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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
2020 x-ens-maths-d__mp

14 maths questions

QII.6 Proof Existence Proof View
6. Show that for all $n \in \mathbb{N}$, for every function $f$ of $S_n$ satisfying $\lim_{x \rightarrow \pm\infty} |f(x)| = \pm\infty$, there exists an element $g \in \mathscr{P}_{n+1}$ such that $f \sim g$ (where $\sim$ is the relation defined in I.2).
QIII.1 Proof Proof That a Map Has a Specific Property View
Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
1. For $m \geq 2$ verify that the map $\operatorname{Opp} : \Sigma_m \rightarrow \Sigma_m$, which to $\sigma \in \Sigma_m$ associates $\eta \in \Sigma_m$ defined by $$\eta(i) = m + 1 - \sigma(i)$$ is a bijection satisfying $\operatorname{Opp}(\operatorname{MD}(m)) = \operatorname{DM}(m)$ and $\operatorname{Opp}(\operatorname{DM}(m)) = \operatorname{MD}(m)$. Verify that if $\sigma \in \Sigma_m$ and if $i, j$ are elements of $\{1, \ldots, m\}$ satisfying $\sigma(j) > \sigma(i)$, $$\sigma(j) - \sigma(i) = 1 + \operatorname{Card}\{k \in \Delta_m \mid \sigma(i) < \sigma(k) < \sigma(j)\}$$
QIII.2 Proof Characterization or Determination of a Set or Class View
Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
2. Under what condition (necessary and sufficient) on $n$ and $k$ is the set $\mathcal{B}(n, k)$ non-empty? Under what condition (necessary and sufficient) on $n$, $s$ and $k$ is the set $\mathcal{C}(n, s, k)$ non-empty?
QIII.3 Proof Proof That a Map Has a Specific Property View
Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
3. In this question and the next, we fix $n \geq 2$, $1 \leq k \leq n-1$ and $1 \leq s \leq n-k$. We propose to construct a bijection from $\mathcal{C}(n, s, k)$ to $\mathcal{B}(n, k)$. Let $\sigma \in \mathcal{C}(n, s, k)$. a. Verify that the number $m$ of integers $j \geq 4$ such that $\sigma(j) > \sigma(3)$ satisfies $m \geq k$. We denote $j_1, \ldots, j_m$ these integers, which we order in such a way that $\sigma(j_1) < \sigma(j_2) < \cdots < \sigma(j_m)$. b. Show that $\xi$ defined by $\xi(1) = \sigma(j_k) + \frac{1}{2}$, $\xi(2) = \sigma(3)$, $\xi(n+1) = \ldots$ satisfies $$\xi(p) > \xi(p+1) \text{ for } p \text{ odd}, \quad \xi(p) < \xi(p+1) \text{ for } p \text{ even}$$ and that the interval $]\xi(2), \xi(1)[$ contains exactly $k$ elements of $\{\xi(3), \ldots, \xi(n+1)\}$. c. We denote $A = \xi(\Delta_{n+1})$ and we set $\bar{\xi} = \beta_A \circ \xi$ (we recall that $\beta_A$ denotes the unique increasing bijection from $A$ to $\Delta_{n+1}$). Show that $\bar{\xi} \in \operatorname{DM}(n+1)$. d. Let $\eta = \operatorname{Opp}(\bar{\xi})$. Verify that $\eta \in \mathcal{B}(n, k)$.
QIII.4 Proof Proof That a Map Has a Specific Property View
Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
We denote by $\Psi_{n,s,k}$ the map from $\mathcal{C}(n,s,k)$ to $\mathcal{B}(n,k)$ defined by $\Psi_{n,s,k}(\sigma) = \eta$.
4. Let $\eta \in \mathcal{B}(n, k)$ and let $\xi = \operatorname{Opp}(\eta)$. a. Verify that the number $m$ of integers $j \geq 3$ such that $\xi(j) > \xi(2)$ satisfies $m \geq k$. We denote these integers by $j_1, \ldots, j_m$, with $\xi(j_1) > \xi(j_2) \cdots > \xi(j_m)$. b. We set $u_2 = \xi(j_k) - \frac{1}{2} > \xi(2)$. Show that the number $m'$ of integers $i \geq 2$ such that $\xi(i) < u_2$ satisfies $m' \geq s$. We denote them by $i_1, \ldots, i_{m'}$, with $\xi(i_1) > \cdots > \xi(i_{m'})$ and we set $u_1 = \xi(i_k) - \frac{1}{2}$. c. By considering the map $\theta$ defined by $$\theta(1) = u_1, \theta(2) = u_2, \theta(3) = \xi(2), \ldots, \theta(n+2) = \xi(n+1)$$ show the existence of $\sigma \in \mathcal{C}(n, s, k)$ satisfying $\Psi_{n,s,k}(\sigma) = \eta$. d. Show that $\Psi_{n,s,k}$ is bijective.
