Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients whose constant term equals 1. Show that there exists an integer $b \geq 1$ such that $Q(bx)$ has integer coefficients.

We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Let $d \in \mathbb{N}^{*}$. Determine an annihilating polynomial of $\Delta_{d}$. Is the endomorphism $\Delta_{d}$ diagonalisable?

We assume in this question that $0 < R_u \leqslant 1$. Let $A \in \mathbb{M}_n(u)$ and $B \in \mathbb{M}_n(u)$ be two symmetric matrices such that $AB = BA$. Show that $AB \in \mathbb{M}_n(u)$.

Show that if $f \in \mathbf{Q}\llbracket x \rrbracket$ is the power series expansion of a rational function with rational coefficients, then $f$ is globally bounded.
(A power series $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ is globally bounded if there exist integers $A, B \geq 1$ such that $A f(Bx) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} (B^n A c_n) x^n$ is a power series with integer coefficients.)

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Justify that if $(f, g) \in \mathcal{E}^{2}$ and $(\lambda, \mu) \in \mathbb{C}^{2}$, then $\lambda f + \mu g \in \mathcal{E}$ and $fg \in \mathcal{E}$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that $\mathscr{V}(A)$ is nonempty.

Show that the power series $$\sum_{m=0}^{\infty} x^{m^2} = \sum_{n=0}^{\infty} c_n x^n$$ where $c_n = 1$ if $n$ is the square of an integer $m \geq 0$ and $c_n = 0$ otherwise, is not the power series expansion of a rational function.

For which values of $d$ do we have $gg^{\prime} = g^{\prime}g$ for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$?

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity, and $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. Let $f \in \mathcal{E}$ whose power series expansion we denote $\sum a_{n} z^{n}$. Show that, for all $k \in \mathbb{Z}$: $$\int_{0}^{1} f(\omega(t)) \omega(t)^{-k} \,\mathrm{d}t = \begin{cases} a_{k} & \text{if } k \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $P \in \mathscr{V}(A)$. Show that $\varphi_A$ divides $P$.

We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $\delta(\boldsymbol{x}, \boldsymbol{y}) = \inf\{ \|\boldsymbol{y} - g \cdot \boldsymbol{x}\| \mid g \in \operatorname{Dep}(\mathbb{R}^{d}) \}$.

• [(a)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $$\|g \cdot \boldsymbol{y} - g \cdot \boldsymbol{x}\| = \|\boldsymbol{y} - \boldsymbol{x}\|.$$
• [(b)] Deduce that $\delta(\boldsymbol{x}, \boldsymbol{y}) = \delta(\boldsymbol{y}, \boldsymbol{x})$.
• [(c)] Show that for all $(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}) \in \mathscr{E}_{d}^{n}(\mathbb{R})^{3}$ and $(g, g^{\prime}) \in (\operatorname{Dep}(\mathbb{R}^{d}))^{2}$, we have $$\|\boldsymbol{z} - g \cdot \boldsymbol{x}\| \leqslant \|\boldsymbol{z} - (gg^{\prime}) \cdot \boldsymbol{y}\| + \|g^{\prime} \cdot \boldsymbol{y} - \boldsymbol{x}\|$$
• [(d)] Deduce that $\delta(\boldsymbol{x}, \boldsymbol{z}) \leqslant \delta(\boldsymbol{x}, \boldsymbol{y}) + \delta(\boldsymbol{y}, \boldsymbol{z})$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that the roots of $\varphi_A$ in $\mathbb{C}$ are exactly the eigenvalues of $A$.

We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$. For all $\boldsymbol{x} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $c(\boldsymbol{x}) = \{ \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R}) \mid \exists g \in \operatorname{Dep}(\mathbb{R}^{d}), g \cdot \boldsymbol{x} = \boldsymbol{y} \}$.

• [(a)] Show that if $c(\boldsymbol{x}) \cap c(\boldsymbol{y}) \neq \emptyset$ then $c(\boldsymbol{x}) = c(\boldsymbol{y})$.
• [(b)] Show that if $c(\boldsymbol{x}) = c(\boldsymbol{y})$ then $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that if $A \in \mathscr{M}_n(\mathbb{R})$ then $\varphi_A$ has real coefficients (that is, $\varphi_A \in \mathbb{R}[X]$).

