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
2013 x-ens-maths__psi

9 maths questions

Q1 Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ We may freely use the formula $T_{0}(0) = \frac{\sqrt{\pi}}{2}$.
a) Show that if $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, the integral defining $T_{m}(x)$ is convergent.
b) What is the interval $A$ of $m \in \mathbb{R}$ such that the integral defining $T_{m}(0)$ is convergent?
c) Calculate $T_{2k}(0)$ and $T_{2k+1}(0)$ for $k \in \mathbb{N}$ (in terms of $k!$ and $(2k)!$).
Q2 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in A$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}$.
b) Let $m \in \mathbb{R}$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}^{*}$.
c) Show that for $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, $$T_{m}(x) \geq e^{-1} \int_{0}^{1} t^{m} e^{-x/t} dt$$ Deduce the value of $\lim_{x \rightarrow 0^{+}} T_{m}(x)$ when $m \notin A$, using the change of variable $w = x/t$.
Q3 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in \mathbb{R}$. Show that $T_{m}$ is of class $C^{1}$ on $\mathbb{R}_{+}^{*}$, and calculate $T_{m}^{\prime}$ in terms of $T_{m-1}$.
b) Let $m \in \mathbb{R}$. Is the function $T_{m}$ of class $C^{\infty}$ on $\mathbb{R}_{+}^{*}$? What is the monotonicity of $T_{m}$ on $\mathbb{R}_{+}^{*}$? Is the function $T_{m}$ convex on $\mathbb{R}_{+}^{*}$?
c) Discuss as a function of $m \in \mathbb{R}$ the right-differentiability of $T_{m}$ at 0.
Q4 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Calculate $T_{m}(x)$ in terms of $T_{m-2}(x)$ and $T_{m-3}(x)$. For this, you may consider the quantity $$\int_{A}^{B} t^{m-1}\left(2t - x/t^{2}\right) e^{-\left(t^{2}+x/t\right)} dt$$ for $0 < A < B$.
b) Let $m \in \mathbb{R}$. Find a relation between $x T_{m}^{\prime\prime\prime}(x)$, $T_{m}^{\prime\prime}(x)$ and $T_{m}(x)$ for $x \in \mathbb{R}_{+}^{*}$.
Q5 Reduction Formulae Perform a Change of Variable or Transformation on a Parametric Integral View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Perform the change of variable $t = 1/u$ in the integral defining $T_{m}$. Justify the calculation carefully.
b) Let $n \in \mathbb{N}$. Justify the existence of the quantity $\int_{0}^{\infty} u^{n} e^{-u} du$ and calculate it.
c) Show that for $m \in \mathbb{N} - \{0,1\}$, $T_{-m}(1) \leq (m-2)!$.
d) Let $k \in \mathbb{N}$. Show that the radius of convergence $R$ of the power series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$ satisfies $R \geq 1$.
e) Let $k \in \mathbb{N}$. Show that for $x \in ]-1,1[$, $$T_{k}(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$$
Q6 Reduction Formulae Bound or Estimate a Parametric Integral View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ For $t, x \in \mathbb{R}_{+}^{*}$, we set $g(t,x) = t^{2} + x/t$.
a) Show that for $x \in \mathbb{R}_{+}^{*}$, the function $t \in \mathbb{R}_{+}^{*} \mapsto g(t,x)$ is convex and admits a unique minimum at $t = M(x)$, which we will determine. Calculate $g(M(x),x)$.
b) Let $m \in \mathbb{R}_{+}$, and $x \in \mathbb{R}_{+}^{*}$. Show using the inequality $M(x) \leq x^{1/3}$ that $$T_{m}(x) \leq \int_{0}^{x^{1/3}} t^{m} e^{-g(t,x)} dt + e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right)$$
c) Show that for every $\varepsilon > 0$ (and $m \in \mathbb{R}_{+}$), we can find $C > 0$ such that $$\forall t \geq 1, \quad t^{m} \leq C t e^{\varepsilon t^{2}}$$ Deduce that for every $\varepsilon > 0$, $$e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right) = O_{x \rightarrow \infty}\left(e^{-\left[\frac{39}{20} - \varepsilon\right] x^{2/3}}\right)$$
d) Show that $$T_{m}(x) = O_{x \rightarrow \infty}\left(x^{\frac{m+1}{3}} e^{-3(x/2)^{2/3}}\right)$$
Q7 Reduction Formulae Bound or Estimate a Parametric Integral View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Show that for $x \in \mathbb{R}_{+}^{*}$, $$T_{-1}(x) \leq \int_{0}^{1} e^{-1/u^{2}} \frac{du}{u} + \int_{1}^{\infty} e^{-xu} \frac{du}{u}$$ Deduce that $T_{-1}(x) \leq 2$ for $x \geq 1$ and that $$T_{-1}(x) \leq 2 + \int_{x}^{1} e^{-w} \frac{dw}{w} \leq 2 - \ln x$$ if $0 < x \leq 1$.
