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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
2014 centrale-maths2__mp

24 maths questions

QI.A.1 Roots of polynomials Polynomial evaluation, interpolation, and remainder View
Determine $T_0, T_1, T_2$ and $T_3$, where the Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $$\forall n \in \mathbb{N}, \quad \forall \theta \in \mathbb{R}, \quad T_n(\cos\theta) = \cos(n\theta)$$
QI.A.2 Roots of polynomials Coefficient and structural properties of special polynomial families View
The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
By noting that for every real $\theta$, we have $e^{in\theta} = \left(e^{i\theta}\right)^n$, show that $$\forall n \in \mathbb{N}, \quad T_n = \sum_{0 \leqslant k \leqslant n/2} \binom{n}{2k} \left(X^2 - 1\right)^k X^{n-2k}$$
Write in Maple or Mathematica language a function $T$ taking as argument a natural integer $n$ and returning the expanded expression of the polynomial $T_n$.
QI.A.3 Roots of polynomials Coefficient and structural properties of special polynomial families View
The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that the sequence $(T_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $$\forall n \in \mathbb{N}, \quad T_{n+2} = 2X T_{n+1} - T_n$$
Deduce, for every natural integer $n$, the degree and the leading coefficient of $T_n$. Find this result again with the expression from question I.A.2.
QI.A.4 Roots of polynomials Location and bounds on roots View
The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that, for every natural integer $n$, the polynomial $T_n$ is split over $\mathbb{R}$, with simple roots belonging to $]-1,1[$. Determine the roots of $T_n$.
QI.B.1 Roots of polynomials Proof of polynomial identity or inequality involving roots View
The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $$\forall n \in \mathbb{N}, \quad U_n = \frac{1}{n+1} T_{n+1}'$$ where $(T_n)_{n \in \mathbb{N}}$ are the Chebyshev polynomials of the first kind defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that $$\forall n \in \mathbb{N}, \quad \forall \theta \in \mathbb{R} \backslash \pi\mathbb{Z}, \quad U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$$
QI.B.2 Roots of polynomials Location and bounds on roots View
The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $U_n = \frac{1}{n+1} T_{n+1}'$, and satisfy $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Deduce the following properties:
a) The sequence $(U_n)_{n \in \mathbb{N}}$ satisfies the same recurrence relation $T_{n+2} = 2X T_{n+1} - T_n$ as the sequence $(T_n)_{n \in \mathbb{N}}$.
b) For every natural integer $n$, the polynomial $U_n$ is split over $\mathbb{R}$ with simple roots belonging to $]-1,1[$. Determine the roots of $U_n$.
QII.A.1 Roots of polynomials Proof of polynomial identity or inequality involving roots View
The Chebyshev polynomials of the first and second kind are defined by $T_n(\cos\theta) = \cos(n\theta)$ and $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Show that $$\begin{cases} T_m \cdot T_n = \frac{1}{2}\left(T_{n+m} + T_{n-m}\right) & \text{for all integers } 0 \leqslant m \leqslant n \\ T_m \cdot U_{n-1} = \frac{1}{2}\left(U_{n+m-1} + U_{n-m-1}\right) & \text{for all integers } 0 \leqslant m < n \end{cases}$$
QII.A.2 Polynomial Division & Manipulation View
The Chebyshev polynomials of the first kind satisfy $T_m \cdot T_n = \frac{1}{2}(T_{n+m} + T_{n-m})$ for $0 \leqslant m \leqslant n$, and $T_m \cdot U_{n-1} = \frac{1}{2}(U_{n+m-1} + U_{n-m-1})$ for $0 \leqslant m < n$.
For $m$ and $n$ natural integers such that $m \leqslant n$, we propose to determine the quotient $Q_{n,m}$ and the remainder $R_{n,m}$ of the Euclidean division of $T_n$ by $T_m$.
a) Suppose $m < n < 3m$. Show that $$Q_{n,m} = 2T_{n-m} \quad \text{and} \quad R_{n,m} = -T_{|n-2m|}$$
b) Determine $Q_{n,m}$ and $R_{n,m}$ when $n$ is of the form $(2p+1)m$ with $p \in \mathbb{N}^*$.
