grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2011 centrale-maths2__mp

29 maths questions

QI.A.1 Matrices Eigenvalue constraints from matrix properties View
Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive if and only if all its eigenvalues are positive.
QI.A.2 Matrices Eigenvalue constraints from matrix properties View
Let $n \in \mathbb{N}^*$ and $A \in \mathcal{S}_n(\mathbb{R})$. Show that $A$ is positive definite if and only if all its eigenvalues are strictly positive.
QI.B.1 Matrices Eigenvalue constraints from matrix properties View
For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.
For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.
QI.B.2 Matrices Determinant and Rank Computation View
For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
In the special cases $n = 1$ and $n = 2$, show directly that any matrix $A \in \mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite.
QI.B.3 Matrices Compute eigenvectors or eigenspaces View
For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
Let $n \in \mathbb{N}^*$. We assume that any matrix of $\mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite. We consider a matrix $A$ of $\mathcal{S}_{n+1}(\mathbb{R})$ satisfying property $\mathcal{P}_{n+1}$ and we assume by contradiction that $A$ is not positive definite.
a) Show then that $A$ admits two linearly independent eigenvectors associated with eigenvalues (not necessarily distinct) that are strictly negative.
b) Deduce that there exists $X \in \mathcal{M}_{n+1,1}(\mathbb{R})$ whose last component is zero and such that ${}^t X A X < 0$.
c) Conclude.
QI.C Matrices True/False or Multiple-Select Conceptual Reasoning View
For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A$ be a matrix of $\mathcal{S}_n(\mathbb{R})$. Do we have the following equivalence: $$A \text{ is positive} \quad \Longleftrightarrow \quad \forall i \in \llbracket 1; n \rrbracket, \operatorname{det}\left(A^{(i)}\right) \geqslant 0 ?$$
QII.A Proof Definite Integral Evaluation (Computational) View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Show that the map $(P, Q) \mapsto \langle P, Q \rangle$ is an inner product on $\mathbb{R}[X]$.
QII.B Chain Rule Higher-Order Derivatives of Products/Compositions View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$
We denote by $P_n^{(n)}$ the polynomial derived $n$ times of $P_n$.
Determine the degree of $P_n^{(n)}$ and calculate $P_n^{(n)}(1)$.
QII.C Indefinite & Definite Integrals Inner Product or Orthogonality Proof via Integration by Parts View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
Let $n \in \mathbb{N}^*$. Show that, for all $Q \in \mathbb{R}_{n-1}[X]$, $\langle Q, L_n \rangle = 0$.
Hint: you may integrate by parts.
QII.D Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
II.D.1) For all $n \in \mathbb{N}$, we set $I_n = \int_0^1 P_n(u) \, du$.
Calculate, for all $n \in \mathbb{N}$, the value of $I_n$.
II.D.2) Deduce for all $n \in \mathbb{N}$ the relation: $\langle L_n, L_n \rangle = \frac{1}{2n+1}$.
QII.E Proof Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions:
i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive;
ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$.
Justify the uniqueness of such a family.
QII.F Proof Definite Integral Evaluation (Computational) View
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the unique family of polynomials such that for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
Calculate $K_0$, $K_1$ and $K_2$.
QIII.A.1 Matrices Determinant and Rank Computation View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Calculate $H_2$ and $H_3$. Show that these are invertible matrices and determine their inverses.
QIII.A.2 Matrices Determinant and Rank Computation View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Show the relation: $$\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$$
Hint: you may start by subtracting the last column of $\Delta_{n+1}$ from all the others.
QIII.A.3 Matrices Determinant and Rank Computation View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Using the relation $\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$, deduce the expression of $\Delta_n$ as a function of $n$ (we will use the quantities $c_m = \prod_{i=1}^{m-1} i!$ for appropriate integers $m$).
QIII.A.4 Matrices Properties of eigenvalues under matrix operations View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Prove that $H_n$ is invertible, then that $\operatorname{det}\left(H_n^{-1}\right)$ is an integer.
QIII.A.5 Matrices Compute eigenvalues of a given matrix View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Demonstrate that $H_n$ admits $n$ real eigenvalues (counted with their multiplicity) that are strictly positive.
QIII.B.1 Matrices Definite Integral Evaluation (Computational) View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm.
