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
2023 centrale-maths1__official

37 maths questions

Q1 Proof Automorphism and Endomorphism Structure View
Let $a \in \mathbb{K}$. For all $p \in \mathbb{K}[X]$, we set $E_a(p) = E_a p = p(X+a)$.
Show that $E_a$ is an automorphism of $\mathbb{K}[X]$.
Q2 Linear transformations View
To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ is an endomorphism of $\mathbb{R}[X]$.
Q3 Proof Proof That a Map Has a Specific Property View
To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ preserves degree and that $J$ is invertible.
Q4 Proof Compute a Base Case or Specific Value of a Parametric Integral View
To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $\int_0^{+\infty} \mathrm{e}^{-t} t^k\,\mathrm{d}t$ exists for all $k \in \mathbb{N}$ and calculate its value.
Q5 Linear transformations View
To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $L$ is an endomorphism of $\mathbb{K}[X]$. Is it invertible?
Q6 Linear transformations View
Let $a \in \mathbb{K}$. Verify that the endomorphisms $I$ and $D$ are shift-invariant, as well as the endomorphisms $E_a$, $J$ and $L$ defined in part I. Are they delta endomorphisms?
Recall: $T$ is shift-invariant if for all $a \in \mathbb{K}$, $E_a \circ T = T \circ E_a$. $T$ is a delta endomorphism if $T$ is shift-invariant and $TX \in \mathbb{K}^*$.
Q7 Proof Algebra and Subalgebra Proofs View
Show that the set of shift-invariant endomorphisms of $\mathbb{K}[X]$ is a subalgebra of $\mathcal{L}(\mathbb{K}[X])$. Is the set of delta endomorphisms of $\mathbb{K}[X]$ closed under addition? under composition?
Q8 Proof Properties and Manipulation of Power Series or Formal Series View
Let $\left(a_k\right)_{k \in \mathbb{N}}$ be a sequence of elements of $\mathbb{K}$. For every polynomial $p \in \mathbb{K}[X]$, show that the expression $$\sum_{k=0}^{+\infty} a_k D^k p$$ makes sense and defines a polynomial of $\mathbb{K}[X]$.
Q9 Proof Properties and Manipulation of Power Series or Formal Series View
Show that, for every sequence $\left(a_k\right)_{k \in \mathbb{N}}$ of elements of $\mathbb{K}$, $\sum_{k=0}^{+\infty} a_k D^k$ is a shift-invariant endomorphism.
Q10 Proof Properties and Manipulation of Power Series or Formal Series View
Let $\left(a_k\right)_{k \in \mathbb{N}}$ and $\left(b_k\right)_{k \in \mathbb{N}}$ be sequences of elements of $\mathbb{K}$ such that $\sum_{k=0}^{+\infty} a_k D^k = \sum_{k=0}^{+\infty} b_k D^k$.
Show that, for all $k \in \mathbb{N}$, $a_k = b_k$.
Q11 Proof Properties and Manipulation of Power Series or Formal Series View
For every $n \in \mathbb{N}$, define the polynomial $q_n = \frac{X^n}{n!}$. Let $T$ be an endomorphism of $\mathbb{K}[X]$.
Show that $T$ is a shift-invariant endomorphism if, and only if, $$T = \sum_{k=0}^{+\infty} \left(T q_k\right)(0) D^k$$
Q12 Proof Matrix Algebra and Product Properties View
Show that two shift-invariant endomorphisms of $\mathbb{K}[X]$ commute.
Q13 Proof Formal power series manipulation (Cauchy product, algebraic identities) View
For all $p \in \mathbb{K}[X]$ non-zero and $a \in \mathbb{K}$, show, using question 11, that $$p(X+a) = \sum_{k=0}^{\deg(p)} \frac{a^k}{k!} p^{(k)}$$ where $p^{(k)}$ denotes the $k$-th derivative of the polynomial $p$. Recognize this formula.
Q14 Proof Integral Involving a Parameter or Operator Identity View
For $p \in \mathbb{K}[X]$, express $Jp$ in terms of the derivatives $p^{(k)}$ ($k \in \mathbb{N}$) of $p$, where $J$ is defined by $Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$.
Q15 Proof Existence or Properties of Functions and Inverses (Proof-Based) View
Prove that the endomorphism $D - I$ is invertible and express $L$ in terms of $(D-I)^{-1}$, where $L$ is defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.
Q16 Proof Proof That a Map Has a Specific Property View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$
Q17 Proof Deduction or Consequence from Prior Results View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$, and let $n(T)$ be the natural number such that $\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$ for every $p \in \mathbb{K}[X]$.
Deduce $\ker(T)$ in terms of $n(T)$.
Q18 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.
Show that the following three assertions are equivalent:
  1. [(1)] $T$ is invertible;
  2. [(2)] $T1 \neq 0$;
  3. [(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$.
Q19 Linear transformations View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$ that is invertible. Show that $T^{-1}$ is still a shift-invariant endomorphism.
Q20 Linear transformations View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a sequence of scalars $\left(\alpha_k\right)_{k \in \mathbb{N}}$ satisfying $\alpha_0 = 0$, $\alpha_1 \neq 0$ and $T = \sum_{k=1}^{+\infty} \alpha_k D^k$.
Q21 Proof Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a unique shift-invariant and invertible endomorphism $U$ such that $T = D \circ U$. Specify $U$ in the case $T = D$, then in the case $T = L$.
Q22 Proof Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
For every polynomial $p \in \mathbb{K}[X]$ non-zero, verify that $\deg(Tp) = \deg(p) - 1$. Deduce $\ker(T)$ and the spectrum of $T$.
Q23 Matrices Diagonalizability determination or proof View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, we denote by $T_n$ the restriction of $T$ to $\mathbb{K}_n[X]$.
Show that $T_n$ is an endomorphism of $\mathbb{K}_n[X]$. Is it diagonalizable?
Q24 Matrices Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, let $T_n$ denote the restriction of $T$ to $\mathbb{K}_n[X]$.
Determine $\operatorname{Im}(T_n)$ in terms of $n \in \mathbb{N}$ and deduce that $T$ is surjective.
Q25 Matrices Properties and Manipulation of Power Series or Formal Series View
We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:
  • $q_0 = 1$;
  • $\forall n \in \mathbb{N}, \deg(q_n) = n$;
  • $\forall n \in \mathbb{N}^*, q_n(0) = 0$;
  • $\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.

Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.
Q26 Proof Functional Equations and Identities via Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that, for every natural number $n$, $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Q27 Matrices Properties and Manipulation of Power Series or Formal Series View
Let $(q_n)_{n \in \mathbb{N}}$ be a sequence of polynomials of $\mathbb{K}[X]$ such that $\forall n \in \mathbb{N}, \deg(q_n) = n$ and $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Show that there exists a unique delta endomorphism $Q$ for which $(q_n)_{n \in \mathbb{N}}$ is the associated sequence of polynomials.
Q28 Proof Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number.
Show that the family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
Q29 Matrices Determinant and Rank Computation View
Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number. The family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
According to question 23, $Q$ induces an endomorphism of $\mathbb{K}_n[X]$ denoted $Q_n$. Give its matrix in the previous basis. Deduce its trace, its determinant and its characteristic polynomial.
Q30 Proof Properties and Manipulation of Power Series or Formal Series View
For $Q = D$, verify that $$\forall n \in \mathbb{N}, \quad q_n = \frac{X^n}{n!}$$
Q31 Proof Properties and Manipulation of Power Series or Formal Series View
For $Q = E_1 - I$, verify that $$\forall n \in \mathbb{N}^*, \quad q_n = \frac{X(X-1)\cdots(X-n+1)}{n!}$$
Q32 Proof Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Prove that, for all $p \in \mathbb{K}[X]$, the expression $\sum_{k=0}^{+\infty} (Q^k p)(0) q_k$ makes sense and defines a polynomial of $\mathbb{K}[X]$, then that $$p = \sum_{k=0}^{+\infty} (Q^k p)(0) q_k$$
Q34 Proof Functional Equations and Identities via Series View
By choosing $Q = E_1 - I$, prove that, if $p$ is a non-constant polynomial, then $$p'(X) = \sum_{k=1}^{\deg(p)} \frac{1}{k} \left( \sum_{j=0}^k (-1)^{j+1} \binom{k}{j} p(X+j) \right)$$ This is the formula for numerical differentiation of polynomials.
Q35 Proof Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
Show that, if there exists $(a_n)_{n \in \mathbb{N}}$ a sequence of scalars such that $T = \sum_{k=0}^{+\infty} a_k D^k$, then $T' = \sum_{k=1}^{+\infty} k a_k D^{k-1}$.
Q36 Proof Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a shift-invariant endomorphism, show that $T'$ is still a shift-invariant endomorphism.
Q37 Proof Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a delta endomorphism, show that $T'$ is a shift-invariant and invertible endomorphism.
Q38 Proof Properties and Manipulation of Power Series or Formal Series View
Let $S$ and $T$ be two endomorphisms of $\mathbb{K}[X]$. The Pincherle derivative of an endomorphism $T$ is defined by $T'(p) = T(Xp) - XT(p)$.
Verify that $(S \circ T)' = S' \circ T + S \circ T'$.