grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2014 x-ens-maths__psi

24 maths questions

Q1a Groups Group Homomorphisms and Isomorphisms View
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$.
Q1b Groups Group Homomorphisms and Isomorphisms View
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.
Q1c Groups Group Homomorphisms and Isomorphisms View
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Determine $\operatorname{Ker}(T)$ and $\operatorname{Im}(T)$.
Q1d Groups Group Homomorphisms and Isomorphisms View
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$ with this norm.
Q1e Groups Group Homomorphisms and Isomorphisms View
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_2 \leq M\|f\|_2$. For this, you may consider the family $(f_n)_{n \geq 2}$ of elements of $E$ such that: (i) $f_n$ is piecewise affine, (ii) $f_n(0) = f_n\left(\frac{1}{2} - \frac{1}{n}\right) = f_n\left(\frac{1}{2} + \frac{1}{n^2}\right) = f_n(1) = 0$ and $f_n\left(\frac{1}{2}\right) = 1$.
Q2a Matrices Matrix Norm, Convergence, and Inequality View
Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Show that $S$ and $V$ belong to $\mathcal{L}(H)$.
Q2b Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Calculate the point spectrum of $S$ and $V$.
Q2c Matrices Matrix Norm, Convergence, and Inequality View
We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Show that $S$ and $V$ belong to $\mathcal{L}(F)$.
Q2d Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the point spectrum of $S$ and $V$ in $F$.
Q2e Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the spectrum of $S$ and $V$ in $F$.
Q3a Matrices Linear Transformation and Endomorphism Properties View
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T \in \mathcal{L}(E)$.
Q3b Differential equations Integral Equations Reducible to DEs View
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Let $f \in E$. By decomposing $T(f)$ into two integrals, show that $T(f)$ is a $C^2$ function and express $(T(f))'$ then $(T(f))''$.
Q3c Matrices Linear Transformation and Endomorphism Properties View
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T$ is injective.
Q3d Second order differential equations Second-order ODE with initial or boundary value conditions View
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that if $\lambda \in \sigma_p(T)$ and $f \in \operatorname{Ker}(T - \lambda Id)$, then $f \in C^2([0,1], \mathbb{R})$ and satisfies the equation $$\lambda f'' + f = 0$$ with the conditions $f(0) = f(1) = 0$.
Q3e Second order differential equations Second-order ODE with initial or boundary value conditions View
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Deduce $\sigma_p(T)$. Calculate the eigenspaces $E_{\lambda} = \operatorname{Ker}(T - \lambda Id)$ associated with each element $\lambda \in \sigma_p(T)$.
Q4a Matrices Matrix Norm, Convergence, and Inequality View
We recall that a pre-Hilbert space $H$ is a normed vector space whose norm is derived from an inner product denoted $\langle .,. \rangle$. We call Hilbert basis of $H$ any family $B = (b_i)_{i \in \mathbb{N}}$ such that: (i) the family is orthonormal: for all $i$ and $j$ in $\mathbb{N}$, $\langle b_i, b_j \rangle = 1$ if $i = j$ and $0$ otherwise. (ii) every element $x$ of $H$ can be written: $x = \sum_{i=0}^{+\infty} \langle x, b_i \rangle b_i$, that is $$\lim_{N \rightarrow +\infty} \left\| x - \sum_{i=0}^{N} \langle x, b_i \rangle b_i \right\| = 0$$
Show that if $B = (b_i)_{i \in \mathbb{N}}$ is a Hilbert basis of $H$, then $$\forall x \in H, \quad \|x\|^2 = \sum_{i=0}^{+\infty} |\langle x, b_i \rangle|^2$$
Q4b Matrices Matrix Norm, Convergence, and Inequality View
Show that $H = l^2(\mathbb{N})$ equipped with the norm $\|.\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$ is a pre-Hilbert space for the inner product: $$\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$$ (justify that this is indeed an inner product) then determine a Hilbert basis of $H$.
Q4c Matrices Matrix Norm, Convergence, and Inequality View
In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $T$ be an operator on $H$. We admit the existence of an operator $\tilde{T} \in \mathcal{L}(H)$ such that $$\forall (x,y) \in H^2, \quad \langle T(x), y \rangle = \langle x, \tilde{T}(y) \rangle$$ Let $B = (b_i)_{i \in \mathbb{N}}$ and $C = (c_i)_{i \in \mathbb{N}}$ be two Hilbert bases of $H$ such that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty$$ Show that $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2 = \sum_{i=0}^{+\infty} \|\tilde{T}(c_i)\|^2$$
Q4d Matrices Matrix Norm, Convergence, and Inequality View
In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $B = (b_i)_{i \in \mathbb{N}}$ be a Hilbert basis of $H$ and $T \in \mathcal{L}(H)$. Show that the quantity (possibly infinite) $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ does not depend on the basis $B$. We denote $$\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ and we set $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$$
Q4e Matrices Matrix Norm, Convergence, and Inequality View
In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, with $S$ the left shift $(Su)_n = u_{n-1}$ if $n \geq 1$, $(Su)_0 = 0$, and $V$ the right shift $(Vu)_n = u_{n+1}$, and $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2 < +\infty\right\}$$ Show that the operators $S$ and $V$ defined in part 2 are not in $\mathcal{L}^2(H)$. Give an example of a non-zero operator in $\mathcal{L}^2(H)$.
Q4f Matrices Matrix Norm, Convergence, and Inequality View
With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
Show that $\mathcal{L}^2(H)$ equipped with $\|.\|_2$ has the structure of a normed vector space.
Q4g Matrices Matrix Norm, Convergence, and Inequality View
With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
Let $L$ and $U$ be in $\mathcal{L}^2(H)$ and $B = (b_i)_{i \in \mathbb{N}}$ a Hilbert basis of $H$. Show that the quantity $$\sum_{i=0}^{+\infty} \langle L(b_i), U(b_i) \rangle$$ is finite, independent of the basis $B$ chosen, and defines an inner product on $\mathcal{L}^2(H)$.
Q4h Matrices Matrix Norm, Convergence, and Inequality View
With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. Show that if $L \in \mathcal{L}^2(H)$, then so is $UL$.
Q4i Matrices Matrix Norm, Convergence, and Inequality View
With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. What happens for $UL$ assuming this time that $U \in \mathcal{L}^2(H)$?