b) Let $L \in [0,1]$, and $\rho \in C([0,L])$. We set $$[F(\rho)](x) = \int_{0}^{L} \rho(y) T_{-1}(|x-y|) dy$$ Show that $[F(\rho)](x)$ is well-defined for $x \in [0,L]$ and that $$\|F(\rho)\|_{\infty} \leq (4L + 2L|\ln L|) \|\rho\|_{\infty}.$$
Q8 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
Let $L > 0$, $\rho \in C([0,L])$, and $g_{0}$ continuous and integrable on $\mathbb{R}_{+}$. We set for $x \in [0,L]$ and $v \in \mathbb{R}^{*}$: $$\begin{gathered} g(x,v) = \frac{2}{\sqrt{\pi}} \frac{e^{-v^{2}}}{|v|} \int_{x}^{L} \rho(y) e^{-\frac{x-y}{v}} dy, \quad \text{if} \quad v < 0, \\ g(x,v) = \frac{2}{\sqrt{\pi}} \frac{e^{-v^{2}}}{|v|} \int_{0}^{x} \rho(y) e^{-\frac{x-y}{v}} dy + g_{0}(v) e^{-\frac{x}{v}}, \quad \text{if} \quad v > 0. \end{gathered}$$
a) Show that $\alpha : x \in [0,L] \mapsto \int_{0}^{\infty} g_{0}(v) e^{-\frac{x}{v}} dv$ defines a function in $C([0,L])$.
b) Show that for $v \in \mathbb{R}^{*}$, the function $x \in [0,L] \mapsto g(x,v)$ is of class $C^{1}$ on $[0,L]$ and $$\begin{aligned} & \forall x \in [0,L], v \in \mathbb{R}^{*}, \quad v \frac{\partial g}{\partial x}(x,v) = \rho(x) \frac{2}{\sqrt{\pi}} e^{-v^{2}} - g(x,v) \\ & \forall v \in \mathbb{R}_{+}^{*}, \quad g(0,v) = g_{0}(v), \quad \forall v \in \mathbb{R}_{-}^{*}, \quad g(L,v) = 0 \end{aligned}$$
Q9 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
We admit in this question the following theorem: Let $L > 0$. If $H$ is a mapping from $C([0,L])$ to $C([0,L])$ satisfying $$\exists k \in [0,1[, \quad \forall \rho_{1}, \rho_{2} \in C([0,L]), \quad \|H(\rho_{1}) - H(\rho_{2})\|_{\infty} \leq k \|\rho_{1} - \rho_{2}\|_{\infty}$$ then there exists a unique $\rho \in C([0,L])$ such that $H(\rho) = \rho$.
a) Let $L > 0$ and $\tilde{\alpha} \in C([0,L])$. Show that if $L \in ]0, 1/20[$, there exists a unique function $\tilde{\rho} \in C([0,L])$ such that $$\forall x \in [0,L], \quad \tilde{\rho}(x) = \tilde{\alpha}(x) + \frac{2}{\sqrt{\pi}} \int_{0}^{L} \tilde{\rho}(y) T_{-1}(|x-y|) dy$$
b) Let $L \in ]0, 1/20[$, and $g_{0}$ continuous and integrable on $\mathbb{R}_{+}$. Show that there exists a function $\tilde{g}$ from $[0,L] \times \mathbb{R}$ to $\mathbb{R}$ such that
  • $\forall v \in \mathbb{R}^{*}$, $\tilde{g}(\cdot, v)$ is of class $C^{1}$ on $[0,L]$
  • $\forall x \in [0,L]$, $\tilde{g}(x, \cdot)$ is integrable on $\mathbb{R}_{+}^{*}$ and $\mathbb{R}_{-}^{*}$
  • $\forall x \in [0,L], v \in \mathbb{R}^{*}$, $v \frac{\partial \tilde{g}}{\partial x}(x,v) = \left(\int_{\mathbb{R}_{+}^{*}} \tilde{g}(x,w) dw + \int_{\mathbb{R}_{-}^{*}} \tilde{g}(x,w) dw\right) \frac{2}{\sqrt{\pi}} e^{-v^{2}} - \tilde{g}(x,v)$
  • $\forall v \in \mathbb{R}_{+}^{*}$, $\tilde{g}(0,v) = g_{0}(v)$, $\quad \forall v \in \mathbb{R}_{-}^{*}$, $\tilde{g}(L,v) = 0$