c) Suppose that $m > 0$ and that $n$ is not the product of $m$ by an odd integer. Show that there exists a unique integer $p \geqslant 1$ such that $|n - 2pm| < m$ and that $$Q_{n,m} = 2\left(T_{n-m} - T_{n-3m} + \cdots + (-1)^{p-1} T_{n-(2p-1)m}\right) \quad \text{and} \quad R_{n,m} = (-1)^p T_{|n-2pm|}$$
QII.B.1 Number Theory GCD, LCM, and Coprimality View
Throughout this subsection II.B, we fix two natural integers $m$ and $n$. The Chebyshev polynomials of the second kind $U_n$ have roots $\cos\left(\frac{k\pi}{n+2}\right)$ for $k = 1, \ldots, n+1$.
Let $h$ be the $\gcd$ in $\mathbb{N}$ of $m+1$ and $n+1$. By examining the common roots of $U_n$ and $U_m$, show that $U_{h-1}$ is a $\gcd$ in $\mathbb{R}[X]$ of $U_n$ and $U_m$.
QII.B.2 Number Theory GCD, LCM, and Coprimality View
Throughout this subsection II.B, we fix two natural integers $m$ and $n$. Let $g > 0$ be the $\gcd$ of $m$ and $n$. We set $m_1 = m/g$ and $n_1 = n/g$.
a) Show that if $m_1$ and $n_1$ are odd, then $T_g$ is a $\gcd$ of $T_n$ and $T_m$.
b) Show that if one of the two integers $m_1$ or $n_1$ is even, then $T_n$ and $T_m$ are coprime.
c) What can be said about the gcds of $T_n$ and $T_m$ when $m$ and $n$ are odd? When $n$ and $m$ are two distinct powers of 2?
QIII.A.1 Groups Group Homomorphisms and Isomorphisms View
We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Show that the family $(T_n)_{n \in \mathbb{N}}$ satisfies property (III.1). One may compare $T_n \circ T_m$ and $T_{mn}$.
QIII.A.2 Groups Algebraic Structure Identification View
We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We denote by $G$ the set of complex polynomials of degree 1.
Verify that $G$ is a group for the law $\circ$.
QIII.B.1 Roots of polynomials Coefficient and structural properties of special polynomial families View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition.
Let $\alpha \in \mathbb{C}$ and let $Q$ be a non-constant complex polynomial that commutes with $P_\alpha$. Show that $Q$ is monic.
QIII.B.2 Roots of polynomials Divisibility and minimal polynomial arguments View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition. Every non-constant polynomial commuting with $P_\alpha$ is monic.
Deduce that, for every integer $n \geqslant 1$, there exists at most one polynomial of degree $n$ that commutes with $P_\alpha$. Determine $\mathcal{C}(X^2)$.
QIII.B.3 Roots of polynomials Factored form and root structure from polynomial identities View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$.
Let $P$ be a complex polynomial of degree 2. Justify the existence and uniqueness of $U \in G$ and $\alpha \in \mathbb{C}$ such that $U \circ P \circ U^{-1} = P_\alpha$. Determine these two elements when $P = T_2$.
QIII.B.4 Roots of polynomials Divisibility and minimal polynomial arguments View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with $P$ under composition. The Chebyshev polynomials $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Justify that $\mathcal{C}(T_2) = \{-1/2\} \cup \{T_n, n \in \mathbb{N}\}$.
QIII.C.1 Roots of polynomials Existence or counting of roots with specified properties View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.
Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.
QIII.C.2 Roots of polynomials Factored form and root structure from polynomial identities View
For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Deduce the Block and Thielmann theorem: if $(F_n)_{n \in \mathbb{N}}$ satisfies (III.1), then there exists $U \in G$ such that $$\forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ X^n \circ U \quad \text{or} \quad \forall n \in \mathbb{N}^*, \quad F_n = U^{-1} \circ T_n \circ U$$
QIV.A Matrices Determinant and Rank Computation View
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of the ring $\mathcal{M}_2(\mathbb{Z})$, equipped with its usual addition and multiplication.
Justify that an element $M$ of $\mathcal{M}_2(\mathbb{Z})$ belongs to $\mathrm{GL}_2(\mathbb{Z})$ if and only if $|\det M| = 1$.
