

Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$, $$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$

$x_i$	4,50	4,55	4,60	4,65	4,70
$\hat{\lambda}^*(x_i)$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$
$x_i$	4,75	4,80	4,85	4,90	4,95
$\hat{\lambda}^*(x_i)$	$5{,}1 \times 10^{-4}$	$5{,}5 \times 10^{-3}$	$1{,}1 \times 10^{-2}$	$1{,}6 \times 10^{-2}$	$2{,}1 \times 10^{-2}$
$x_i$	5,00	5,05	5,10	5,15	5,20
$\hat{\lambda}^*(x_i)$	$2{,}6 \times 10^{-2}$	$3{,}1 \times 10^{-2}$	$3{,}6 \times 10^{-2}$	$4{,}1 \times 10^{-2}$	$4{,}6 \times 10^{-2}$
$x_i$	5,25	5,30	5,35	5,40	5,45
$\hat{\lambda}^*(x_i)$	$5{,}1 \times 10^{-2}$	$5{,}6 \times 10^{-2}$	$6{,}1 \times 10^{-2}$	$6{,}6 \times 10^{-2}$	$7{,}1 \times 10^{-2}$

5. Give a procedure for computing $\operatorname{Card} \operatorname{MD}(n)$ by recursion.

We consider the case where $I = [-1,1]$ and $w(x) = 1$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, orthogonal for $\langle f, g \rangle = \int_{-1}^1 f(x)g(x)\,\mathrm{d}x$).
Determine the first four orthogonal polynomials $(p_0, p_1, p_2, p_3)$ associated with the weight $w$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$ and $M(Q) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\ln\left|Q(e^{i\theta})\right|d\theta\right)$. Compute the limit of $\varphi'$ at $0^+$ then deduce that: $$M_p(Q)^{1/p} \underset{p \rightarrow 0^+}{\longrightarrow} M(Q).$$

Write the matrix $H$ of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, that is, the matrix with general term $h_{i,j} = \phi\left(X^i, X^j\right)$ where the indices $i$ and $j$ vary between 0 and $n-1$.

$\mathbf{15}$ ▷ Let $M \in \mathcal{M}_n(\mathbf{R})$, which can also be considered as a complex matrix, and let the application $\delta_M : \mathbf{R} \rightarrow \mathbf{R},\ t \mapsto \delta_M(t) = \operatorname{det}\left(I_n + tM\right)$. Using a Taylor expansion to order 1, show that $\delta_M$ is differentiable at 0 and compute $\delta_M'(0)$.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Explicitly determine $\operatorname{proj}_C$ in the following cases: $$\text{i) } C = \mathbb{R}_+^d, \quad \text{ii) } C = \left\{y \in \mathbb{R}^d : \|y\| \leqslant 1\right\}$$ $$\text{iii) } C = \left\{y \in \mathbb{R}^d : \sum_{i=1}^d y_i \leqslant 1\right\}, \quad \text{iv) } C = [-1,1]^d$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Using the Cauchy-Schwarz inequality, show that for all functions $f \in E$, $$\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } U ( f ) ( x ) = 0$$

We fix a choice of $\lambda$ such that $P_a(x) = x - \lambda x(x-a)(x-1)$ satisfies $P([0,1])=[0,1]$ and $P$ is increasing on $[0,1]$. Let $\left(P_a^{\circ n}\right)_{n\geq 0}$ be the sequence of polynomials defined recursively by $P_a^{\circ 0}(x) = x$ and $P_a^{\circ n+1}(x) = P_a\left(P_a^{\circ n}(x)\right)$.
Show that $P_a^{\circ n}$ converges uniformly to 1 on every compact subset of $]a,1]$ and uniformly to 0 on every compact subset of $[0,a[$.