Let $n \in \mathbb{N}$. Show that there exists a unique polynomial $\Pi_n \in \mathbb{R}_n[X]$ such that $$\left\|\Pi_n - f\right\| = \min_{Q \in \mathbb{R}_n[X]} \|Q - f\|$$
QIII.B.2 Matrices Uniform or Pointwise Convergence of Function Series/Sequences View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm. For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Show that the sequence $\left(\left\|\Pi_n - f\right\|\right)_{n \in \mathbb{N}}$ is decreasing and converges to 0.
QIII.B.3 Matrices Structured Matrix Characterization View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$
Show that $H_n$ is the matrix of the inner product $\langle \cdot, \cdot \rangle$, restricted to $\mathbb{R}_{n-1}[X]$, in the canonical basis of $\mathbb{R}_{n-1}[X]$.
QIII.B.4 Matrices Linear System and Inverse Existence View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Calculate the coefficients of $\Pi_n$ using the matrix $H_{n+1}^{-1}$ and the reals $\langle f, X^i \rangle$.
QIII.B.5 Matrices Definite Integral Evaluation (Computational) View
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Determine explicitly $\Pi_2$ when $f$ is the function defined for all $t \in [0,1]$ by $f(t) = \frac{1}{1+t^2}$.
QIV.A.1 Matrices Matrix Entry and Coefficient Identities View
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Calculate $s_1$, $s_2$ and $s_3$. Conjecture in general the value of $s_n$ as a function of $n$.
QIV.A.2 Matrices Linear System and Inverse Existence View
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Let $n \in \mathbb{N}^*$.
a) Show that there exists a unique $n$-tuple of real numbers $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ satisfying the following system of $n$ linear equations in $n$ unknowns: $$\left\{\begin{array}{ccccccc} a_0^{(n)} + & \frac{a_1^{(n)}}{2} + \cdots + \frac{a_{n-1}^{(n)}}{n} = & 1 \\ \frac{a_0^{(n)}}{2} + \frac{a_1^{(n)}}{3} + \cdots + \frac{a_{n-1}^{(n)}}{n+1} = & 1 \\ \vdots & \vdots & & \vdots \\ \frac{a_0^{(n)}}{n} + \frac{a_1^{(n)}}{n+1} + \cdots + \frac{a_{n-1}^{(n)}}{2n-1} = & 1 \end{array}\right.$$
b) Show that $s_n = \sum_{p=0}^{n-1} a_p^{(n)}$.
QIV.A.3 Matrices Definite Integral Evaluation (Computational) View
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$, where $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ is the unique solution of the system in IV.A.2.
Show that $$\forall Q = \alpha_0 + \alpha_1 X + \cdots + \alpha_{n-1} X^{n-1} \in \mathbb{R}_{n-1}[X], \quad \langle S_n, Q \rangle = \sum_{p=0}^{n-1} \alpha_p$$
QIV.A.5 Proof Evaluation of a Finite or Infinite Sum View
The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the unique family of polynomials such that for all $p \in \mathbb{N}$, the degree of $K_p$ equals $p$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_p\right)_{0 \leqslant p \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
For all $p \in \llbracket 0; n-1 \rrbracket$, calculate $K_p(1)$.
QIV.B.1 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View
For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
Let $p \in \mathbb{N}^*$. Show that $\binom{2p}{p}$ is an even integer.
Deduce that, if $n \in \mathbb{N}^*$ and $p \in \llbracket 1; n \rrbracket$, then $\binom{n+p}{p}\binom{n}{p}$ is an even integer.
QIV.B.2 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View
For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
For all $n \in \mathbb{N}$, show that we can write: $$K_n = \sqrt{2n+1} \, \Lambda_n$$ where $\Lambda_n$ is a polynomial with integer coefficients that we will make explicit.
Among the coefficients of $\Lambda_n$, which ones are even?
QIV.B.3 Matrices Matrix Entry and Coefficient Identities View
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$. For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E, and $K_n = \sqrt{2n+1}\,\Lambda_n$ where $\Lambda_n$ is a polynomial with integer coefficients.
Let $n \in \mathbb{N}^*$.
a) Calculate $h_{i,i}^{(-1,n)}$ for all $i \in \llbracket 1; n \rrbracket$; we will give in particular a very simple expression of $h_{1,1}^{(-1,n)}$ and $h_{n,n}^{(-1,n)}$ as a function of $n$.
b) Calculate $h_{i,j}^{(-1,n)}$ for all pairs $(i,j) \in \llbracket 1; n \rrbracket^2$; deduce that the coefficients of $H_n^{-1}$ are integers.
c) Show that $h_{i,j}^{(-1,n)}$ is divisible by 4 for all pairs $(i,j) \in \llbracket 2; n \rrbracket^2$.