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
2012 centrale-maths1__psi

33 maths questions

QI.A Sequences and Series Convergence/Divergence Determination of Numerical Series View
What inclusion exists between the sets $E$ and $E^{\prime}$, where $E$ is the set of real numbers $x$ for which the application $t \mapsto f(t)e^{-\lambda(t)x}$ is integrable on $\mathbb{R}^+$, and $E^{\prime}$ is the set of real numbers $x$ for which the integral $\int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt$ converges?
QI.B Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that if $E$ is not empty, then $E$ is an unbounded interval of $\mathbb{R}$.
QI.C Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Show that if $E$ is not empty, then $Lf$ is continuous on $E$, where for $x \in E^{\prime}$, $$Lf(x) = \int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt.$$
QII.A Sequences and Series Convergence/Divergence Determination of Numerical Series View
Compare $E$ and $E^{\prime}$ in the case where $f$ is positive.
QII.B Sequences and Series Convergence/Divergence Determination of Numerical Series View
In the three following cases, determine $E$.
II.B.1) $f(t) = \lambda^{\prime}(t)$, with $\lambda$ assumed to be of class $C^1$.
II.B.2) $f(t) = e^{t\lambda(t)}$.
II.B.3) $f(t) = \dfrac{e^{-t\lambda(t)}}{1+t^2}$.
QII.C Sequences and Series Evaluation of a Finite or Infinite Sum View
In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$.
II.C.1) Determine $E$. What is the value of $Lf(0)$?
II.C.2) Prove that $Lf$ is differentiable.
II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have $$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$
II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$.
Show that for all $x \geqslant 0$, we have $$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$
II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.
QIII.A Sequences and Series Power Series Expansion and Radius of Convergence View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$.
Show that $f$ extends by continuity at 0.
QIII.B Sequences and Series Convergence/Divergence Determination of Numerical Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Determine $E$.
QIII.C Sequences and Series Power Series Expansion and Radius of Convergence View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Using a series expansion, show that for all $x > 0$, we have $$Lf(x) = \frac{1}{2x^2} - \frac{1}{x} + \sum_{n=1}^{+\infty} \frac{1}{(n+x)^2}.$$
QIII.D Sequences and Series Limit Evaluation Involving Sequences View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Does $Lf(x) - \dfrac{1}{2x^2} + \dfrac{1}{x}$ admit a finite limit at $0^+$?
QIV.A Sequences and Series Properties and Manipulation of Power Series or Formal Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that if $E$ is not empty and if $\alpha$ is its infimum (we agree that $\alpha = -\infty$ if $E = \mathbb{R}$), then $Lf$ is of class $C^{\infty}$ on $]\alpha, +\infty[$ and express its successive derivatives using an integral.
QIV.B Sequences and Series Evaluation of a Finite or Infinite Sum View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
In the particular case where $f(t) = e^{-at}t^n$ for all $t \in \mathbb{R}^+$, with $n \in \mathbb{N}$ and $a \in \mathbb{R}$, make explicit $E$, $E^{\prime}$ and calculate $Lf(x)$ for $x \in E^{\prime}$.
QIV.C Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$: $$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$
IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion: $$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$
IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion: $$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$
QIV.D Sequences and Series Convergence/Divergence Determination of Numerical Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $f$ admits a finite limit $l$ at $+\infty$.
IV.D.1) Show that $E$ contains $\mathbb{R}^{+*}$.
IV.D.2) Show that $xLf(x)$ tends to $l$ at $0^+$.
QV.A Sequences and Series Convergence/Divergence Determination of Numerical Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E$ does not contain 0.
QV.B Sequences and Series Convergence/Divergence Determination of Numerical Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E = ]0, +\infty[$.
QV.C Sequences and Series Convergence/Divergence Determination of Numerical Series View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E^{\prime}$ contains 0.
QV.D Sequences and Series Evaluation of a Finite or Infinite Sum View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Calculate $(Lf)^{\prime}(x)$ for $x \in E$.
QV.E Differential equations Eigenvalue Problems and Operator-Based DEs View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Deduce from V.D the expression of $(Lf)(x)$ for $x \in E$.
QV.F Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
We denote for $n \in \mathbb{N}$ and $x \geqslant 0$, $$f_n(x) = \int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t} e^{-xt}\,dt.$$ Show that $\sum_{n \geqslant 0} f_n$ converges uniformly on $[0, +\infty[$.
QV.G Differential equations Eigenvalue Problems and Operator-Based DEs View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
What is the value of $Lf(0)$?
