Translate the statement "For every real number $x$, there exists a real number $y$ such that $x ^ { 3 } + 3 y - 2 = 0$.

``$\mathrm { x } = 1$'' is ``$\mathrm { x } ^ { 2 } - 2 x + 1 = 0$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

Let $\alpha , \beta$ be two planes. Then a necessary and sufficient condition for $\alpha \parallel \beta$ is
A. There are infinitely many lines in $\alpha$ that are parallel to $\beta$
B. There are two intersecting lines in $\alpha$ that are parallel to $\beta$
C. $\alpha$ and $\beta$ are both parallel to the same line
D. $\alpha$ and $\beta$ are both perpendicular to the same plane

Let the set $M = \{ ( i , j , s , t ) \mid i \in \{ 1,2 \} , j \in \{ 3,4 \} , s \in \{ 5,6 \} , t \in \{ 7,8 \} \}$. For a given finite sequence $A$ and sequence $\Omega : \omega _ { 1 } , \omega _ { 2 } , \cdots , \omega _ { k } , \omega _ { k } = \left( i _ { k } , j _ { k } , s _ { k } , t _ { k } \right) \in M$, define transformation $T$: add 1 to columns $i _ { 1 } , j _ { 1 } , s _ { 1 } , t _ { 1 }$ of sequence $A$ to obtain sequence $T _ { 1 } ( A )$; add 1 to columns $i _ { 2 } , j _ { 2 } , s _ { 2 } , t _ { 2 }$ of sequence $T _ { 1 } ( A )$ to obtain sequence $T _ { 2 } T _ { 1 } ( A )$; repeat the above operations to obtain sequence $T _ { k } \cdots T _ { 2 } T _ { 1 } ( A )$, denoted as $\Omega ( A )$.
(3) If $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 }$ is even, prove that ``$\Omega ( A )$ is a constant sequence'' is a necessary and sufficient condition for ``$a _ { 1 } + a _ { 2 } = a _ { 3 } + a _ { 4 } = a _ { 5 } + a _ { 6 } = a _ { 7 } + a _ { 8 }$''.

We say that $\varphi$ is non-degenerate if and only if $E^{\perp\varphi} = \{0\}$.
Show that $\varphi$ is non-degenerate if and only if $h$ is an isomorphism.

Let $q$ be a quadratic form on $E$. Let $E'$ be a second $\mathbb{K}$-vector space of dimension $n$, and let $q'$ be a quadratic form on $E'$.
We call an isometry from $(E,q)$ to $(E',q')$ any isomorphism $f$ from $E$ to $E'$ satisfying: for all $x \in E$, $q'(f(x)) = q(x)$. We will say that $(E,q)$ and $(E',q')$ are isometric if and only if there exists an isometry from $(E,q)$ to $(E',q')$.
Show that $(E,q)$ and $(E',q')$ are isometric if and only if there exists a basis $e$ of $E$ and a basis $e'$ of $E'$ such that $\operatorname{mat}(q,e) = \operatorname{mat}(q',e')$.

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.
Let $F$ be a vector subspace of $E$. We denote by $\varphi_F$ the restriction of $\varphi$ to $F^2$. We will say that $F$ is singular if and only if $\varphi_F$ is degenerate.
Show that $F$ is non-singular if and only if one of the following properties is verified:

• $F \cap F^\perp = \{0\}$;
• $E = F \oplus F^\perp$;
• $F^\perp$ is non-singular.

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 $n = 2p$. Show that $(E,q)$ is an Artin space if and only if there exists a vector subspace $F$ of $E$ of dimension $p$ such that $q_{/F} = 0$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that $P \in \mathbb{R}_n[X]$ is integer-valued on the integers if and only if its coordinates in the basis $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ are integers.

Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $m = 2n+1$ if and only if the $x_i$ are the roots of $p_{n+1}$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathscr{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathcal{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

Let $E$ be a non-empty subset of $\mathbb{R}^d$. Using the definitions of $E^+$ and $E^{++}$ from question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. With $E^+$ and $E^{++}$ as defined in question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geq 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geq 0, i = 1, \ldots, k$.

Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. The standard symplectic form is $b_s(x,y) = \langle x, j(y) \rangle$ where $j$ is canonically associated with $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $R$ is $B ^ { 2 m } ( R )$ and the symplectic cylinder of radius $R'$ is $Z ^ { 2 m } ( R' ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant R'^{ 2 } \right\}$. Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.

Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.

Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.

Let $P$ be a polynomial function not identically zero with real coefficients. Show that the restriction of $P$ to $\mathbb { R } _ { + } ^ { * }$ belongs to $E$ if and only if $P ( 0 ) = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Let $a$ and $b$ be two real numbers. Show that the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & a \mathrm { e } ^ { t } + b \end{array} \right.$ belongs to $E$ if and only if $a = b = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.
Show that the following three assertions are equivalent:

1. [(1)] $T$ is invertible;
2. [(2)] $T1 \neq 0$;
3. [(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$.

Prove Theorem A: Let $\|\cdot\|$ be a norm on the $\mathbb{R}$-vector space $\mathbb{R}^2$. If $$\|x+y\|^2 + \|x-y\|^2 \geq 4$$ for all $x, y \in \mathbb{R}^2$ satisfying $\|x\| = \|y\| = 1$, then $\|\cdot\|$ comes from an inner product on $\mathbb{R}^2$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.