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
2023 x-ens-maths-a__mp

24 maths questions

Q1 Matrices Algebra and Subalgebra Proofs View
a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$. b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible. c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.
Q2 Groups Subgroup and Normal Subgroup Properties View
a) Show that $N(ZZ') = N(Z)N(Z')$ for all $Z, Z' \in \mathbb{H}$. b) Show that $S$ is a subgroup of $\mathbb{H}^\times$ and that $\frac{1}{\sqrt{N(Z)}}Z \in S$ for all $Z \in \mathbb{H}^\times$.
Q3 Complex Numbers Arithmetic Algebraic Structure and Abstract Properties of Complex Numbers View
a) Show that for all $x, y, z, t \in \mathbb{R}$ we have $$N(xE + yI + zJ + tK) = x^2 + y^2 + z^2 + t^2.$$ b) Show that for all $U \in \mathbb{H}^{\mathrm{im}}$ we have $U^2 = -N(U)E$ and that $$\mathbb{H}^{\mathrm{im}} = \left\{ U \in \mathbb{H} \mid U^2 \in \left]-\infty, 0\right] E \right\}.$$
Q4 Proof Proof of Set Membership, Containment, or Structural Property View
Show that $S$ is a closed and path-connected subset of $\mathbb{H}$.
Q5 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $U, V \in \mathbb{H}^{\mathrm{im}}$. a) Show that $U$ and $V$ are orthogonal if and only if $UV + VU = 0$. In this case show that $UV \in \mathbb{H}^{\mathrm{im}}$ and that the determinant of the family $(U, V, UV)$ in the basis $(I, J, K)$ of $\mathbb{H}^{\mathrm{im}}$ is non-negative. b) Show that if $(U, V)$ is an orthonormal family in $\mathbb{H}^{\mathrm{im}}$, then $(U, V, UV)$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$.
Q6 Groups Group Homomorphisms and Isomorphisms View
Show that $\alpha$ is a group morphism and describe its kernel, where $$\begin{aligned} \alpha : S \times S & \longrightarrow \mathrm{GL}(\mathbb{H}) \\ (u, v) & \longmapsto (Z \mapsto uZv^{-1}) \end{aligned}$$
Q7 Groups Group Homomorphisms and Isomorphisms View
Show that $\alpha$ is continuous and that the image of $\alpha$ is contained in $\mathrm{SO}(\mathbb{H})$. One may begin by showing that $\alpha(u,v) \in \mathrm{O}(\mathbb{H})$ for $(u,v) \in S \times S$.
Q8 Groups Group Homomorphisms and Isomorphisms View
Let $\theta \in \mathbb{R}$ and $v \in \mathbb{H}^{\mathrm{im}} \cap S$, and let $u = (\cos\theta)E + (\sin\theta)v$. a) Show that $u \in S$ and that $u^{-1} = (\cos\theta)E - (\sin\theta)v$. b) Let $w \in \mathbb{H}^{\mathrm{im}} \cap S$ be a vector orthogonal to $v$. Describe the matrix of $C_u$ in the direct orthonormal basis $(v, w, vw)$ of $\mathbb{H}^{\mathrm{im}}$.
Q9 Groups Group Homomorphisms and Isomorphisms View
Show that the map $u \mapsto C_u$ induces a surjective group morphism $S \rightarrow \mathrm{SO}(\mathbb{H}^{\mathrm{im}})$ and describe its kernel.
Q10 Groups Subgroup and Normal Subgroup Properties View
a) Deduce that $\alpha(S \times S) = \mathrm{SO}(\mathbb{H})$. b) Show that $N := \alpha(S \times \{E\})$ is a subgroup of $\mathrm{SO}(\mathbb{H})$, then that $gng^{-1} \in N$ for all $n \in N$ and $g \in \mathrm{SO}(\mathbb{H})$ and that $\{\pm\mathrm{id}\} \subsetneq N \subsetneq \mathrm{SO}(\mathbb{H})$.