QVI.A Proof Deduction or Consequence from Prior Results View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $g$ be a continuous application from $[0,1]$ to $\mathbb{R}$. We assume that for all $n \in \mathbb{N}$, we have $$\int_0^1 t^n g(t)\,dt = 0.$$
VI.A.1) What can we say about $\displaystyle\int_0^1 P(t)g(t)\,dt$ for $P \in \mathbb{R}[X]$?
VI.A.2) Deduce from this that $g$ is the zero application.
QVI.B Differential equations Eigenvalue Problems and Operator-Based DEs View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $f$ be fixed such that $E$ is non-empty, $x \in E$ and $a > 0$. We set $h(t) = \displaystyle\int_0^t e^{-xu} f(u)\,du$ for all $t \geqslant 0$.
VI.B.1) Show that $Lf(x+a) = a\displaystyle\int_0^{+\infty} e^{-at} h(t)\,dt$.
VI.B.2) We assume that for all $n \in \mathbb{N}$, we have $Lf(x + na) = 0$.
Show that, for all $n \in \mathbb{N}$, the integral $\displaystyle\int_0^1 u^n h\!\left(-\frac{\ln u}{a}\right)du$ converges and that it is zero.
VI.B.3) What do we deduce for the function $h$?
QVI.C Proof Proof That a Map Has a Specific Property View
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that the application which associates to $f$ the function $Lf$ is injective.
QVII.A Differential equations Eigenvalue Problems and Operator-Based DEs View
We assume that $f$ is positive and that $E$ is neither empty nor equal to $\mathbb{R}$. We denote by $\alpha$ its infimum.
VII.A.1) Show that if $Lf$ is bounded on $E$, then $\alpha \in E$.
VII.A.2) If $\alpha \notin E$, what can we say about $Lf(x)$ when $x$ tends to $\alpha^+$?
QVII.B Differential equations Eigenvalue Problems and Operator-Based DEs View
In this question, $f(t) = \cos t$ and $\lambda(t) = \ln(1+t)$.
VII.B.1) Determine $E$.
VII.B.2) Determine $E^{\prime}$.
VII.B.3) Show that $Lf$ admits a limit at $\alpha$, the infimum of $E$, and determine it.
QVIII.A Sequences and Series Convergence/Divergence Determination of Numerical Series View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients and we use the transformation $L$ applied to elements of $\mathcal{P}$ for the study of an operator $U$.
Let $P$ and $Q$ be two elements of $\mathcal{P}$.
Show that the integral $\displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, where $\bar{P}$ is the polynomial whose coefficients are the conjugates of those of $P$, converges.
QVIII.B Matrices Bilinear and Symplectic Form Properties View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients. We denote for every pair $(P,Q) \in \mathcal{P}^2$, $$\langle P, Q \rangle = \int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt.$$
Verify that $\langle \cdot, \cdot \rangle$ defines an inner product on $\mathcal{P}$.
QVIII.C Matrices Linear Transformation and Endomorphism Properties View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$.
We denote by $D$ the differentiation endomorphism and $U$ the endomorphism of $\mathcal{P}$ defined by $$U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right).$$
Verify that $U$ is an endomorphism of $\mathcal{P}$.
QVIII.D Integration by Parts Inner Product or Orthogonality Proof via Integration by Parts View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that for all $P$ and $Q$ in $\mathcal{P}$, we have $$\langle U(P), Q \rangle = \langle P, U(Q) \rangle.$$
QVIII.E Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that $U$ admits eigenvalues in $\mathbb{C}$, that they are real and that two eigenvectors associated with distinct eigenvalues are orthogonal.
QVIII.F Second order differential equations Verifying a particular solution satisfies a second-order ODE View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it.
VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify.
VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?
QVIII.G Second order differential equations Solving non-homogeneous second-order linear ODE View
$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
We consider on $[0, +\infty[$ the differential equation $$(E_n): \quad tP^{\prime\prime} + (1-t)P^{\prime} + nP = 0$$ with $n \in \mathbb{N}$ and unknown $P \in \mathcal{P}$.
VIII.G.1) By applying the transformation $L$ with $\lambda(t) = t$ to $(E_n)$, show that if $P$ is a solution of $(E_n)$ on $[0, +\infty[$, then its image $Q$ by $L$ is a solution of a differential equation $(E_n^{\prime})$ of order 1 on $]1, +\infty[$.
VIII.G.2) Solve the equation $(E_n^{\prime})$ on $]1, +\infty[$ and deduce from this the eigenvalues and eigenvectors of the endomorphism $U$.
VIII.G.3) What is the relationship between the above and the polynomial functions defined for $n \in \mathbb{N}$ by $P_n(t) = e^t D^n\left(e^{-t}t^n\right)$?