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
2018 x-ens-maths2__mp

13 maths questions

Q5 Sequences and Series Derive a Reduction/Recurrence Formula via Integration by Parts View
We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
We denote $$g_{j} = \int_{-1}^{1} P_{j}(x)^{2}\,dx, \quad I_{j} = \int_{-1}^{1} \left(1 - x^{2}\right)^{j}\,dx$$
(a) Establish a relation between $g_{j}$ and $I_{j}$.
(b) Find a relation between $I_{j}$ and $I_{j-1} - I_{j}$, and deduce a recurrence relation for the sequence $\left(I_{j}\right)_{j \in \mathbb{N}}$.
(c) Deduce the value of $I_{j}$, then that of $g_{j}$.
Q6 Sequences and Series Matrix Decomposition and Factorization View
We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. The subspace of $\mathbb{R}_{n}[X]$ formed by even polynomials is denoted $\Pi_{n}$, and that of odd polynomials is denoted $J_{n}$.
(a) Show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is a basis of $\mathbb{R}_{n}[X]$.
(b) Deduce that the family $\left(P_{2j}\right)_{0 \leqslant j \leqslant \frac{n}{2}}$ is a basis of $\Pi_{n}$, while the family $\left(P_{2j+1}\right)_{0 \leqslant j \leqslant \frac{n-1}{2}}$ is a basis of $J_{n}$.
Q7 Polynomial Division & Manipulation Factored form and root structure from polynomial identities View
We choose an even polynomial in $B_{N}$ (see question 2(c)), and we denote it $R_{N}$.
Show that there exist non-negative integers $r, s, t \geqslant 0$, real numbers $c_{1}, \ldots, c_{r}$ different from $\pm 1$, non-zero reals $\rho_{1}, \ldots, \rho_{s}$ and complex numbers $w_{1}, \ldots, w_{t}$ that are neither real nor purely imaginary, such that $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
Q8 Polynomial Division & Manipulation Direct Proof of an Inequality View
We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
We decide to replace all $\rho_{k}$ by zeros. We thus replace the corresponding factors of $R_{N}$, $$\frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}},$$ by factors $X^{2}$. We thus obtain a new polynomial $S_{N}$ of the same degree as $R_{N}$.
Show that $0 \leqslant S_{N}(x) \leqslant R_{N}(x)$ for all $x \in [-1,1]$, then that $S_{N} \in B_{N}$.
Q9 Polynomial Division & Manipulation Direct Proof of an Inequality View
We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$ After replacing all $\rho_k$ by zeros we obtained $S_N$.
Similarly, in the list of $c_{j}$, we decide to replace those that do not belong to $[-1,1]$ by zeros. We thus replace the corresponding factors of $S_{N}$, $$\frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}}$$ by factors $X^{2}$. We thus obtain a new polynomial $T_{N}$.
Show that $0 \leqslant T_{N}(x) \leqslant S_{N}(x)$ for all $x \in [-1,1]$, then that $T_{N} \in B_{N}$.
Q10 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let $w \in \mathbb{C}$ be a number that is neither real nor purely imaginary.
(a) Show that the equation $$\left|\frac{z-1}{z+1}\right| = \left|\frac{w-1}{w+1}\right|$$ defines a circle in the complex plane, which passes through $w$. Verify that the interval $]-1,1[$ intersects this circle at a unique point; we denote this point by $y$. We will express $y$ in terms of the number $$\lambda = \left|\frac{w-1}{w+1}\right|.$$
(b) Show the inequality $$\left|\frac{1-w}{1-y}\right| > 1.$$
(c) Show that the equation $$\left|\frac{z-w}{z-y}\right| = \left|\frac{1-w}{1-y}\right|$$ defines a circle in the complex plane, which passes through $1$ and through $-1$.
Deduce that, for all $x \in [-1,1] \setminus \{y\}$, we have $$\left|\frac{w-x}{y-x}\right| \geqslant \left|\frac{w-1}{y-1}\right| = \left|\frac{w+1}{y+1}\right|$$
Q11 Polynomial Division & Manipulation Location and bounds on roots View
We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
Using the results of questions 8, 9, and 10, conclude that $R_{N}$ has all its roots in the interval $[-1,1]$.
Q12 Polynomial Division & Manipulation Coefficient and structural properties of special polynomial families View
We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$).
Show that $\deg R_{N} = 2n$.
Q13 Polynomial Division & Manipulation Multiplicity and derivative analysis of roots View
We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$, with all roots in $[-1,1]$).
Show that $R_{N}$ is the square of a polynomial: $R_{N}(X) = U_{N}(X)^{2}$ where $U_{N}(1) = 1$ and $U_{N}(-1) = \pm 1$. What can we say about the parity of $U_{N}$?
Q14 Sequences and Series Proof of polynomial identity or inequality involving roots View
We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We assume in this question that $U_{N}$ is even; we thus have $U_{N} \in \Pi_{n}$. In $\Pi_{n}$, the equation $P(1) = 1$ defines an affine subspace denoted $H_{n}$.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
(a) Show that $$\left\|U_{N}\right\|_{2} = \min\left\{\|P\|_{2} \mid P \in H_{n}\right\}$$
(b) Deduce that there exists a real number $\mu$ such that for all integers $0 \leqslant j \leqslant \frac{n}{2}$, we have $\left\langle U_{N}, P_{2j} \right\rangle = \mu$. (One may consider polynomials $P \in H_{n}$ of the form $U_{N} + t\left(P_{2j} - P_{2k}\right)$ with $t \in \mathbb{R}$.)
(c) Express $U_{N}$ in the basis of $P_{2j}$. Deduce that $$\frac{1}{\mu} = \sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}$$
(d) Establish in this case the formula $$a_{N} = \left(\sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}\right)^{-1}.$$
Q15 Proof by induction Proof of polynomial identity or inequality involving roots View
We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We now assume that $U_{N}$ is odd.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Express $a_{N}$ again in terms of the $g_{\ell}$.
Q16 Proof by induction Evaluation of a Finite or Infinite Sum View
We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Discuss, depending on the parity of $n$, the value of $a_{N}$. We will give its explicit value.
Q17 Proof by induction Evaluation of a Finite or Infinite Sum View
We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$.
Give the explicit formula for $R_{N}$, in terms of the polynomials $P_{j}$.