grandes-ecoles

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
2024 mines-ponts-maths2__mp

23 maths questions

Q1 Matrices Diagonalizability and Similarity View
Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar. Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar.
Q2 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Justify that an adjacency matrix of a non-empty graph is diagonalizable.
Q3 Matrices Determinant and Rank Computation View
Show that an adjacency matrix of a non-empty graph is never of rank 1.
Q4 Matrices Determinant and Rank Computation View
Show that an adjacency matrix of a graph whose non-isolated vertices form a star-type graph is of rank 2 and represent an example of a graph whose adjacency matrix is of rank 2 and which is not of the previous type.
Q5 Proof Direct Proof of a Stated Identity or Equality View
Let $G$ be a graph and $G ^ { \prime }$ a copy of $G$. Justify that $\chi _ { G } = \chi _ { G ^ { \prime } }$.
Q6 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $G = ( S , A )$ be a graph with $| S | = n \geq 2$. We write $\chi _ { G } ( X ) = X ^ { n } + \sum _ { k = 0 } ^ { n - 1 } a _ { k } X ^ { k }$. Give the value of $a _ { n - 1 }$ and express $a _ { n - 2 }$ in terms of $| A |$.
Q7 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
Deduce the characteristic polynomial of a graph with $n$ vertices whose non-isolated vertices form a star with $d$ branches with $1 \leq d \leq n - 1$. Then determine the eigenvalues and eigenvectors of an adjacency matrix of this graph.
Q8 Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View
Let $G _ { 1 } = \left( S _ { 1 } , A _ { 1 } \right)$ and $G _ { 2 } = \left( S _ { 2 } , A _ { 2 } \right)$ be two non-empty graphs such that $S _ { 1 }$ and $S _ { 2 }$ are disjoint, that is, such that $S _ { 1 } \cap S _ { 2 } = \varnothing$. Let $s _ { 1 } \in S _ { 1 }$ and let $s _ { 2 } \in S _ { 2 }$.
We define the graph $G = ( S , A )$ with $S = S _ { 1 } \cup S _ { 2 }$ and $A = A _ { 1 } \cup A _ { 2 } \cup \left\{ \left\{ s _ { 1 } , s _ { 2 } \right\} \right\}$.
Show that : $$\chi _ { G } = \chi _ { G _ { 1 } } \times \chi _ { G _ { 2 } } - \chi _ { G _ { 1 } \backslash s _ { 1 } } \times \chi _ { G _ { 2 } \backslash s _ { 2 } }$$
Q9 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Determine the characteristic polynomial of the double star with $d _ { 1 } + d _ { 2 } + 2$ vertices, consisting respectively of two disjoint stars with $d _ { 1 }$ and $d _ { 2 }$ branches, to which an additional edge has been added connecting the two centers of the two stars. What is the rank of the adjacency matrix of this double star?
Q10 Probability Definitions Finite Equally-Likely Probability Computation View
Let $G = ( S , A ) \in \Omega _ { n }$. Determine the probability $\mathbf { P } ( \{ G \} )$ of the elementary event $\{ G \}$ in terms of $p _ { n } , q _ { n } , N$ and $a = \operatorname { card } ( A )$. Then recover the fact that $\mathbf { P } \left( \Omega _ { n } \right) = 1$.
Q11 Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds View
Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that $\mathbf{P}(X > 0) \leq \mathbf{E}(X)$.
Q12 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that if $\mathbf{E}(X) \neq 0$, then $\mathbf{P}(X = 0) \leq \frac{\mathbf{V}(X)}{(\mathbf{E}(X))^{2}}$. Hint: note that $(X = 0) \subset (|X - \mathbf{E}(X)| \geq \mathbf{E}(X))$.
Q13 Binomial Distribution Justify Binomial Model and State Parameters View
What is the distribution followed by the random variable $A _ { n }$ representing the number of edges of a graph of $\Omega _ { n }$?
