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-maths__pc

17 maths questions

QI.1 Matrices Combinatorial Structures on Permutation Matrices/Groups View
What is the cardinality of $\mathcal{M}_{n}(\{-1,1\})$? Is this set a vector subspace of $\mathcal{M}_{n}(\mathbb{R})$?
QI.2 Matrices Matrix Entry and Coefficient Identities View
Show that for every matrix $A$ in $\mathcal{M}_{n}(\{-1,1\})$, the set $S(A)$ is included in $\{-n^{2}, \ldots, n^{2}\}$. Show that the inclusion is strict (one may think of a parity argument), and show that $S(A)$ is a symmetric set, in the sense that an integer $k$ is in $S(A)$ if and only if $-k$ is in $S(A)$.
QI.3 Matrices Matrix Algebra and Product Properties View
Let $A$ and $B$ be in $\mathcal{M}_{n}(\{-1,1\})$. Suppose that there exist diagonal matrices $C$ and $D$ containing only 1s and $-1$s on the diagonal, such that $B = CAD$. Show that $S(A) = S(B)$.
QI.4 Matrices Determinant and Rank Computation View
In this question only, we assume $n = 2$, and we denote $$I = \left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \quad \text{and} \quad J = \left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$$ Calculate $S(I)$ and $S(J)$, and deduce $S(A)$ for all $A \in \mathcal{M}_{2}(\{-1,1\})$.
QI.5 Proof Determinant and Rank Computation View
Let $A \in \mathcal{M}_{n}(\{-1,1\})$. Show that the following statements are equivalent:
(a) $n^{2} \in S(A)$.
(b) There exist $X$ and $Y$ in $\{-1,1\}^{n}$ such that $A = X\,{}^{t}Y$.
(c) $A$ is a rank 1 matrix.
QI.6 Proof Determinant and Rank Computation View
Deduce the proportion, among matrices of $\mathcal{M}_{n}(\{-1,1\})$, of matrices $A$ that satisfy $n^{2} \in S(A)$.
QII.2 Probability Definitions Probability Inequality and Tail Bound Proof View
Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$ Let $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$.
Let $t \in \mathbb{R}$. Show that for all $\lambda > 0$, we have the inequality $$\mathbb{P}\left(S_{k} \geqslant t\right) \leqslant \exp(k\varphi(\lambda) - \lambda t).$$
QII.3 Normal Distribution Probability Inequality and Tail Bound Proof View
Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$
Deduce Hoeffding's inequality for $S_{k}$: for all $t > 0$, we have $$\mathbb{P}\left(S_{k} \geqslant t\right) \leqslant \exp\left(-\frac{t^{2}}{2k}\right).$$
QII.6 Proof Probability Inequality and Tail Bound Proof View
We recall the notation $\underline{M}(n) = \min\left\{M(A) \mid A \in \mathcal{M}_{n}(\{-1,1\})\right\}$. Show that for all $n \geqslant 1$, we have $$\underline{M}(n) \leqslant 2\sqrt{\ln 2}\, n^{3/2}.$$
Hint: one may begin by showing that for all $\varepsilon > 0$, there exists a matrix $A$ in $\mathcal{M}_{n}(\{-1,1\})$ such that $$M(A) \leqslant (2\sqrt{\ln 2} + \varepsilon)\, n^{3/2}.$$
QIII.1 Matrices Matrix Entry and Coefficient Identities View
For $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$ and $Y = (y_{i})_{1 \leqslant i \leqslant n} \in \{-1,1\}^{n}$, we denote $$g_{A}(Y) = \max\left\{{}^{t}X A Y \mid X \in \{-1,1\}^{n}\right\}.$$
Show that the function $g_{A}$ can be rewritten as $$g_{A}(Y) = \sum_{i=1}^{n} \left|\sum_{j=1}^{n} a_{i,j} y_{j}\right|.$$
QIII.2 Discrete Probability Distributions Compute Expectation of a Binomial Sum (Algebraic Evaluation) View
We introduce a uniformly distributed random variable $Z : \Omega \rightarrow \{-1,1\}^{n}$. For $\omega \in \Omega$, we denote by $Z_{i}(\omega)$ the coordinates of $Z(\omega)$. Show that for all $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$, we have $$\forall i \in \{1, \ldots, n\}, \quad \mathbb{E}\left[\left|\sum_{j=1}^{n} a_{i,j} Z_{j}\right|\right] = \frac{1}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|,$$ where $\binom{n}{k}$ denotes the binomial coefficient. Deduce $$\mathbb{E}\left[g_{A}(Z)\right] = \frac{n}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|.$$
QIII.3 Binomial Theorem (positive integer n) Functional Equations and Identities via Series View
We introduce a uniformly distributed random variable $Z : \Omega \rightarrow \{-1,1\}^{n}$.
(a) Show that for $m \in \{0, \ldots, n-1\}$, we have $$\sum_{k=0}^{m} (n - 2k) \binom{n}{k} = n \binom{n-1}{m}.$$
(b) Deduce that for all $A \in \mathcal{M}_{n}(\{-1,1\})$, $$\mathbb{E}\left[g_{A}(Z)\right] = \frac{n^{2}}{2^{n-1}} \binom{n-1}{\left\lfloor \frac{n}{2} \right\rfloor}$$ where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor of $\frac{n}{2}$.
QIII.4 Proof Asymptotic Equivalents and Growth Estimates for Sequences/Series View
(a) Show that $$\underline{M}(n) \geqslant \frac{n^{2}}{2^{n-1}} \binom{n-1}{\left\lfloor \frac{n}{2} \right\rfloor}.$$
(b) Show next, using Stirling's formula recalled in the preamble, that this lower bound is equivalent to $C n^{\alpha}$ as $n$ tends to infinity, for constants $C$ and $\alpha > 0$ that one will make explicit. Compare with the upper bound for $\underline{M}(n)$ obtained in question 6 of Part II.
QIV.1 Standard Integrals and Reverse Chain Rule Evaluate a Closed-Form Expression Using the Reduction Formula View
For $n \in \mathbb{N}$, we set $$I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx.$$ Determine by induction $I_{n}$ for all $n \in \mathbb{N}$.
QIV.2 Standard Integrals and Reverse Chain Rule Perform a Change of Variable or Transformation on a Parametric Integral View
For $n \in \mathbb{N}$, we set $I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx$.
Show that for $n \geqslant 1$, we have $$I_{n} = \left(\frac{n}{e}\right)^{n} \sqrt{n} \int_{-\sqrt{n}}^{+\infty} \left(1 + \frac{x}{\sqrt{n}}\right)^{n} e^{-x\sqrt{n}}\, dx.$$
QIV.3 Stationary points and optimisation Partial derivatives and multivariable differentiation View
Let $U$ be the open set of $\mathbb{R}^{2}$ defined by $$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$ and let $f$ be the function defined on $U$ by $$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$
(a) Show that for $(t, x) \in U$, we have $$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$
(b) For $x > 0$, show that we have $$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$ For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.
QV.1 Proof Matrix Norm, Convergence, and Inequality View
We fix $A \in \mathcal{M}_{n}(\{-1,1\})$ and denote $$m(A) := \min(S(A) \cap \mathbb{N}).$$
For $Y \in \{-1,1\}^{n}$, show that we have $$\min\left\{\left|{}^{t}X A Y\right| \mid X \in \{-1,1\}^{n}\right\} \leqslant n$$ and deduce $m(A) \leqslant n$.