QIII.5 Proof Computation of a Limit, Value, or Explicit Formula View
5. Give a procedure for computing $\operatorname{Card} \operatorname{MD}(n)$ by recursion.
QIV.1 Sequences and Series Functional Equations and Identities via Series View
1. We denote $E_n = \operatorname{Card} \operatorname{MD}(n)$ and $\mathscr{I}_n$ the set of odd numbers in $\Delta_n$. a. Prove that for $n \geq 1$: $$E_{n+1} = \sum_{i \in \mathscr{I}_{n+1}} \binom{n}{i-1} E_{i-1} E_{n+1-i}$$ b. Deduce that for $n \geq 1$: $$2E_{n+1} = \sum_{i=0}^{n} \binom{n}{i} E_i E_{n-i}$$
QIV.2 Taylor series Recursive or implicit derivative computation for series coefficients View
2. a. Show that the radius of convergence of the power series $\sum \frac{E_n}{n!} x^n$ is $\geq 1$. b. For $|x| < 1$, we denote by $f(x)$ the sum of the preceding power series. Prove that $$2f'(x) = f^2(x) + 1, \quad \forall x \in ]-1, 1[$$ c. Deduce that $f(x) = \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1}{\cos x} + \tan x, \quad \forall x \in ]-1, 1[$, then that $$\frac{1}{\cos x} = \sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!} x^{2n}, \quad \tan x = \sum_{n=0}^{\infty} \frac{E_{2n+1}}{(2n+1)!} x^{2n+1}, \quad \forall x \in ]-1, 1[$$
QIV.3 Taylor series Recursive or implicit derivative computation for series coefficients View
3. For a function $f : \mathbb{R} \rightarrow \mathbb{R}$ of class $C^\infty$ and $n \in \mathbb{N}$, we denote by $f^{(n)}$ the derivative of order $n$ of $f$, with the convention $f^{(0)} = f$. We denote by $D : \mathbb{R}[X] \rightarrow \mathbb{R}[X]$ the unique linear map such that $$D(X^0) = 0, \quad D(X^k) = k(X^{k-1} + X^{k+1}), \quad \forall k \in \mathbb{N}^*$$ For $n \in \mathbb{N}^*$, we denote by $D^n$ the composition of order $n$ of $D$, with the convention $D^0 = \mathrm{Id}$. a. Let $P_n = D^n(X)$. Prove that for $n \in \mathbb{N}$, $\tan^{(n)}(x) = P_n(\tan x)$ for $x \in ]-\frac{\pi}{2}, \frac{\pi}{2}[$. b. For $m \in \mathbb{N}^*$, let $V_m$ be the subspace of $\mathbb{R}[X]$ generated by $\{X, \ldots, X^m\}$. Let $\iota_m$ be the canonical injection of $V_m$ into $\mathbb{R}[X]$ and let $\tau_m : \mathbb{R}[X] \rightarrow V_m$ be the linear projection defined by $\tau_m(X^k) = X^k$ if $k \in \{1, \ldots, m\}$ and $\tau_m(X^k) = 0$ otherwise. Finally, we set $\delta_m = \tau_m \circ D \circ \iota_m$. Verify that $\delta_m$ is a linear map from $V_m$ to $V_m$ and write its matrix $M_m$ in the basis $(X, \ldots, X^m)$.
QIV.4 Matrices Eigenvalue and Characteristic Polynomial Analysis View
4. Let $C_m \in \mathbb{R}[Y]$ be the characteristic polynomial of $M_m$. a. Verify that $C_1 = Y$, $C_2 = Y^2 - 2$ and $$C_m = Y C_{m-1} - m(m-1) C_{m-2}, \quad m \geq 3$$ b. Calculate the determinant of $M_m$. c. Prove that, if $e_m$ denotes the integer part of $m/2$, $$C_m = \sum_{k=0}^{e_m} (-1)^k c_{m,k} Y^{m-2k}$$ with $$c_{m,0} = 1; \quad c_{m,k} = \sum_{(a_1, \ldots, a_k) \in J_k(m)} a_1(a_1+1) a_2(a_2+1) \cdots a_k(a_k+1), \quad 1 \leq k \leq e_m;$$ where $J_k(m)$ denotes the set of $k$-tuples of integers from $\{1, \ldots, m-1\}$ such that $a_i + 2 \leq a_{i+1}$ for $1 \leq i \leq k-1$.