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and we introduce for all $(\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$ $$J(\tau, R) = \sum_{i=1}^{n} |\boldsymbol{y}_{i} - (R\boldsymbol{x}_{i} + \tau)|^{2} = \|\boldsymbol{y} - g \cdot \boldsymbol{x}\|^{2}$$ where $g = (\tau, R)$. We denote $\overline{\boldsymbol{x}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{x}_{i}$ and $\overline{\boldsymbol{y}} = \frac{1}{n} \sum_{i=1}^{n} \boldsymbol{y}_{i}$.

• [(a)] Show that $J(\tau, R) = \left(\sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}} - R(\boldsymbol{x}_{i} - \overline{\boldsymbol{x}})|^{2}\right) + n|\overline{\boldsymbol{y}} - R\overline{\boldsymbol{x}} - \tau|^{2}$.
• [(b)] Deduce that for all $R \in \mathrm{SO}_{d}(\mathbb{R})$, the map $\tau \mapsto J(\tau, R)$ from $\mathbb{R}^{d}$ to $\mathbb{R}$ has a unique minimum, denoted $\tau(R)$, which we will express explicitly.

We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. Thus we have $$\varphi_A(X) = (X - \lambda_1)^{m_1} \cdots (X - \lambda_\ell)^{m_\ell}$$ with $m = m_1 + \cdots + m_\ell$.
Show that the map $$T : P \in \mathbb{C}_{m-1}[X] \mapsto \left(P(\lambda_1), P'(\lambda_1), \cdots, P^{(m_1-1)}(\lambda_1), \cdots, P(\lambda_\ell), P'(\lambda_\ell), \cdots, P^{(m_\ell-1)}(\lambda_\ell)\right) \in \mathbb{C}^m$$ is an isomorphism and deduce that there exists a unique polynomial $Q \in \mathbb{C}_{m-1}[X]$ such that $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$

We equip $\mathscr{M}_{d}(\mathbb{R})$ with the topology associated with the norm $\|M\| = \sqrt{\langle M, M \rangle}$.

• [(a)] Show that the map $f : \mathscr{M}_{d}(\mathbb{R}) \rightarrow \mathscr{M}_{d}(\mathbb{R})$ defined by $f(M) = M^{T}M$ is continuous.
• [(b)] Show that $\mathrm{SO}_{d}(\mathbb{R})$ is a closed bounded subset of $\mathscr{M}_{d}(\mathbb{R})$.

We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$.
Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and $J(\tau, R) = \sum_{i=1}^{n} |\boldsymbol{y}_{i} - (R\boldsymbol{x}_{i} + \tau)|^{2}$. For all $R \in \mathrm{SO}_{d}(\mathbb{R})$, $\tau(R)$ denotes the unique minimizer of $\tau \mapsto J(\tau, R)$.

• [(a)] Show that there exists $R_{*} \in \mathrm{SO}_{d}(\mathbb{R})$ such that $J(\tau(R_{*}), R_{*}) \leqslant J(\tau, R)$ for all $(\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$.
• [(b)] Show that $R_{*}$ is not necessarily unique.

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$. Show that if $V_{n}(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{x}_{i} - \overline{\boldsymbol{x}}|^{2}$ and $V_{n}(\boldsymbol{y}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}}|^{2}$ then $$\delta(\boldsymbol{x}, \boldsymbol{y})^{2} = nV_{n}(\boldsymbol{x}) + nV_{n}(\boldsymbol{y}) - 2\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z(\boldsymbol{x}, \boldsymbol{y}), R \rangle$$ where $Z(\boldsymbol{x}, \boldsymbol{y})$ is a matrix that we will specify.

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
We now assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Establish the result at rank $n$ if $n$ is even.

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
We now assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Let $n = 2m+1$ with $m \in \mathbb{N}$. Justify that $\prod_{\substack{m+1 < p \leqslant 2m+1 \\ p \text{ prime}}} p$ divides $\binom{2m+1}{m}$ and show that $\binom{2m+1}{m} \leqslant 4^{m}$.

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
Conclude.

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
We assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Establish the result at rank $n$ if $n$ is even.

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
We assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Let $n = 2m+1$ with $m \in \mathbb{N}$. Justify that $\prod_{\substack{m+1 < p < 2m+1 \\ p \text{ prime}}} p$ divides $\binom{2m+1}{m}$ and show that $\binom{2m+1}{m} \leqslant 4^m$.