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
2010 centrale-maths1__pc

23 maths questions

QI.A.1 Trig Graphs & Exact Values Trigonometric Identity Proof or Derivation View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.
b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.
c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.
d) Specify the parity properties of $F_n$ as a function of $n$.
QI.A.2 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Calculate $F_{n+1}(x) + F_{n-1}(x)$ for all $n \in \mathbb{N}^*$ and all $x \in D$.
QI.A.3 Polynomial Division & Manipulation Formal power series manipulation (Cauchy product, algebraic identities) View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Deduce from the above that $F_n$ extends to $\mathbb{R}$ as a unique polynomial function, whose degree and leading coefficient should be specified.
QI.A.5 Sequences and series, recurrence and convergence Recursive or implicit derivative computation for series coefficients View
Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Determine two real numbers $a$ and $b$ such that $$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$
QI.B.1 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $n \in \mathbb{N}$.
a) Show that the function $F_n$ is of class $C^\infty$ on $\mathbb{R}$.
b) For $x \in ]-1,1[$, give a simple expression for $F_n'(x)$. Justify the calculation carefully.
QI.B.2 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.
QI.B.3 Differentiating Transcendental Functions Higher-order or nth derivative computation View
Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds: $$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$
QI.C.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function: $$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$ is integrable on the interval $]-1,1[$.
QI.C.2 Indefinite & Definite Integrals Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.
QI.C.3 Indefinite & Definite Integrals Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:
  • [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
  • [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.
QI.C.4 Indefinite & Definite Integrals Prove Orthogonality or Algebraic Relations Between Integral-Defined Objects View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$. For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ and $T_0(x) = 1$. The space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $(f|g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
Calculate $(T_m | T_n)$ for all $(m, n) \in \mathbb{N} \times \mathbb{N}$. What can we deduce from this?
QII.A.1 Trig Proofs Direct Proof of an Inequality View
We seek to show that the inequality $|\sin(n\theta)| \leqslant n \sin(\theta)$ is satisfied for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2}\right]$.
a) Show that $\sin(n\theta) \leqslant n \sin(\theta)$ for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2n}\right]$.
b) Show that, for all $\theta \in \left[0, \frac{\pi}{2}\right]$, we have $\sin(\theta) \geqslant \frac{2}{\pi} \theta$.
c) Deduce that: $$\forall \theta \in \left[\frac{\pi}{2n}, \frac{\pi}{2}\right], \quad 1 \leqslant n \sin(\theta)$$
d) Conclude.
e) For which values of $\theta \in \left[0, \frac{\pi}{2}\right]$ do we have $|\sin(n\theta)| = n \sin(\theta)$?
QII.A.2 Stationary points and optimisation Higher-order or nth derivative computation View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.
QII.A.3 Proof Location and bounds on roots View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Let $n \in \mathbb{N}^*$. Show that the polynomial function $T_n$ has exactly $n$ distinct zeros all belonging to $]-1,1[$. For $j \in \{1, 2, \ldots, n\}$, we denote by $x_{n,j}$ the $j$-th zero of $T_n$ in increasing order. Give the value of $x_{n,j}$.
QII.A.4 Proof Proof of polynomial identity or inequality involving roots View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Let $n \in \mathbb{N}^*$ and $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$. Show that: $$\frac{T_n'(x)}{T_n(x)} = \sum_{j=1}^{n} \frac{1}{x - x_{n,j}}$$
QII.A.5 Proof Polynomial evaluation, interpolation, and remainder View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
Let $n \in \mathbb{N}^*$, $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$ and $P \in E_{n-1}$.
a) Show that: $$P(x) = \sum_{j=1}^{n} \frac{P(x_{n,j})}{T_n'(x_{n,j})} \frac{T_n(x)}{x - x_{n,j}}$$
b) Deduce that: $$P(x) = \frac{2^{n-1}}{n} \sum_{j=1}^{n} (-1)^{n-j} \sqrt{1 - x_{n,j}^2}\, P(x_{n,j}) \frac{T_n(x)}{x - x_{n,j}}$$
QII.B.1 Proof Direct Proof of an Inequality View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Show that, for all $x \in [x_{n,1}, x_{n,n}]$, we have $$\sqrt{1 - x^2} \geq \frac{1}{n}$$
QII.B.2 Proof Bounding or Estimation Proof View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$
QII.B.3 Proof Deduction or Consequence from Prior Results View
Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
QII.C Proof Direct Proof of an Inequality View
Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)
QIII.A.1 Sequences and series, recurrence and convergence Convergence/Divergence Determination of Numerical Series View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$.
Show that the numerical series $\sum_{n \in \mathbb{N}} n^j \alpha_n$ is convergent.
QIII.A.2 Sequences and series, recurrence and convergence Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$. We set, for $n \in \mathbb{N}$, $$R_n(j) = \sum_{p=n+1}^{+\infty} p^j \alpha_p$$
Show that the sequence $(R_n(j))_{n \in \mathbb{N}}$ has rapid decay.
QIII.B.1 Sequences and series, recurrence and convergence Uniform or Pointwise Convergence of Function Series/Sequences View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.