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
2021 centrale-maths2__mp

21 maths questions

Q1 Proof Prove a general algebraic or analytic statement by induction View
For all $n$ in $\mathbb{N}$, determine the degree of $T_n$, then show that $\left(T_k\right)_{0 \leqslant k \leqslant n}$ is a basis of $\mathbb{C}_n[X]$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.
Q2 Proof by induction Proof by Induction or Recursive Construction View
Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.
Q3 Proof Deduction or Consequence from Prior Results View
Deduce that, for all $n \in \mathbb{N}$ and $P \in \mathbb{C}_n[X]$, the function from $\mathbb{R}$ to $\mathbb{C}$, $\theta \mapsto P(\cos\theta)$ is in $\mathcal{S}_n$.
Recall that $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$
Q4 Proof Recurrence Relations and Sequence Properties View
For $n \in \mathbb{N}$, calculate $\left\|T_n\right\|_{L^\infty([-1,1])}$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.
Q5 Proof Direct Proof of a Stated Identity or Equality View
Show that, for all $n \in \mathbb{N}$, $\left\|T_n'\right\|_{L^\infty([-1,1])} = n^2$.
One may begin by establishing that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $|\sin(n\theta)| \leqslant n|\sin\theta|$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.
Q6 Factor & Remainder Theorem Direct Proof of a Stated Identity or Equality View
Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$
Q7 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View
Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and, for all $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. If $\lambda \in \mathbb{C}$, verify that $X - 1$ divides $P_\lambda$.
Q8 Factor & Remainder Theorem Remainder Theorem with Composed or Shifted Arguments View
Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda$ be the quotient of $P_\lambda = P(\lambda X) - P(\lambda)$ by $X-1$: $$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$ Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.
Q9 Factor & Remainder Theorem Factorization and Root Analysis View
Let $n$ be a non-zero natural number. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$R(X) = \prod_{k=1}^{2n}(X - \omega_k)$$
Q10 Factor & Remainder Theorem Polynomial evaluation, interpolation, and remainder View
Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X-1} \in \mathbb{C}_{2n-1}[X]$. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Using formula $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)} \tag{I.1}$$ show that $$\forall \lambda \in \mathbb{C}, \quad Q_\lambda(X) = -\frac{1}{2n} \sum_{k=1}^{2n} \frac{P(\lambda\omega_k) - P(\lambda)}{\omega_k - 1} \frac{X^{2n}+1}{X - \omega_k} \omega_k$$ then deduce that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1-\omega_k)^2} - \frac{P(\lambda)}{2n} \sum_{k=1}^{2n} \frac{2\omega_k}{(1-\omega_k)^2}.$$
Q11 Factor & Remainder Theorem Proof of polynomial identity or inequality involving roots View
Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1-\omega_k)^2} + nP(\lambda) \tag{I.2}$$ One may apply equality (I.2) to the polynomial $X^{2n}$.
Q12 Factor & Remainder Theorem Roots of Unity and Cyclotomic Properties View
Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$, where $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$ Show that there exists $U \in \mathbb{C}_{2n}[X]$ such that, for all $\theta \in \mathbb{R}$, $f(\theta) = \mathrm{e}^{-\mathrm{i}n\theta} U(\mathrm{e}^{\mathrm{i}\theta})$.
Q13 Factor & Remainder Theorem Roots of Unity and Cyclotomic Properties View
Let $n$ be a non-zero natural number. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Let $f \in \mathcal{S}_n$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1-\omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$
Q14 Factor & Remainder Theorem Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Let $f \in \mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})} \tag{I.4}$$
Q15 Proof Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Deduce from questions 3 and 14 that $$\forall P \in \mathbb{C}_n[X], \quad \forall x \in [-1,1], \quad \left|P'(x)\sqrt{1-x^2}\right| \leqslant n \|P\|_{L^\infty([-1,1])}$$
Q16 Proof Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Show that $$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$ One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.
Q17 Proof Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that $$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$ One may consider the polynomial $S_t(X) = R(tX)$.
Q18 Proof Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$
Q19 Proof Existence Proof View
Let $n$ be a non-zero natural number. Can there be equality in the inequality $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}?$$
Q20 Proof Proof That a Map Has a Specific Property View
Let $f \in L^1(\mathbb{R})$, where the Fourier transform of $f$ is defined by $$\forall \xi \in \mathbb{R}, \quad \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i}x\xi} \,\mathrm{d}x$$ Show that, for every function $f \in L^1(\mathbb{R})$, $\hat{f}$ is defined and continuous on $\mathbb{R}$.
Q21 Proof Proof That a Map Has a Specific Property View
Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.