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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
2021 centrale-maths2__official

29 maths questions

Q1 Proof by induction 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 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 Sequences and Series 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 Proof 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 $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. For all $\lambda$ in $\mathbb{C}$, we denote by $Q_\lambda$ the quotient of $P_\lambda$ 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 Roots of polynomials 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 $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. For all $\lambda$ in $\mathbb{C}$, $Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$. 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 (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 Roots of polynomials 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 $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} + nP(\lambda)$$ One may apply equality (I.2) to the polynomial $X^{2n}$.
Q12 Complex numbers 2 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 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
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 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})}$$
Q15 Sequences and Series 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 Sequences and Series 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 Sequences and Series 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 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$
Q23 Sequences and Series Evaluation of a Finite or Infinite Sum View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. The convolution product of $f$ and $g$ is defined by $$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(t) g(x-t) \,\mathrm{d}t$$
Show that $f * g$ is defined on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(x-t) g(t) \,\mathrm{d}t = (g * f)(x)$$
Q24 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f * g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.
Q25 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f * g)^{(k)} = f * (g^{(k)})$.
Q27 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$
Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.
Q28 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin ]-1,1[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$
Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.
Q29 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $\theta$ be the unique antiderivative of $\psi$ vanishing at 0. Show that $\theta$ is of class $\mathcal{C}^\infty$, constant on $]-\infty, -1]$ (we denote this constant by $A$) and constant on $[1, +\infty[$ (we denote this constant by $B$). Verify that $A \neq B$.
Q30 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Construct a function $\rho \in \mathcal{C}^\infty(\mathbb{R})$, constant equal to 1 on $[-1,1]$ and constant equal to 0 on $\mathbb{R} \setminus [-2,2]$.
Q31 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $r$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that, for all real $x$, $$r(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \mathrm{e}^{\mathrm{i}x\xi} \rho(\xi) \,\mathrm{d}\xi$$ where $\rho \in \mathcal{C}^\infty(\mathbb{R})$ is constant equal to 1 on $[-1,1]$ and constant equal to 0 on $\mathbb{R} \setminus [-2,2]$.
Show that $r$ is differentiable on $\mathbb{R}$ and give an expression for its derivative function (possibly involving an integral).
Q32 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $r$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that, for all real $x$, $$r(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \mathrm{e}^{\mathrm{i}x\xi} \rho(\xi) \,\mathrm{d}\xi$$ where $\rho \in \mathcal{C}^\infty(\mathbb{R})$ is constant equal to 1 on $[-1,1]$ and constant equal to 0 on $\mathbb{R} \setminus [-2,2]$.
Show that $x \mapsto x^2 r(x)$ is bounded on $\mathbb{R}$ and deduce that $r$ is integrable and bounded on $\mathbb{R}$.
Q33 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$, where $r(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \mathrm{e}^{\mathrm{i}x\xi} \rho(\xi) \,\mathrm{d}\xi$.
We admit that $f * r_\lambda$ is integrable. Show that $f = \lambda f * r_\lambda$.
Q34 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$.
Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^\star$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$