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
2018 centrale-maths1__pc

41 maths questions

Q1 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Show that $g_{\sigma}$ is integrable on $\mathbb{R}$.
Q2 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
Assuming that $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \mathrm{d}x = \sqrt{\pi}$, give the value of $\int_{-\infty}^{+\infty} g_{\sigma}(x) \mathrm{d}x$.
Q3 Curve Sketching Sketching a Curve from Analytical Properties View
Study the variations of $g_{\sigma}$. Show that the second derivative of $g_{\sigma}$ vanishes and changes sign at exactly two points. Give the shape of the graph of $g_{\sigma}$ and mark the two points mentioned.
Q4 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.
Q5 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.
Q6 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.
Q7 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that, for any real $\xi$, $\mathcal{F}\left(f^{\prime}\right)(\xi) = 2\mathrm{i}\pi\xi \mathcal{F}(f)(\xi)$.
Q8 Standard Integrals and Reverse Chain Rule Convergence and Estimation of Improper Integrals View
Show that, for any natural integer $p$, the function $x \mapsto x^{2p} \exp\left(-x^{2}\right)$ is integrable on $\mathbb{R}$.
Q9 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
We denote $M_{p} = \int_{-\infty}^{+\infty} x^{2p} \exp\left(-x^{2}\right) \mathrm{d}x$. For $p$ a natural integer, give a relation between $M_{p+1}$ and $M_{p}$ and deduce that, for all $p \in \mathbb{N}$, $$M_{p} = \frac{\sqrt{\pi}(2p)!}{2^{2p} p!}$$
Q10 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
Show that, for any real $\xi$, there exists a real sequence $\left(c_{p}(\xi)\right)_{p \in \mathbb{N}}$ such that $$\forall x \in \mathbb{R}, \quad \exp\left(-x^{2}\right) \cos(2\pi \xi x) = \sum_{p=0}^{+\infty} c_{p}(\xi) \exp\left(-x^{2}\right) x^{2p}$$
Q11 Moment generating functions Compute MGF or characteristic function for a named distribution View
Deduce that, for any real $\xi$, $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \sqrt{\pi} \exp\left(-\pi^{2} \xi^{2}\right)$.
Q12 Moment generating functions Compute MGF or characteristic function for a named distribution View
We set $\sigma^{\prime} = \frac{1}{2\pi\sigma}$. Show that there exists a real $\mu$ such that $\mathcal{F}\left(g_{\sigma}\right) = \mu g_{\sigma^{\prime}}$. The value of $\mu$ need not be made explicit.
Q13 Differential equations Verification that a Function Satisfies a DE View
Show that the function $\left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \times \mathbb{R} \rightarrow \mathbb{R} \\ & (t, x) \mapsto g_{\sqrt{\sigma^{2}+2t}}(x) \end{aligned}\right.$ satisfies conditions i and iii, where:
  • [i.] the diffusion equation: $\forall(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \frac{\partial f}{\partial t}(t, x) = \frac{\partial^{2} f}{\partial x^{2}}(t, x)$;
  • [iii.] the boundary condition: $\forall x \in \mathbb{R},\ \lim_{t \rightarrow 0^{+}} f(t, x) = g_{\sigma}(x)$.
Q14 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f$ be a function satisfying the diffusion equation, the three domination conditions, and the boundary condition $\lim_{t\to 0^+} f(t,x) = g_\sigma(x)$. Justify that, for any real $t > 0$ and any real $\xi$, the function $x \mapsto f(t, x) \exp(-2\mathrm{i}\pi \xi x)$ is integrable on $\mathbb{R}$.
Q15 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We define $\hat{f}$ on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ by: $\forall(t, \xi) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \hat{f}(t, \xi) = \int_{-\infty}^{+\infty} f(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$. Show that, for any real number $\xi$, $\lim_{t \rightarrow 0^{+}} \hat{f}(t, \xi) = \widehat{g_{\sigma}}(\xi)$. One may use any sequence $\left(t_{n}\right)_{n \in \mathbb{N}}$ of strictly positive reals converging to zero.
Q16 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = \int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$.
Q17 Differential equations First-Order Linear DE: General Solution View
By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.
Q18 Differential equations First-Order Linear DE: General Solution View
Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.
Q19 Taylor series Identify a closed-form function from its Taylor series View
Using question 15, determine, for any real $\xi$, the value of $K(\xi)$.
Q20 Taylor series Construct series for a composite or related function View
Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$, $$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$
Q21 Taylor series Extract derivative values from a given series View
Give the value of $\nu_{\sigma}$.
Q22 Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables View
Let $t$ be a strictly positive real number. Using questions 20 and 12, and the result that if $u$ and $v$ are functions from $\mathbb{R}$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}$ and satisfying $\mathcal{F}(u) = \mathcal{F}(v)$, then $u = v$, deduce the existence of a real $\lambda_{t,\sigma}$ such that $$f(t, \cdot) = \lambda_{t,\sigma} g_{\sqrt{\sigma^{2}+2t}}$$
Q23 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
Show that the function $I : \left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \rightarrow \mathbb{R} \\ & t \mapsto \int_{-\infty}^{+\infty} f(t, x) \mathrm{d}x \end{aligned}\right.$ is constant. One may use the result of question 17.