Q11 Groups Algebra and Subalgebra Proofs View
Show that $\mathrm{Aut}(\mathbb{H})$ is a subgroup of $\mathrm{GL}(\mathbb{H})$, containing $\alpha(u,u)$ for all $u \in S$.
Q12 Groups Group Homomorphisms and Isomorphisms View
Show that $(f(I), f(J), f(K))$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$ for all $f \in \mathrm{Aut}(\mathbb{H})$.
Q13 Groups Automorphism and Endomorphism Structure View
a) Show that the restriction map to $\mathbb{H}^{\mathrm{im}}$ induces a group isomorphism $$\mathrm{Aut}(\mathbb{H}) \simeq \mathrm{SO}(\mathbb{H}^{\mathrm{im}}).$$ b) Show that $$\mathrm{Aut}(\mathbb{H}) = \{\alpha(u,u) \mid u \in S\}.$$
Q14 Matrices Existence or properties of extrema via abstract/theoretical argument View
We denote $\|\cdot\|_2$ the canonical Euclidean norm on $\mathbb{R}^2$ and we denote $$\mathcal{C} := \left\{ x \in \mathbb{R}^2 \mid \|x\|_2 = 1 \right\}$$ We fix an arbitrary norm $\|\cdot\|$ on $\mathbb{R}^2$ and we denote $$\mathcal{K} = \left\{ A \in M_2(\mathbb{R}) \mid \forall x \in \mathbb{R}^2,\ \|x\|_2 \geq \|Ax\| \right\}.$$ a) Show that $\mathcal{K}$ is a compact and convex subset of $M_2(\mathbb{R})$. b) Show that there exists $A \in \mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$.
Q15 Matrices Existence or properties of extrema via abstract/theoretical argument View
We fix an element $A$ of $\mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$. Show that $\det A > 0$ and that there exists $x \in \mathcal{C}$ such that $\|Ax\| = 1$.
Q17 Matrices Existence Proof View
Using the above, show that there exists a basis $(e_1, e_2)$ of $\mathbb{R}^2$ such that $\|Ax\| = \|x\|_2$ for $x \in \{e_1, e_2\}$.
Q18 Matrices Direct Proof of a Stated Identity or Equality View
Let $T$ be a closed subset of $\mathcal{C}$, such that there exist $x, y \in T$ with $y \notin \{-x, x\}$. We assume that for all $a, b \in T$ with $b \notin \{-a, a\}$, we have that $\dfrac{b-a}{\|b-a\|_2}$ and $\dfrac{b+a}{\|b+a\|_2}$ belong to $T$. Show that $T = \mathcal{C}$.
Q19 Matrices Proof of Equivalence or Logical Relationship Between Conditions View
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 $\mathbb{R}^2$.
Q20 Matrices Ring and Field Structure View
Let $A$ be an algebraic $\mathbb{R}$-algebra without zero divisors. a) Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. b) Show that if $x \in A \setminus \mathbb{R}$, then $\mathbb{R} + \mathbb{R}x$ is an $\mathbb{R}$-algebra isomorphic to $\mathbb{C}$.
Q21 Matrices Ring and Field Structure View
We assume that $A$ is not isomorphic to one of the algebras $\mathbb{R}$ or $\mathbb{C}$. Show that there exists $i_A \in A$ such that $i_A^2 = -1$.
Q22 Matrices Algebra and Subalgebra Proofs View
We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map $$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$ We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$. a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$. b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.
Q23 Matrices Algebra and Subalgebra Proofs View
Show that $\ker(T + \mathrm{id}) = U$ and deduce that $\ker(T - \mathrm{id}) \neq \{0\}$.
Q24 Matrices Algebra and Subalgebra Proofs View
We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$. a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$. b) Show that $\beta^2 \in ]-\infty, 0[$. c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.
Q26 Matrices Algebra and Subalgebra Proofs View
Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.