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
2020 centrale-maths1__psi

38 maths questions

Q1 Independent Events View
Show that $S_n$ and $X_{n+1}$ are independent.
Q2 Probability Generating Functions Explicit computation of a PGF or characteristic function View
Explicitly calculate the generating function $G_{X_1}$ of the random variable $X_1$, where $X_1$ follows a Poisson distribution with parameter $1/2$.
Q3 Probability Generating Functions PGF of sum of independent variables View
Justify that $\forall t \in \mathbb{R}, G_{S_n}(t) = \left(G_{X_1}(t)\right)^n$.
Q4 Poisson distribution View
Show that the random variable $S_n$ follows a Poisson distribution and specify its parameter.
Q5 Poisson distribution View
Verify that, for all $n \in \mathbb{N}^*$, $$n! \left(\frac{2}{n}\right)^n P\left(S_n > n\right) = \mathrm{e}^{-n/2} \sum_{k=1}^{\infty} \frac{n! n^k}{(n+k)!} \left(\frac{1}{2}\right)^k.$$
Q6 Poisson distribution Divisibility and Divisor Analysis View
Let $n \in \mathbb{N}^*$. Show that for all $k \in \mathbb{N}^*$, $$\left(\frac{n}{n+k}\right)^k \leqslant \frac{n! n^k}{(n+k)!} \leqslant 1.$$
Q7 Sequences and series, recurrence and convergence Uniform or Pointwise Convergence of Function Series/Sequences View
Show that the series of functions $\sum u_k$ where for all $k \in \mathbb{N}^*$, the function $u_k$ is defined on $[0, +\infty[$ by $u_k : x \mapsto (1 + kx)^{-k}(1/2)^k$ is normally convergent on $[0, +\infty[$.
Q8 Sequences and series, recurrence and convergence Limit Evaluation Involving Sequences View
Deduce that for all $n \in \mathbb{N}^*, \sum_{k \geqslant 1} \left(1 + \frac{k}{n}\right)^{-k} \left(\frac{1}{2}\right)^k$ converges and that $$\lim_{n \rightarrow +\infty} \sum_{k=1}^{\infty} \left(1 + \frac{k}{n}\right)^{-k} \left(\frac{1}{2}\right)^k = 1.$$
Q9 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Deduce that, when $n$ tends to $+\infty$, $$P\left(S_n > n\right) \sim \frac{\mathrm{e}^{-n/2}}{n!} \left(\frac{n}{2}\right)^n.$$
Q10 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Deduce, using Stirling's formula, that there exists a real $\alpha \in ]0,1[$ such that $P\left(S_n > n\right) = O\left(\alpha^n\right)$.
Q11 Matrices Matrix Norm, Convergence, and Inequality View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that, for all $x \in \mathbb{R}^n$, $$\left\{\begin{array}{l} x \geqslant 0 \Longrightarrow Ax \geqslant 0 \\ x \geqslant 0 \text{ and } x \neq 0 \Longrightarrow Ax > 0. \end{array}\right.$$
Q12 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that $\forall k \in \mathbb{N}^*, A^k > 0$.
Q13 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Deduce that $\rho(A) > 0$ then show that $\rho\left(\frac{A}{\rho(A)}\right) = 1$.
Q14 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.
Q15 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Show that $|x| \leqslant A|x|$.
Q16 Matrices Matrix Norm, Convergence, and Inequality View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. We assume that $|x| < A|x|$. Show that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$.
Q17 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. We assume that $|x| < A|x|$ and that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$. We set $B = \frac{1}{1+\varepsilon} A$. Show that for all $k \geqslant 1, B^k A|x| \geqslant A|x|$.
Q18 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We set $B = \frac{1}{1+\varepsilon} A$ for some $\varepsilon > 0$. Determine $\lim_{k \rightarrow +\infty} B^k$.
Q19 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Conclude (that 1 is an eigenvalue of $A$).
Q20 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $A$ admits a strictly positive eigenvector associated with the eigenvalue 1.
Q21 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that 1 is the only eigenvalue of $A$ with modulus 1.
One may admit without proof that if $z_1, z_2, \ldots, z_k$ are non-zero complex numbers such that $|z_1 + \cdots + z_k| = |z_1| + \cdots + |z_k|$, then $\forall j \in \llbracket 1, k \rrbracket, \exists \lambda_j \in \mathbb{R}^+$ such that $z_j = \lambda_j z_1$.
Q22 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $\dim\left(\ker\left(A - I_n\right)\right) = 1$.
Q23 Matrices Eigenvalue and Characteristic Polynomial Analysis View
By combining the results of sub-parts II.B and II.C, justify that we have proved Proposition 1: If $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix, then $\rho(A)$ is a dominant eigenvalue of $A$. The associated eigenspace $\ker\left(A - \rho(A) I_n\right)$ is one-dimensional and is spanned by a strictly positive eigenvector.
Q24 Matrices Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $\lambda \in S = \operatorname{sp}(A) \setminus \{\rho(A)\}$. Let $Y \in \ker\left(A - \lambda I_n\right)$. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to 0.
Q25 Matrices Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $Y \in \mathcal{M}_{n,1}(\mathbb{R})$ be a positive vector. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to the projection of $Y$ onto $E_{\rho(A)}(A)$ parallel to $\bigoplus_{\lambda \in S} E_\lambda(A)$. Verify that, if it is non-zero, this latter vector (the projection of $Y$) is strictly positive.
