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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
2011 centrale-maths1__psi

24 maths questions

QI.A Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View
Show that the function $t \rightarrow e^{-t} t^{x-1}$ is integrable on $]0, +\infty[$ if, and only if, $x > 0$.
QI.B Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.
QI.C Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.
QI.D Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Calculate $\Gamma(n)$ for all natural integers $n$, $n \geqslant 1$.
QII.A Integration by Parts Prove an Integral Identity or Equality View
For all integers $k \geqslant 2$, we set: $$u_{k} = \ln k - \int_{k-1}^{k} \ln t \, dt$$ Using two integrations by parts, show that: $$u_{k} = \frac{1}{2}(\ln k - \ln(k-1)) - \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$
QII.B Sequences and Series Convergence/Divergence Determination of Numerical Series View
For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Justify the convergence of the series $\sum_{k \geqslant 2} w_{k}$.
Deduce that there exists a real number $a$ such that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + v_{n}$$ where $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
QII.C Integration by Parts Prove an Integral Inequality or Bound View
For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Using yet another integration by parts, show that: $$\left| w_{k} - \frac{1}{12} \int_{k-1}^{k} \frac{\mathrm{~d}t}{t^{2}} \right| \leqslant \frac{1}{6} \int_{k-1}^{k} \frac{dt}{t^{3}}$$
QII.D Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ and $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
Deduce that $$\left| v_{n} - \frac{1}{12n} \right| \leqslant \frac{1}{12n^{2}}$$ then that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + \frac{1}{12n} + O\left(\frac{1}{n^{2}}\right)$$
QIII.A Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View
We denote by $(f_{n})_{n \geqslant 1}$ the sequence of functions defined on $]0, +\infty[$ by: $$f_{n}(t) = \begin{cases} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} & \text{if } t \in ]0, n[ \\ 0 & \text{if } t \geqslant n \end{cases}$$ Show that for all integers $n$, $n \geqslant 1$, the function $f_{n}$ is continuous and integrable on $]0, +\infty[$.
QIII.B Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View
We define for all real $x > 0$ the sequence $(I_{n}(x))_{n \geqslant 1}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt$$ Show that, for all $x > 0$, $$\lim_{n \rightarrow +\infty} I_{n}(x) = \Gamma(x)$$
QIII.C Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Show that, for all integers $n$, $n \geqslant 0$, $$\forall x > 0, \quad J_{n+1}(x) = \frac{n+1}{x} J_{n}(x+1)$$
QIII.D Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Deduce that, for all $x > 0$, $$J_{n}(x) = \frac{n!}{x(x+1) \cdots (x+n-1)(x+n)}$$
QIII.E Reduction Formulae Derive a Product or Series Representation from Reduction Formulae View
We define for all real $x > 0$ the sequences $(I_{n}(x))_{n \geqslant 1}$ and $(J_{n}(x))_{n \geqslant 0}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt, \qquad J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Establish Euler's identity: $$\forall x > 0, \quad \Gamma(x) = \lim_{n \rightarrow +\infty} \frac{n! \, n^{x}}{x(x+1) \cdots (x+n)}$$
QIV.A Curve Sketching Sketching a Curve from Analytical Properties View
The function $h$ is defined on $\mathbb{R}$ by $$h : \mathbb{R} \longrightarrow \mathbb{R}, \quad u \longmapsto u - [u] - 1/2$$ where $[u]$ denotes the integer part of $u$.
Carefully draw the graph of the application $h$ on the interval $[-1, 1]$.
QIV.B Sequences and Series Properties and Manipulation of Power Series or Formal Series View
The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Show that the function $H$ defined on $\mathbb{R}$ by: $$H(x) = \int_{0}^{x} h(t) \, dt$$ is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.
QIV.C Integration by Parts Prove an Integral Inequality or Bound View
The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Using integration by parts, justify, for $x > 0$, the convergence of the following integral: $$\int_{0}^{+\infty} \frac{h(u)}{u+x} du$$
QIV.D Sequences and Series Convergence/Divergence Determination of Numerical Series View
The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Is the application $u \longmapsto \dfrac{h(u)}{u+x}$ integrable on $\mathbb{R}_{+}$?
QIV.E Sequences and Series Properties and Manipulation of Power Series or Formal Series View
The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Let $\varphi$ be the application defined for all $x > 0$ by: $$\varphi(x) = \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ By repeating the integration by parts from question IV.C, prove that the application $\varphi$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{*}$ and that for all $x > 0$, $$\varphi^{\prime}(x) = -\int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$
QV.A Integration by Parts Prove an Integral Identity or Equality View
We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Show that for all natural integers $i$: $$\int_{x+i}^{x+i+1} \ln t \, dt = \ln(x+i) - \int_{i}^{i+1} \frac{u-i-1}{u+x} du$$
QV.B Sequences and Series Evaluation of a Finite or Infinite Sum View
We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Deduce that: $$F_{n}(x) = G_{n}(x) - \int_{0}^{n+1} \frac{h(u)}{u+x} du$$ where $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$
QV.C Sequences and Series Limit Evaluation Involving Sequences View
We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ and $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$
V.C.1) Using Stirling's formula, show that: $$\lim_{n \rightarrow +\infty} G_{n}(x) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi}$$
V.C.2) Deduce that: $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$
QV.D Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Using the identity $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ show that for all strictly positive real $x$, $$\frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)} = \ln x + \frac{1}{2x} + \int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$
QVI.A Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$
VI.A.1) Let $f$ be a function of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{4}$. Show that $f$ admits a maximum on $\Omega$. We then denote $a = (a_{1}, a_{2}, a_{3}, a_{4}) \in \Omega$ a point at which this maximum is attained.
VI.A.2) Show that if $(x_{1}, x_{2}, x_{3}, x_{4}) \in \Omega$ then $x_{3}$ and $x_{4}$ can be written in the form $$\begin{aligned} & x_{3} = u x_{1} + v x_{2} + w \\ & x_{4} = u^{\prime} x_{1} + v^{\prime} x_{2} + w^{\prime} \end{aligned}$$ where we shall give explicitly $u, v, u^{\prime}, v^{\prime}$ in terms of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$.
VI.A.3) Assuming that none of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ is zero, deduce that $$\begin{aligned} & \frac{\partial f}{\partial x_{1}}(a) + u \frac{\partial f}{\partial x_{3}}(a) + u^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \\ & \frac{\partial f}{\partial x_{2}}(a) + v \frac{\partial f}{\partial x_{3}}(a) + v^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \end{aligned}$$
VI.A.4) Show that the vector subspace of $\mathbb{R}^{4}$ spanned by the vectors $(1, 0, u, u^{\prime})$ and $(0, 1, v, v^{\prime})$ admits a supplementary orthogonal subspace spanned by the vectors $(1,1,1,1)$ and $(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4})$.
VI.A.5) Deduce the existence of two real numbers $\alpha, \beta$ such that for all $i \in \{1,2,3,4\}$ we have $$\frac{\partial f}{\partial x_{i}}(a) = \alpha + \beta \varepsilon_{i}$$
QVI.C Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
For all $i \in \{1,2,3,4\}$, we set $$\theta(N_{i}) = \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du$$
VI.C.1) Show that for all $i \in \{1,2,3,4\}$ $$0 < \theta(N_{i}) < \frac{1}{N_{i}}$$
VI.C.2) Show the existence of a strictly positive real $K$ such that for all $i \in \{1,2,3,4\}$ $$N_{i} = K e^{\mu \varepsilon_{i}} e^{-\theta(N_{i})}$$