QIV.5 Number Theory Combinatorial Number Theory and Counting View
5. In the remainder of this part, $p$ denotes a fixed odd prime integer. One may use without proof Wilson's theorem: $$(p-1)! + 1 \equiv 0 \quad [p]$$ We denote by $\mathbb{Z}_p$ the field $\mathbb{Z}/p\mathbb{Z}$ and if $a \in \mathbb{Z}$, we denote by $\bar{a}$ its class in $\mathbb{Z}_p$. For $1 \leq k \leq e_p$, we denote by $\mathcal{P}_k$ the set of subsets $P$ with $k$ elements of $\mathbb{Z}_p$ satisfying the condition $$\forall \alpha \in P, \quad \alpha + 1 \notin P.$$ a. For $P = \{\alpha_1, \ldots, \alpha_k\} \in \mathcal{P}_k$ and $\alpha \in \mathbb{Z}_p$, we set $\tau_\alpha(P) = \{\alpha_1 + \alpha, \ldots, \alpha_k + \alpha\}$. Show that the map $\alpha \mapsto \tau_\alpha$ is a homomorphism from $(\mathbb{Z}_p, +)$ to the group of bijections of $\mathcal{P}_k$. b. We define a relation $\mathscr{R}$ between elements of $\mathcal{P}_k$ as follows: if $A, B$ are in $\mathcal{P}_k$, $A \mathscr{R} B$ if and only if there exists $\alpha \in \mathbb{Z}_p$ such that $B = \tau_\alpha(A)$. Show that $\mathscr{R}$ is an equivalence relation on $\mathcal{P}_k$, and that each equivalence class has cardinality $p$ and admits a representative of the form $\{\bar{0}, \bar{a}_2, \ldots, \bar{a}_k\}$ with $0 < a_2 < \cdots < a_k < p$. We choose such a representative for each class and we denote by $R$ the set of representatives thus chosen. c. Prove that $$\overline{c_{p-1,k}} = \sum_{\{0, \ldots, a_k\} \in R} \sum_{1 \leq \ell \leq p-1} \bar{\ell}\, \overline{\ell+1}\, \overline{a_2 + \ell}\, \overline{a_2 + \ell + 1} \cdots \overline{a_k + \ell}\, \overline{a_k + \ell + 1}$$
QIV.6 Number Theory Modular Arithmetic Computation View
6. a. For $q \in \mathbb{N}$, we set $S_q = \sum_{\ell=0}^{p-1} \ell^q$. Observe that $p$ divides $\sum_{\ell=0}^{p-1} ((\ell+1)^{q+1} - \ell^{q+1})$ and deduce by recursion that $p$ divides $S_q$ for $0 \leq q \leq p-2$. b. Let $Z = [z_{ij}]$ and $Z' = [z'_{ij}]$ be two square matrices of order $N$ with entries in $\mathbb{Z}$. We define the relation $Z \equiv Z'[p]$ by $z_{ij} \equiv z'_{ij}[p]$ for $1 \leq i, j \leq N$. Prove that $$(M_{p-1})^{(p-1)} \equiv (-1)^{(p-1)/2} \mathrm{Id} \quad [p].$$ c. What can be said about a polynomial $Q$ with integer coefficients such that $Q(M_{p-1}) \equiv 0[p]$?
QIV.7 Number Theory Arithmetic Functions and Multiplicative Number Theory View
7. We recall that $\bar{E}_n$ denotes the class of $E_n = \operatorname{Card} \operatorname{MD}(n)$ in $\mathbb{Z}_p$. a. Show that $E_{2n+1} \equiv u_{2n}[p]$, where $u_m$ is the coefficient of the term $X$ in the decomposition of $\delta_{p-1}^m(X)$ in the basis $(X, \ldots, X^{p-1})$. b. Prove that the sequence $(\bar{E}_{2n+1})_{n \in \mathbb{N}}$ is periodic, with minimal period $(p-1)/2$ if $p \equiv 1\,[4]$ and minimal period $(p-1)$ if $p \equiv 3\,[4]$. c. Indicate the modifications to be made to the preceding questions to show an analogous result for the sequence $(\bar{E}_{2n})_{n \in \mathbb{N}}$.
QV.1 Proof Existence Proof View
We denote by $\widehat{S}$ the set of $f \in S_*$ satisfying $\lim_{x \rightarrow -\infty} f(x) = -\infty$ and $\lim_{x \rightarrow +\infty} f(x) = +\infty$. We denote by $\operatorname{Mi}(f)$ the set of minima of $f$ and by $\operatorname{Ma}(f)$ the set of maxima of $f$, so $E(f) = \operatorname{Mi}(f) \cup \operatorname{Ma}(f)$.
1. Let $f \in \widehat{S}$. a. Verify that $\operatorname{Card} \operatorname{Mi}(f) = \operatorname{Card} \operatorname{Ma}(f)$ and that for $y \in \mathbb{R}$, $f^{-1}([-\infty, y])$ is the union of non-empty open intervals that are pairwise disjoint. We denote by $\mathscr{I}(y)$ their set. b. Show that for every element $M$ of $\operatorname{Ma}(f)$, there exists a unique element $m$ of $\operatorname{Mi}(f)$ such that $f(m) < f(M)$ and $m > M$.