Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \backslash A$, show that there exists $p \in \mathbb{R}^d \backslash \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geqslant 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$

• a) Show that $p \cdot z \geqslant 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geqslant 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
• b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
• c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|$$

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leq \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

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 $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \setminus A$, show that there exists $p \in \mathbb{R}^d \setminus \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A.$$

Let $A = \operatorname{co}(E)$ where $E$ is the subset of $\mathbb{R}^3$ defined by $$E = \{(0,0,1),(0,0,-1)\} \cup \{(1 + \cos(\theta), \sin(\theta), 0), \theta \in [0, 2\pi]\}$$ show that $\operatorname{Ext}(A)$ is non-empty and is not closed.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $\operatorname{Ext}(K)$ is non-empty (one may reduce to the case where $0 \in K$ and reason on the dimension of $K$).

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geq 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$
a) Show that $p \cdot z \geq 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geq 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|.$$

Let $C _ { 1 } , \ldots , C _ { n }$ be $n$ column matrices in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$, with $C _ { 1 }$ non-zero.
Prove that, if the family $\left( C _ { 1 } , \ldots , C _ { n } \right)$ is linearly dependent, then there exists a unique $j \in \llbracket 1 , n - 1 \rrbracket$ such that $$\left\{ \begin{array} { l } \left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent } \\ C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \end{array} \right.$$

Let $d \in \llbracket 1 , n \rrbracket , \left( U _ { 1 } , \ldots , U _ { d } \right)$ be a linearly independent family in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $H = \operatorname { Vect } \left( U _ { 1 } , \ldots , U _ { d } \right)$.
Prove that there exist integers $i _ { 1 } , \ldots , i _ { d }$ satisfying $1 \leqslant i _ { 1 } < \cdots < i _ { d } \leqslant n$ such that the application $$\left\lvert \, \begin{array} { c c c } H & \rightarrow & \mathcal { M } _ { d , 1 } ( \mathbb { R } ) \\ \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) & \mapsto & \left( \begin{array} { c } x _ { i _ { 1 } } \\ \vdots \\ x _ { i _ { d } } \end{array} \right) \end{array} \right.$$ is bijective.
One may consider the rank of the matrix in $\mathcal { M } _ { n , d } ( \mathbb { R } )$ whose columns are $U _ { 1 } , \ldots , U _ { d }$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. For all $k \in \llbracket 0, K-1 \rrbracket$, show that there exist at least $K - k$ distinct real numbers in $[0,1]$ at which the function $f^{(k)} - P^{(k)}$ vanishes.

We fix $f \in \mathcal{C}^K([0,1])$ and denote by $P$ the polynomial determined in question Q7. For all $k \in \llbracket 0, K-1 \rrbracket$, show that there exist at least $K - k$ distinct real numbers in $[0,1]$ at which the function $f^{(k)} - P^{(k)}$ vanishes.

Let $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by $$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$ For all $x \in I$, show that there exists $c \in I$ such that $$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$ For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by $$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$ where the real number $K$ is chosen so that $r ( x ) = 0$.

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 $D = f - g$.
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 $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.

Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. 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 $a$ 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 $[0,1]$.

We denote by $\mathcal{C}([-1,1]^2)$ the space of continuous functions from $[-1,1]^2$ to $\mathbb{C}$ and $\mathcal{T}([-1,1]^2)$ the subspace generated by the functions $$e_{u,v} : (s,t) \mapsto e^{i\pi us}e^{i\pi vt}, \quad (u,v) \in \mathbb{Z}^2.$$ Let $a,b,c,d \in [-1,1]$ such that $a < b$ and $c < d$. Show that for every $\varepsilon < \min\left(\frac{b-a}{2}, \frac{d-c}{2}\right)$, there exists $f_\varepsilon \in \mathcal{T}([-1,1]\times[-1,1])$ satisfying the following properties:

1. $f_\varepsilon(s,t) \in [0,1]$ for all $(s,t) \in [-1,1]^2$,
2. $f_\varepsilon(s,t) \leq \varepsilon$ for $(s,t) \notin [a,b]\times[c,d]$,
3. $f_\varepsilon(s,t) \geq 1-\varepsilon$ for $(s,t) \in [a+\varepsilon,b-\varepsilon]\times[c+\varepsilon,d-\varepsilon]$.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Let $S_{1,2}$ be the subgroup of $\Gamma$ generated by $s_{w_1}$ and $s_{w_2}$. Let $v\in\mathcal{H}$.
Show that there exists $g\in S_{1,2}$ such that $$B(gv,w_1)\geq 0 \quad \text{and} \quad B(gv,w_2)\geq 0.$$

Show that for all $v\in\mathcal{H}$, there exists $g\in\Gamma$ such that $gv\in T$.

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 $d(gv_0,v)\leq r$.