QIV.B Roots of polynomials Proof of polynomial identity or inequality involving roots View
We introduce the Dickson polynomials of the first and second kind, $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$, defined in the form of polynomial functions of two variables by $$D_0(x,a) = 2 \quad D_1(x,a) = x \quad E_0(x,a) = 1 \quad E_1(x,a) = x$$ then, for every integer $n \in \mathbb{N}$, $$D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a) \quad \text{and} \quad E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$$
Justify the following relation with Chebyshev polynomials $$\forall (x,a) \in \mathbb{C}^2, \quad D_n\left(2xa, a^2\right) = 2a^n T_n(x) \quad \text{and} \quad E_n\left(2xa, a^2\right) = a^n U_n(x)$$ as well as the following two relations, valid for every natural integer $n$ and every $(x,a) \in \mathbb{C}^* \times \mathbb{C}$ $$D_n\left(x + \frac{a}{x}, a\right) = x^n + \frac{a^n}{x^n} \quad \text{and} \quad \left(x - \frac{a}{x}\right) E_n\left(x + \frac{a}{x}, a\right) = \left(x^{n+1} - \frac{a^{n+1}}{x^{n+1}}\right)$$
QIV.C.1 Matrices Matrix Power Computation and Application View
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The Dickson polynomials $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$ satisfy the recurrences $D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a)$ and $E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$ with $D_0 = 2, D_1(x,a) = x, E_0 = 1, E_1(x,a) = x$.
Let $B \in \mathrm{GL}_2(\mathbb{Z})$. We denote, in this question only, $\sigma = \operatorname{Tr} B$ and $\nu = \det B$. Show for every $n \geqslant 2$, the equality $$B^n = E_{n-1}(\sigma, \nu) \cdot B - \nu E_{n-2}(\sigma, \nu) \cdot I_2$$ where $I_2$ is the identity matrix of order 2.
Establish that $\operatorname{Tr}(B^n) = D_n(\sigma, \nu)$.
QIV.C.2 Matrices Matrix Power Computation and Application View
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. For $B \in \mathrm{GL}_2(\mathbb{Z})$ with $\sigma = \operatorname{Tr} B$ and $\nu = \det B$, we have $B^n = E_{n-1}(\sigma,\nu) \cdot B - \nu E_{n-2}(\sigma,\nu) \cdot I_2$ and $\operatorname{Tr}(B^n) = D_n(\sigma,\nu)$.
We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Deduce that if $A$ is an $n$-th power ($n \geqslant 2$) in $\mathrm{GL}_2(\mathbb{Z})$, then there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that
i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. One will briefly justify that $E_{n-1}(\sigma, \nu)$ is indeed an integer.
ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.
QIV.C.3 Matrices Matrix Power Computation and Application View
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Let $A$ be an element of $\mathrm{GL}_2(\mathbb{Z})$ for which there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ satisfying: i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.
For simplicity, we denote $p = E_{n-1}(\sigma, \nu)$. We then define a matrix $B = \begin{pmatrix} r & s \\ t & u \end{pmatrix}$ with $$r = \frac{1}{2}\left(\sigma + \frac{a-d}{p}\right) \quad s = \frac{b}{p} \quad t = \frac{c}{p} \quad u = \frac{1}{2}\left(\sigma - \frac{a-d}{p}\right)$$
a) By introducing a complex root of the polynomial $X^2 - \sigma X + \nu$ and using the relation $D_n(x + a/x, a) = x^n + a^n/x^n$, show that $$\tau^2 - 4\delta = p^2(\sigma^2 - 4\nu) \quad \text{then} \quad ru - st = \nu$$ Deduce that $B$ belongs to $\mathrm{GL}_2(\mathbb{Z})$.
b) Show that $A = B^n$.
QIV.C.4 Matrices Matrix Power Computation and Application View
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The necessary and sufficient condition for $A \in \mathrm{GL}_2(\mathbb{Z})$ to be an $n$-th power is the existence of $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that $E_{n-1}(\sigma,\nu)$ divides $b$, $c$ and $a-d$, and $\tau = D_n(\sigma,\nu)$, $\delta = \nu^n$.
Show that the matrix $A = \begin{pmatrix} 7 & 10 \\ 5 & 7 \end{pmatrix}$ is a cube in $\mathrm{GL}_2(\mathbb{Z})$ and determine a matrix $B \in \mathrm{GL}_2(\mathbb{Z})$ such that $B^3 = A$.