Q14 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Show that if $p _ { n } = o \left( \frac { 1 } { n ^ { 2 } } \right)$ in the neighborhood of $+ \infty$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( A _ { n } > 0 \right) = 0$.
Q15 Continuous Probability Distributions and Random Variables Combinatorial Probability and Limiting Probability View
Show that if $\frac { 1 } { n ^ { 2 } } = \mathrm { o} \left( p _ { n } \right)$ in the neighborhood of $+ \infty$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( A _ { n } > 0 \right) = 1$.
Q16 Modelling and Hypothesis Testing View
Deduce a property $\mathcal { P } _ { n }$ and its associated threshold function.
Q17 Probability Definitions Probability Using Set/Event Algebra View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$
Q18 Combinations & Selection Combinatorial Identity or Bijection Proof View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $\mathcal{C}_0$ be the set of copies of $G_0$ whose vertices are included in $\llbracket 1, n \rrbracket$: $$\mathcal { C } _ { 0 } = \left\{ H \mid H \text { is a copy of } G _ { 0 } \text { and } H = \left( S _ { H } , A _ { H } \right) \text { with } S _ { H } \subset \llbracket 1 , n \rrbracket \right\}$$ Let $S _ { 0 } ^ { \prime }$ be a fixed set of cardinality $s _ { 0 }$. We denote by $c _ { 0 }$ the number of graphs whose vertex set is $S _ { 0 } ^ { \prime }$ and which are copies of $G _ { 0 }$. Express the cardinality of $\mathcal { C } _ { 0 }$ using $c _ { 0 }$ and using a simple upper bound for $c _ { 0 }$, justify that the cardinality of $\mathcal { C } _ { 0 }$ is less than $n ^ { s _ { 0 } }$.
Q19 Discrete Random Variables Expectation and Variance via Combinatorial Counting View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable defined on $\mathcal{E}_n$ such that for $G \in \Omega_n$, the integer $X_n^0(G)$ equals the number of copies of $G_0$ contained in $G$. Express $X _ { n } ^ { 0 }$ using random variables of the type $X _ { H }$, and show that : $$\mathbf { E } \left( X _ { n } ^ { 0 } \right) = \sum _ { H \in \mathcal { C } _ { 0 } } \mathbf { P } ( H \subset G ) \leq n ^ { s _ { 0 } } p _ { n } ^ { a _ { 0 } } .$$
Q20 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable counting the number of copies of $G_0$ contained in $G \in \Omega_n$, and let $$\omega _ { 0 } = \min _ { \substack { H \subset G _ { 0 } \\ a _ { H } \geq 1 } } \frac { s _ { H } } { a _ { H } }$$ Deduce that if $p _ { n } = \mathrm { o } \left( n ^ { - \omega _ { 0 } } \right)$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( X _ { n } ^ { 0 } > 0 \right) = 0$. Hint: one may introduce $H _ { 0 } \subset G _ { 0 }$ achieving the minimum giving $\omega _ { 0 }$.
Q21 Discrete Random Variables Expectation and Variance via Combinatorial Counting View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. Show that the expectation $\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right)$ satisfies : $$\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } p _ { n } ^ { 2 a _ { 0 } - a _ { H \cap H ^ { \prime } } } .$$
Q22 Probability Definitions Proof of a Probability Identity or Inequality View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote: $$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$ Show that $\Sigma _ { 0 } \leq \left( \mathbf{E} \left( X _ { n } ^ { 0 } \right) \right) ^ { 2 }$.
Q23 Probability Definitions Proof of a Probability Identity or Inequality View
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote: $$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$ Let $k \in \llbracket 1 , s _ { 0 } \rrbracket$; show that : $$\Sigma _ { k } \leq \sum _ { H \in \mathcal { C } _ { 0 } } \binom { s _ { 0 } } { k } \binom { n - s _ { 0 } } { s _ { 0 } - k } c _ { 0 } p _ { n } ^ { 2 a _ { 0 } } p _ { n } ^ { - \frac { k } { \omega _ { 0 } } }$$