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
2024 x-ens-maths__pc

20 maths questions

Q1 Proof Determinant and Rank Computation View
Let $R \in \mathrm{O}_{d}(\mathbb{R})$. Verify that $\operatorname{det}(R) \in \{-1, +1\}$.
Q2 Proof Matrix Norm, Convergence, and Inequality View
Verify that $(A, B) \mapsto \langle A, B \rangle$ is an inner product on the vector space $\mathscr{M}_{d}(\mathbb{R})$. We denote by $\|A\| = \sqrt{\langle A, A \rangle}$ the associated norm.
Q3 Proof Matrix Algebra and Product Properties View
  • [(a)] Show that for all $u, v \in \mathbb{R}^{d}$ and $A \in \mathscr{M}_{d}(\mathbb{R})$, we have $\langle u, Av \rangle_{\mathbb{R}^{d}} = \langle uv^{T}, A \rangle$.
  • [(b)] Show that $\operatorname{tr}(AB) = \operatorname{tr}(BA)$ for $A, B \in \mathscr{M}_{d}(\mathbb{R})$.
  • [(c)] Deduce that for all $A, B$ and $C$ in $\mathscr{M}_{d}(\mathbb{R})$ we have $$\langle A, BC \rangle = \langle B^{T} A, C \rangle = \langle AC^{T}, B \rangle.$$
Q4 Proof Matrix Norm, Convergence, and Inequality View
Let $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ be a diagonal matrix with positive coefficients and let $R \in \mathrm{O}_{d}(\mathbb{R})$.
  • [(a)] Show that for all $1 \leqslant i \leqslant d$, we have $|R_{ii}| \leqslant 1$ where $R_{ii}$ is the $i$-th diagonal coefficient of $R$.
  • [(b)] Deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D)$.
Q5 Proof Binary Operation Properties View
For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
  • [(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
  • [(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
  • [(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.
Q8 Proof Binary Operation Properties View
For which values of $d$ do we have $gg^{\prime} = g^{\prime}g$ for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$?
Q10 Proof Proof That a Map Has a Specific Property View
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 $\delta(\boldsymbol{x}, \boldsymbol{z}) \leqslant \delta(\boldsymbol{x}, \boldsymbol{y}) + \delta(\boldsymbol{y}, \boldsymbol{z})$.
Q11 Proof Group Actions and Surjectivity/Injectivity of Maps View
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 $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.
Q12 Proof Direct Proof of a Stated Identity or Equality View
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.
Q13 Proof Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
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 $\mathscr{M}_{d}(\mathbb{R})$.
Q14 Proof Limit Evaluation Involving Sequences View
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.
Q15 Proof Computation of a Limit, Value, or Explicit Formula View
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.
Q16 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix. We denote $\mathrm{S} = Z^{T}Z$. Show that there exists a decreasing family $(\lambda_{i})_{1 \leqslant i \leqslant d}$ of strictly positive real numbers and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Su_{i} = \lambda_{i} u_{i}$ for all $1 \leqslant i \leqslant d$.
Q17 Invariant lines and eigenvalues and vectors Matrix Decomposition and Factorization View
Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with $\mathrm{S} = Z^{T}Z$, and let $(\lambda_{i})_{1 \leqslant i \leqslant d}$ be the decreasing family of strictly positive eigenvalues of $S$ with associated orthonormal basis $(u_{1}, \ldots, u_{d})$. We consider $v_{i} = \frac{1}{\sqrt{\lambda_{i}}} Z u_{i}$ for all $1 \leqslant i \leqslant d$.
  • [(a)] Show that $(v_{1}, \ldots, v_{d})$ is an orthonormal basis of $\mathbb{R}^{d}$.
  • [(b)] Verify that if $U = (u_{1} | \ldots | u_{d})$, $V = (v_{1} | \ldots | v_{d})$ and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$ then $Z = VDU^{T}$.
Q18 Invariant lines and eigenvalues and vectors Matrix Decomposition and Factorization View
Express in the form $Z = VDU^{T}$ (specifying your choices of $U$, $V$ and $D$) for the matrices $$Z_{1} = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix} \quad \text{and} \quad Z_{2} = \begin{pmatrix} 0 & 1 \\ -2 & 0 \end{pmatrix}.$$
Q19 Invariant lines and eigenvalues and vectors Projection and Orthogonality View
Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with singular value decomposition $Z = VDU^{T}$ where $U = (u_{1}|\ldots|u_{d})$, $V = (v_{1}|\ldots|v_{d})$ are orthogonal matrices and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$. We assume that $\operatorname{det}(Z) > 0$.
  • [(a)] Show that if $R \in \mathrm{SO}_{d}(\mathbb{R})$ then $V^{T}RU \in \mathrm{SO}_{d}(\mathbb{R})$.
  • [(b)] Show that $$\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z, R \rangle = \sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle D, R \rangle$$
Q20 Invariant lines and eigenvalues and vectors Matrix Decomposition and Factorization View
Give the value of $\delta(\boldsymbol{x}, \boldsymbol{y})$ as a function of $V_{n}(\boldsymbol{x})$, $V_{n}(\boldsymbol{y})$ and the singular values of $Z(\boldsymbol{x}, \boldsymbol{y})$ in the case where $\operatorname{det}(Z(\boldsymbol{x}, \boldsymbol{y})) > 0$.
Q21 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$.
  • [(a)] Show that if $\lambda$ is an eigenvalue of $R$ then $\lambda \in \{+1, -1\}$.
  • [(b)] Show that $\operatorname{det}(R + I) = \operatorname{det}(R) \operatorname{det}(I + R^{T})$.
  • [(c)] Deduce that if $\operatorname{det}(R) = -1$ then $\operatorname{det}(R + I) = 0$.
Q22 Invariant lines and eigenvalues and vectors Projection and Orthogonality View
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$.
  • [(a)] Show that there exists an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $u_{d}^{T} Rx = 0$ for all $x \in E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$.
  • [(b)] Deduce that $R(E_{1}) \subset E_{1}$ and then that $R(E_{1}) = E_{1}$.
Q23 Invariant lines and eigenvalues and vectors Projection and Orthogonality View
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $R(E_{1}) = E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$. We consider a matrix $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d}) \in \mathscr{M}_{d}(\mathbb{R})$ diagonal with diagonal entries $\alpha_{i} \geqslant 0$ in decreasing order. We denote $U = (u_{1} | \ldots | u_{d})$.
  • [(a)] Verify that $\langle D, R \rangle = \langle S, R^{\prime} \rangle$ where $R^{\prime} = U^{T}RU$ and $S = U^{T}DU$.
  • [(b)] Show that if $R_{0} = (R_{ij}^{\prime})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$ then $R_{0} \in \mathrm{O}_{d-1}(\mathbb{R})$.