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-maths1__pc

15 maths questions

Q1 Proof Direct Proof of an Inequality View

Let $\lambda$ be a real number in the interval $]0,1[$, and let $a$ and $b$ be two non-negative real numbers. Show that $$\lambda a + (1-\lambda) b \geq a^{\lambda} b^{1-\lambda}$$ (one may introduce a certain auxiliary function and justify its concavity). Moreover, show that for all real $u > 1$, $$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

Q2 Proof Direct Proof of an Inequality View

Let $a$ and $b$ be two non-negative real numbers and $\lambda$ a real number in $]0,1[$. Show that $$(a+b)^{\lambda} \leq a^{\lambda} + b^{\lambda}$$

Q3 Proof Change of Variable and Integral Evaluation View

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
We denote $F = \int_{-\infty}^{+\infty} f(x)\,dx$ and $G = \int_{-\infty}^{+\infty} g(x)\,dx$. Show that for all $t$ in the interval $]0,1[$ there exists a unique real number denoted $u(t)$ and a unique real number denoted $v(t)$ such that $$\frac{1}{F} \int_{-\infty}^{u(t)} f(x)\,dx = t, \quad \frac{1}{G} \int_{-\infty}^{v(t)} g(x)\,dx = t$$ (One may study the variations of the function: $u \mapsto \frac{1}{F} \int_{-\infty}^{u} f(x)\,dx$).

Q4 Proof Compute derivative of transcendental function View

Q5 Proof Substitution within a Multi-Part Proof or Derivation View

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the image set of the application $w$ defined on $]0,1[$ by $$\forall t \in ]0,1[, \quad w(t) = \lambda u(t) + (1-\lambda) v(t),$$ is equal to $\mathbb{R}$. Then prove that $w$ defines a change of variable from $]0,1[$ to $\mathbb{R}$. Using this and $\int_{-\infty}^{+\infty} h(w)\,dw$, show that $f$, $g$ and $h$ satisfy the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}.$$

Q6 Proof Direct Proof of an Inequality View

We set $\Psi(u) = \exp(-u^{2})$ for all real $u$. Prove that for all $x, y \in \mathbb{R}$, $$\Psi(\lambda x + (1-\lambda) y) \geq \Psi(x)^{\lambda} \Psi(y)^{1-\lambda}$$

Q7 Proof Bounding or Estimation Proof View

Q8 Proof Bounding or Estimation Proof View

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $\epsilon \in ]0,1[$, $f_{\epsilon} = f + \epsilon\Psi$ and $g_{\epsilon} = g + \epsilon\Psi$. Show that $$\forall x, y \in \mathbb{R}, \quad f_{\epsilon}(x)^{\lambda} g_{\epsilon}(y)^{1-\lambda} \leq h(z) + \epsilon^{\Lambda}\left(\|f\|_{\infty}^{\lambda} + \|g\|_{\infty}^{1-\lambda}\right)\left(\Psi_{M}(z)\right)^{\Lambda} + \epsilon\Psi(z)$$ where $z = \lambda x + (1-\lambda) y$. One should begin by applying the inequality from question 2, then the two preceding questions. (We recall that $f(x) = 0$ if $|x| > M$ and that $g(y) = 0$ if $|y| > M$).

Q9 Proof Deduction or Consequence from Prior Results View

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. Deduce that if $f$ and $g$ are zero outside a bounded interval then the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied.

Q10 Proof Direct Proof of an Inequality View

Let $n \in \mathbb{N}^{*}$. We denote by $\chi_{n} : \mathbb{R} \rightarrow \mathbb{R}$ the continuous function that equals 1 on $[-n, n]$, equals 0 on $]-\infty, -n-1] \cup [n+1, +\infty[$ and is affine on each of the two intervals $[-n-1, -n]$ and $[n, n+1]$.
Show that: $$\forall x, y \in \mathbb{R},\quad \chi_{n}(x)^{\lambda} \chi_{n}(y)^{1-\lambda} \leq \chi_{n+1}(\lambda x + (1-\lambda) y)$$

Q11 Proof Deduction or Consequence from Prior Results View

Show that the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied (if you choose to use the dominated convergence theorem then carefully verify that its conditions of validity are satisfied).

Q12 Proof Proof That a Map Has a Specific Property View

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by $$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$ is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

Q13 Proof Deduction or Consequence from Prior Results View

Let $\lambda \in ]0,1[$ and $f, g, h$ be functions from $\mathbb{R}^{2}$ to $\mathbb{R}_{+}$ that are continuous with bounded support and such that $$\forall X \in \mathbb{R}^{2}, \forall Y \in \mathbb{R}^{2}, \quad h(\lambda X + (1-\lambda) Y) \geq f(X)^{\lambda} g(Y)^{1-\lambda}$$ Show that $$\iint_{\mathbb{R}^{2}} h(x,y)\,dx\,dy \geq \left(\iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy\right)^{\lambda} \left(\iint_{\mathbb{R}^{2}} g(x,y)\,dx\,dy\right)^{1-\lambda}.$$

Q15 Proof Definite Integral Evaluation (Computational) View

We consider a rectangle $]a,b[ \times ]c,d[$ of the plane $\mathbb{R}^{2}$, with $a < b$ and $c < d$. Calculate the real number $V(]a,b[ \times ]c,d[)$. What does it represent? (One may use functions of the type $$(x,y) \mapsto f(x,y) = \phi(x)\varphi(y)$$ where $\phi$ and $\varphi$ are well-chosen continuous and piecewise affine functions).

Q16 Proof Integral Inequalities and Limit of Integral Sequences View

Let $\mathcal{A}$ and $\mathcal{B}$ be two open bounded non-empty subsets of $\mathbb{R}^{2}$ and $\lambda \in ]0,1[$. Verify that $\lambda\mathcal{A} + (1-\lambda)\mathcal{B}$ is an open bounded subset of $\mathbb{R}^{2}$. Then show that $$V(\lambda\mathcal{A} + (1-\lambda)\mathcal{B}) \geq V(\mathcal{A})^{\lambda} V(\mathcal{B})^{1-\lambda}$$ To prove this inequality, you will use the following admitted result. For all $f \in C(\mathcal{A})$ and $g \in C(\mathcal{B})$, the function $h$ determined by: $$\forall Z \in \mathbb{R}^{2},\quad h(Z) = \sup\left\{f(X)^{\lambda} g(Y)^{1-\lambda} \,/\, X, Y \in \mathbb{R}^{2},\, Z = \lambda X + (1-\lambda) Y\right\}$$ defines a continuous function on $\mathbb{R}^{2}$.