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

2022 x-ens-maths__psi

22 maths questions

Q1 Proof Existence Proof View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leqslant \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

Q3 Proof Direct Proof of an Inequality View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have $$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2$$ and deduce that $\operatorname{proj}_C$ is continuous.

Q4 Proof Computation of a Limit, Value, or Explicit Formula View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Explicitly determine $\operatorname{proj}_C$ in the following cases: $$\text{i) } C = \mathbb{R}_+^d, \quad \text{ii) } C = \left\{y \in \mathbb{R}^d : \|y\| \leqslant 1\right\}$$ $$\text{iii) } C = \left\{y \in \mathbb{R}^d : \sum_{i=1}^d y_i \leqslant 1\right\}, \quad \text{iv) } C = [-1,1]^d$$

Q6 Proof Existence Proof View

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that there exist $p \in \mathbb{R}^d$ and $\varepsilon > 0$ such that $$p \cdot x \leqslant p \cdot y - \varepsilon, \forall (x, y) \in C \times D$$ (we say that $C$ and $D$ can be strictly separated).

Q7 Proof Direct Proof of a Stated Identity or Equality View

Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by: $$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$ show that $$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$ (so that $C$ is an intersection of closed half-spaces).

Q8 Proof Existence Proof View

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \backslash A$, show that there exists $p \in \mathbb{R}^d \backslash \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A$$

Q9 Proof Proof of Set Membership, Containment, or Structural Property View

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$. Let $I \in \mathbb{N}^*, x_1, \ldots, x_I \in A^I$ and $(\lambda_1, \ldots, \lambda_I) \in \mathbb{R}_+^I$ such that $\sum_{i=1}^I \lambda_i = 1$, show that:

a) $\sum_{i=1}^I \lambda_i x_i \in A$,
b) if $x := \sum_{i=1}^I \lambda_i x_i \in \operatorname{Ext}(A)$ then $x_i = x$ for all $i \in \{1, \ldots, I\}$ such that $\lambda_i > 0$.

Q10 Proof Proof of Set Membership, Containment, or Structural Property View

Let $E$ be a subset of $\mathbb{R}^d$. Recall that $$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$ Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Q11 Proof Characterization or Determination of a Set or Class View

Let $A = \operatorname{co}(E)$ where $E$ is the subset of $\mathbb{R}^3$ defined by $$E = \{(0,0,1),(0,0,-1)\} \cup \{(1+\cos(\theta), \sin(\theta), 0), \theta \in [0, 2\pi]\}$$ show that $\operatorname{Ext}(A)$ is non-empty and is not closed.

Q12 Matrices Determinant and Rank Computation View

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

Q15 Proof Direct Proof of a Stated Identity or Equality View

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $K = \operatorname{co}(\operatorname{Ext}(K))$.

Q16 Proof Proof of Set Membership, Containment, or Structural Property View

Let $E$ be a non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by $$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geqslant 0, \forall x \in E\right\}$$ and its bi-polar cone by $$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geqslant 0, \forall p \in E^+\right\}.$$ Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.

Q17 Proof Proof of Equivalence or Logical Relationship Between Conditions View

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.

Q18 Proof Proof of Set Membership, Containment, or Structural Property View

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:

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

Q19 Proof Direct Proof of an Inequality View

Q20 Proof Existence Proof View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geqslant 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$

a) Show that $p \cdot z \geqslant 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geqslant 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Q21 Proof Direct Proof of a Stated Identity or Equality View

For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set $$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$ Show that for all $x \in \mathbb{R}^d$, we have $$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\}$$ and $$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$

Q22 Proof Proof of Set Membership, Containment, or Structural Property View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Denote by $C$ the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Show that $C$ is non-empty, convex, closed and bounded.

Q23 Proof Existence Proof View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|$$

Q24 Matrices Linear System and Inverse Existence View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ be as in question 23. Let $K$ be the set of $y \in \mathbb{R}^d$ such that $$My = b, \quad y_i = 0 \ \forall i \in I_0(\bar{x}), \quad q_i y_i \geqslant 0 \ \forall i \in \{1, \ldots, d\}$$ Show that $K$ is non-empty and included in $C$.

Q25 Matrices Linear Transformation and Endomorphism Properties View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Show that if $y \in \operatorname{Ext}(K)$ then $$h \in \operatorname{Ker}(M) \text{ and } I_0(y) \subset I_0(h) \Rightarrow h = 0.$$

Q26 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is at most $k$.