Q26 Matrices Diagonalizability and Similarity View
Justify that for all integer $k \geqslant 1$, $A^k$ is similar in $\mathcal{M}_n(\mathbb{C})$ to a triangular matrix, whose diagonal coefficients we will specify.
Q27 Roots of polynomials Matrix Power Computation and Application View
Show that $\lim_{k \rightarrow +\infty} \frac{\operatorname{tr}\left(A^{k+1}\right)}{\operatorname{tr}\left(A^k\right)} = \rho(A)$.
Q28 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. Justify that $\forall i \in \llbracket 0, N \rrbracket, \sum_{j=0}^{N} q_{i,j} = 1$.
Q29 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We denote $\Pi_n = \begin{pmatrix} P(X_n = 0) \\ \vdots \\ P(X_n = N) \end{pmatrix} \in \mathcal{M}_{N+1,1}(\mathbb{R})$. Justify that, for all $n \in \mathbb{N}^*, \Pi_{n+1} = Q^\top \Pi_n$.
Q30 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We have $\Pi_{n+1} = Q^\top \Pi_n$. Deduce that the distribution of $X_1$ completely determines the distributions of all random variables $X_n, n \in \mathbb{N}^*$.
Q31 Discrete Probability Distributions Eigenvalue and Characteristic Polynomial Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$ and $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N} \in \mathcal{M}_{N+1}(\mathbb{R})$. Justify that $A(t)$ possesses a dominant eigenvalue $\gamma(t) > 0$.
Q32 Discrete Probability Distributions Matrix Power Computation and Application View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$, $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N}$, $z_j(t) = P(X_1 = j)\mathrm{e}^{jt}$, $Z(t) = \begin{pmatrix} z_0(t) \\ \vdots \\ z_N(t) \end{pmatrix}$, and $Y^{(n)}(t) = (A(t))^{n-1} Z(t)$ so that $E\left(\mathrm{e}^{tS_n}\right) = \sum_{j=0}^N Y_j^{(n)}(t)$. Let $\gamma(t)$ be the dominant eigenvalue of $A(t)$. Show that $\lim_{n \rightarrow +\infty} \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n} = \lambda(t)$ where $\lambda(t) = \ln(\gamma(t))$.
Q35 Discrete Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. We admit that the convergence of the sequence of functions $\left(t \mapsto \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n}\right)_{n \in \mathbb{N}^*}$ towards $t \mapsto \ln(\gamma(t))$ is uniform on $\mathbb{R}^+$. Let $\varepsilon > 0$. Show that there exists a rank $n_0 \in \mathbb{N}^*$ such that, for all $t \in \mathbb{R}^+$ and for all $n \in \mathbb{N}^*$, $$n \geqslant n_0 \Longrightarrow \ln\left(E\left(\mathrm{e}^{tS_n}\right)\right) \leqslant n(\lambda(t) + \varepsilon).$$
Q36 Discrete Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Using Markov's inequality applied to the random variable $e^{tS_n}$, show that for $a > 1$, $n \geqslant n_0$ and $t \geqslant 0$, $$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-ntam} \mathrm{e}^{n(\lambda(t) + \varepsilon)}.$$
Q37 Discrete Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Deduce that for $n \geqslant n_0$, $$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-n\left(\lambda^*(am) - \varepsilon\right)}.$$
Q38 Discrete Random Variables Probability Inequality and Tail Bound Proof View
Give a concrete meaning to $m = \lim_{n \rightarrow +\infty} \frac{1}{n} E(S_n)$ in relation to the industrial process studied and interpret the inequality $P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-n(\lambda^*(am) - \varepsilon)}$. One may establish an intuitive link with the law of large numbers.
Q39 Discrete Random Variables Deduction or Consequence from Prior Results View
We have two finite sequences of real numbers $0 = t_1 < t_2 < \cdots < t_K$ ($K \geqslant 2$) and $x_1 < x_2 < \cdots < x_L$ ($L \geqslant 2$). The formula from question 32 applied at $t_i$ with $n$ sufficiently large allows us to estimate $\lambda(t_i)$ by an approximate value $\hat{\lambda}(t_i)$. Justify that for all $i \in \{1, \ldots, L\}$, $$\hat{\lambda}^*\left(x_i\right) = \max_{1 \leqslant j \leqslant K} \left(t_j x_i - \hat{\lambda}\left(t_j\right)\right)$$ constitutes a reasonable approximate value of $\lambda^*\left(x_i\right)$.
Q40 Discrete Random Variables Computation of a Limit, Value, or Explicit Formula View
Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$, $$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$
$x_i$4,504,554,604,654,70
$\hat{\lambda}^*(x_i)$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$
$x_i$4,754,804,854,904,95
$\hat{\lambda}^*(x_i)$$5{,}1 \times 10^{-4}$$5{,}5 \times 10^{-3}$$1{,}1 \times 10^{-2}$$1{,}6 \times 10^{-2}$$2{,}1 \times 10^{-2}$
$x_i$5,005,055,105,155,20
$\hat{\lambda}^*(x_i)$$2{,}6 \times 10^{-2}$$3{,}1 \times 10^{-2}$$3{,}6 \times 10^{-2}$$4{,}1 \times 10^{-2}$$4{,}6 \times 10^{-2}$
$x_i$5,255,305,355,405,45
$\hat{\lambda}^*(x_i)$$5{,}1 \times 10^{-2}$$5{,}6 \times 10^{-2}$$6{,}1 \times 10^{-2}$$6{,}6 \times 10^{-2}$$7{,}1 \times 10^{-2}$