Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Deduce that $Q = 0$, then that $W = \frac { 1 } { 2 ^ { n - 1 } } T _ { n }$.
One may consider the sum of the inequalities from the previous question and exploit question 6 applied to suitable data.

We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.

We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
We admit that every matrix in $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$. Deduce that, if $M \in \mathcal{M}_{n}(\mathbf{C})$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$, there exists an element $P_{M,X,Y}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad X^{T}R_{z}(M)Y = \frac{P_{M,X,Y}(z)}{\chi_{M}(z)}.$$

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
We admit the equality $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t = \int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}t\right) \mathrm{d}u$$ We also admit that, for $u \in [-\pi, \pi]$ such that $f_{u}$ is not constant, the set of points in $]-\pi, \pi[$ where the function $f_{u}^{\prime}$ vanishes is finite.
Deduce the inequality $$(3) \quad V(f) \leq 2\pi n \|f\|_{\infty}.$$

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$: $$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$ and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, \mathbf{P})$ and for all subset $A$ of $E$: $$\left|\mu_X(A) - \mu_Y(A)\right| \leqslant \mathbf{E}\left(\left|\mathbf{1}_{\{X \in A\}} - \mathbf{1}_{\{Y \in A\}}\right|\right)$$ and deduce that $\left\|\mu_X - \mu_Y\right\| \leqslant \mathbf{P}(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

Let $b \in \mathbb{R}$ such that $\cos(b) \in ]0,1[$. Show that the sequence of functions $(f_{b,n})_{n\in\mathbb{N}}$ defined by $$f_{b,n}(t) = P_{\cos(b)}^{\circ n}\left(\cos^2\left(\frac{\pi}{2}t\right)\right)$$ converges uniformly to 1 on every compact subset of $]-\cos(b), \cos(b)[$ and converges uniformly to 0 on every compact subset of $[-1,-\cos(b)[\cup]\cos(b),1]$.

Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$, $$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$, and let $n(T)$ be the natural number such that $\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$ for every $p \in \mathbb{K}[X]$.
Deduce $\ker(T)$ in terms of $n(T)$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, and for $X \in E$, the functions $\psi_X : t \mapsto H_t X$ and $\varphi_X : t \mapsto \|H_t X\|^2$. Deduce that $\varphi_X$ is differentiable and express $\varphi_X'(t)$ in terms of $q_u$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. By reasoning by contradiction, show that if $A$ is relatively compact then $A$ is equicontinuous.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:
Theorem 1: The following two properties are equivalent: - (P1) $A$ is relatively compact. - (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.
Show that $(P1) \Rightarrow (P2)$.

We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$, and $(g_n)_{n \in \mathbb{N}}$ a subsequence converging pointwise on $K$ to $g$.
(a) Show that $g$ is continuous on $K$.
(b) Show that the sequence $(g_n)_{n \in \mathbb{N}}$ converges uniformly to $g$ on $K$. (Hint: you may reason by contradiction.)
(c) Deduce that $(P2) \Rightarrow (P1)$.

Let $A$ be a commutative ring. Show that if $A$ has property (F), then it has property (TF).

Let $n \in \mathbb { N } ^ { * }$. Verify that
$$\frac { 2 n \ln ( 2 ) } { \ln ( 2 n ) } - 1 \geqslant \frac { n \ln ( 2 ) } { \ln ( 2 n ) }$$
then deduce that
$$\pi ( 2 n ) \geqslant n \frac { \ln ( 2 ) } { \ln ( 2 n ) }$$

We admit, only in this question, that $\zeta ( 2 ) = \frac { \pi ^ { 2 } } { 6 }$. Show that $\pi$ is an irrational number.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Deduce that $\psi$ is differentiable on $\mathbb{R}_+^*$ and that $m(h) = \frac{u(h) - h}{\beta}$ for all $h \in \mathbb{R}_+^*$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Deduce that there exists $n_0 \in \mathbb{N}$ such that, for all $n \geqslant n_0$, $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. We assume that $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior, and that the set $Q = \{\ell \in \mathbb{R}^n : \langle \ell, x \rangle \leq 1\ \forall x \in V\}$ is a polytope of $\mathbb{R}^n$. Deduce that $\operatorname{Conv}(V)$ is a polytope.

Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$, and $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$ the linear form defined recursively, satisfying $\chi_n(\mathbb{1}_{P^\circ}) = (-1)^{\operatorname{dim} P}$ for every polytope $P$.
Deduce Euler's formula $\sum_F (-1)^{\operatorname{dim} F} = 1$ where $F$ ranges over the faces of $P$.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients.
Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Deduce that there exists a nonzero matrix $Q$ and $\epsilon > 0$ such that $\{M + tQ, t \in [-\epsilon, \epsilon]\} \subset B_n$, and conclude that every vertex of $B_n$ is of the form $P^\sigma$.

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The operator $p_f$ is defined as the unique minimizer of $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$. Let $x, y \in \mathbb{R}$, $\tilde{x} := p_f(x)$, $\tilde{y} := p_f(y)$. We choose $v := \tilde{y} - \tilde{x}$ in inequality $$2\tau(f(\tilde{x}) + f(\tilde{y}) - f(\tilde{x} + tv) - f(\tilde{y} - tv)) \leq |\tilde{x} + tv - x|^2 + |\tilde{y} - tv - y|^2 - |\tilde{x} - x|^2 - |\tilde{y} - y|^2$$ Show that the left-hand side is positive for all $t \in [0,1]$. Deduce that $$|\tilde{x} - \tilde{y}|^2 \leq (x-y)(\tilde{x} - \tilde{y}).$$

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. We have a convergent subsequence $x_{\varphi(n)} \rightarrow x_{**}$ as $n \rightarrow \infty$, where $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ is strictly increasing and $x_{**} \in \mathbb{R}$. Show that $x_{\varphi(n)+1} \rightarrow x_{**}$ as $n \rightarrow \infty$, then deduce that $p_f(x_{**}) = x_{**}$.