Q24 Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables View
Deduce that, for any strictly positive real $t$, $f(t, \cdot) = g_{\sqrt{\sigma^{2}+2t}}$.
Q25 Connected Rates of Change Pointwise Limit of a Difference Quotient View
Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Give the limit, as $\theta$ tends to zero, of $\frac{f(t+\theta, x) - f(t, x)}{\theta}$.
Q26 Connected Rates of Change Pointwise Limit of a Difference Quotient View
Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Show that $\lim_{h \rightarrow 0} \frac{f(t, x+h) - 2f(t, x) + f(t, x-h)}{h^{2}} = \frac{\partial^{2} f}{\partial x^{2}}(x, t)$.
Q27 Matrices Linear System and Inverse Existence View
Let $\tau$ be a strictly positive real and $q$ a natural integer greater than or equal to 2. We set $\delta = \frac{1}{q+1}$ and $r = \frac{\tau}{\delta^{2}}$. The numerical scheme imposes, for any natural integer $n$ and any $k \in \llbracket 1, q \rrbracket$: $$\frac{f_{n+1}(k) - f_{n}(k)}{\tau} = \frac{f_{n}(k+1) - 2f_{n}(k) + f_{n}(k-1)}{\delta^{2}}$$ as well as $f_{n}(0) = f_{n}(q+1) = 0$. We set $F_{n} = \left(\begin{array}{c} f_{n}(1) \\ \vdots \\ f_{n}(q) \end{array}\right)$, $I_{q}$ is the identity matrix of order $q$, $B$ is the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise, and $A = (1-2r)I_{q} + rB$. Show that, for all $n \in \mathbb{N}$, $F_{n+1} = A F_{n}$.
Q28 Matrices Diagonalizability and Similarity View
With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), justify that the matrices $A$ and $B$ are diagonalizable over $\mathbb{R}$ and that, for all $n \in \mathbb{N}$, $F_{n} = A^{n} F_{0}$.
Q29 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), show that the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ is bounded regardless of the choice of $F_{0}$ if and only if the eigenvalues of $A$ belong to $[-1, 1]$.
Q30 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
Let $\lambda$ be an eigenvalue of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and let $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ be an associated eigenvector. By considering a coefficient of $Y$ whose absolute value is maximal, show that $\lambda \in [-2, 2]$ and justify the existence of an element $\theta$ of $[0, \pi]$, such that $\lambda = 2\cos\theta$.
Q31 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
Let $\lambda$ be an eigenvalue of $B$ and $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ an associated eigenvector. Show that, if we impose $y_{0} = y_{q+1} = 0$, then, for all $k \in \llbracket 1, q \rrbracket$, $y_{k-1} - \lambda y_{k} + y_{k+1} = 0$.
Q32 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Using the results of Q30 and Q31, deduce that there exists $j \in \llbracket 1, q \rrbracket$ such that $\lambda = 2\cos\frac{j\pi}{q+1}$.
Q33 Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View
Determine the spectrum of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and a basis of eigenvectors of $B$.
Q34 Matrices Matrix Norm, Convergence, and Inequality View
With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.
Q35 Binomial Distribution Derive or Prove a Binomial Distribution Identity View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables (taking values in $\{1,-1\}$ each with probability $1/2$). For every $n \in \mathbb{N}^{*}$, we set $Y_{n} = \frac{1}{2}\left(X_{n}+1\right)$ and $Z_{n} = \sum_{j=1}^{n} Y_{j}$. Determine the distribution of the random variable $Y_{n}$ and that of the random variable $Z_{n}$.
Q36 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote, for every integer $n \geqslant 1$, $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ do not have the same parity, then $\mathbb{P}\left(S_{n} = k\right) = 0$.
Q37 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ have the same parity, $\mathbb{P}\left(S_{n} = k\right) = \binom{n}{(k+n)/2} \frac{1}{2^{n}}$.
Q38 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.
Q39 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. Show that, for every real number $\delta$, $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$ is equivalent to $\frac{\delta^{2}}{\tau}$, as $\tau$ tends to 0 from above.
Q40 Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities View
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For every $n \in \mathbb{N}^{*}$ and every $k \in \mathbb{Z}$, by setting $p_{n}(k) = \mathbb{P}\left(S_{n} = k\right)$, show that $$\frac{p_{n+1}(k) - p_{n}(k)}{\tau} = \frac{\delta^{2}}{2\tau} \frac{p_{n}(k+1) - 2p_{n}(k) + p_{n}(k-1)}{\delta^{2}}$$
Q41 Modelling and Hypothesis Testing View
Using the result of Q40, deduce a probabilistic interpretation of the stability condition studied in Part III (i.e., the condition on $r = \frac{\tau}{\delta^2}$ found in Q34).