Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.

Let $a$ be a real number in the open interval $]0,1[$. Show that there exists $\lambda > 0$ such that the polynomial $$P(x) = x - \lambda x(x-a)(x-1)$$ satisfies the following two properties:

1. $P([0,1]) = [0,1]$,
2. $P$ is increasing on $[0,1]$.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$, show that $$y = \operatorname{proj}_C(x) \Longleftrightarrow y \in C \text{ and } (x - y) \cdot (z - y) \leqslant 0, \forall z \in C.$$

Show that, for all $i$ and $k$ in $\llbracket 1 , n \rrbracket$, $$L _ { i } \left( a _ { k } \right) = \begin{cases} 1 & \text { if } k = i \\ 0 & \text { otherwise } \end{cases}$$ where $L_i(X) = \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \frac { X - a _ { j } } { a _ { i } - a _ { j } }$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x).$$

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that $u$ is an orthonormal basis of $V$.
(b) Show that for $k \in \llbracket 1, p-1 \rrbracket$, we have $u_k^{\prime} \in \operatorname{Vect}(u_{k+1}, \ldots, u_p)^{\perp}$. (Hint: One may consider the map $t \mapsto u_k(t) = \frac{u_k + t u_l}{\|u_k + t u_l\|}$ for all $t \in \mathbb{R}$ and $l \in \llbracket k+1, p \rrbracket$ as well as its derivative.)
(c) Show that $u_{k+1} \in \left(\operatorname{Vect}(u_1, \ldots, u_k) + \operatorname{Vect}(u_1^{\prime}, \ldots, u_k^{\prime})\right)^{\perp}$ for all $k$ element of $\llbracket 1, p-1 \rrbracket$.
(d) Deduce that the subspaces $W_k = \operatorname{Vect}(u_k, u_k^{\prime})$ for $k \in \llbracket 1, p \rrbracket$ are pairwise orthogonal.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have $$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2$$ and deduce that $\operatorname{proj}_C$ is continuous.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have $$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2,$$ and deduce that $\operatorname{proj}_C$ is continuous.

Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.

Show that, for all $i \in \llbracket 1 , n \rrbracket$ and all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$\left\langle L _ { i } , P \right\rangle = P \left( a _ { i } \right).$$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M.$$

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?

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

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

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$. The $\omega$-orthogonal of $F$ is defined as $$F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}.$$ Justify that $F ^ { \omega }$ is a vector subspace of $E$.

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that $D - C$ is a convex closed subset of $\mathbb{R}^d$ not containing $0$.

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?

Deduce that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$P = \sum _ { i = 1 } ^ { n } P \left( a _ { i } \right) L _ { i }.$$

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Show that for all $e \in E^p$, we have $\Omega_p(e) \in \mathscr{A}_p(E, \mathbb{R})$.
(b) Verify that for all $(e, u) \in E^p \times E^p$, we have $\Omega_p(e)(u) = \Omega_p(u)(e)$.
(c) Show that $\Omega_p \in \mathscr{A}_p(E, \mathscr{A}_p(E, \mathbb{R}))$.

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that there exist $p \in \mathbb{R}^d$ and $\varepsilon > 0$ such that $$p \cdot x \leqslant p \cdot y - \varepsilon, \forall (x, y) \in C \times D$$ (we say that $C$ and $D$ can be strictly separated).

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that there exist $p \in \mathbb{R}^d$ and $\varepsilon > 0$ such that $$p \cdot x \leqslant p \cdot y - \varepsilon, \forall (x, y) \in C \times D.$$ (we say that $C$ and $D$ can be strictly separated).

For every $x \in E$, we denote by $\omega ( x , \cdot )$ the linear map from $E$ to $\mathbb { R }$, $y \mapsto \omega ( x , y )$ and we consider
$$d _ { \omega } : \left\lvert \, \begin{array} { c c c } E & \rightarrow & \mathcal { L } ( E , \mathbb { R } ) \\ x & \mapsto & \omega ( x , \cdot ) \end{array} \right.$$
Show that $d _ { \omega }$ is an isomorphism.

Show that, for any polynomial $P$ of degree at most $n - 2$, $$\sum _ { i = 1 } ^ { n } \frac { P \left( a _ { i } \right) } { \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \left( a _ { i } - a _ { j } \right) } = 0 .$$

For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Show that there exists $r>0$ satisfying the following two properties:

1. for all $g\in\Gamma(n\ln(2))$, there exists $v\in\Delta(n)$ such that $d(gv_0,v)\leq r$,
2. for all $v\in\Delta(n)$, there exists $g\in\Gamma(n\ln(2))$ such that $d(gv_0,v)\leq r$.