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
2023 mines-ponts-maths1__psi

23 maths questions

Q1 Groups Symplectic and Orthogonal Group Properties View
Show that a matrix $S \in S _ { n } ( \mathbf { R } )$ belongs to $S _ { n } ^ { + } ( \mathbf { R } )$ if, and only if, $\operatorname { Sp } ( S ) \subset \mathbf { R } _ { + }$.
Q2 Matrices Structured Matrix Characterization View
Show that $S _ { n } ^ { + } ( \mathbf { R } )$ and $S _ { n } ^ { + + } ( \mathbf { R } )$ are convex subsets of $M _ { n } ( \mathbf { R } )$. Are they vector subspaces of $M _ { n } ( \mathbf { R } )$ ?
Q3 Matrices Matrix Decomposition and Factorization View
Show that, if $A \in S _ { n } ^ { + + } ( \mathbf { R } )$, there exists $S \in S _ { n } ^ { + + } ( \mathbf { R } )$ such that $A = S ^ { 2 }$.
Q4 Proof by induction Prove a general algebraic or analytic statement by induction View
Let $I$ be an interval of $\mathbf { R }$. Let $f : I \rightarrow \mathbf { R }$ be a convex function. Show that, for all $p \in \mathbf { N } ^ { \star }$, for all $\left( \lambda _ { 1 } , \ldots , \lambda _ { p } \right) \in \left( \mathbf { R } _ { + } \right) ^ { p }$ such that $\sum _ { i = 1 } ^ { p } \lambda _ { i } = 1$ and for all $\left( x _ { 1 } , \ldots , x _ { p } \right) \in I ^ { p }$, we have:
$$f \left( \sum _ { i = 1 } ^ { p } \lambda _ { i } x _ { i } \right) \leq \sum _ { i = 1 } ^ { p } \lambda _ { i } f \left( x _ { i } \right)$$
Hint: You may proceed by induction on $p$.
Q5 Proof by induction Prove a general algebraic or analytic statement by induction View
Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. Show the inequality $\frac { \operatorname { Tr } ( M ) } { n } \geq \operatorname { det } ^ { 1 / n } ( M )$.
Hint: You may show that $x \mapsto - \ln ( x )$ is convex on $\mathbf { R } _ { + } ^ { \star }$.
Q7 Matrices Matrix Norm, Convergence, and Inequality View
Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. In the rest of this part, you may use without proof the inequality below $\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbf { R } _ { + } \right) ^ { n }$,
$$2 \max \left\{ x _ { 1 } , \ldots , x _ { n } \right\} \left( \frac { 1 } { n } \sum _ { k = 1 } ^ { n } x _ { k } - \prod _ { k = 1 } ^ { n } x _ { k } ^ { 1 / n } \right) \geq \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \left( x _ { k } - \prod _ { j = 1 } ^ { n } x _ { j } ^ { 1 / n } \right) ^ { 2 }$$
Deduce that
$$\frac { \operatorname { Tr } ( M ) } { n } - \operatorname { det } ^ { 1 / n } ( M ) \geq \frac { \left\| M - \operatorname { det } ^ { 1 / n } ( M ) I _ { n } \right\| _ { 2 } ^ { 2 } } { 2 n \| M \| _ { 2 } }$$
Q8 Matrices Matrix Decomposition and Factorization View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $B \in S _ { n } ( \mathbf { R } )$. Show that there exists a diagonal matrix $D \in M _ { n } ( \mathbf { R } )$ and $Q \in G L _ { n } ( \mathbf { R } )$ such that $B = Q D Q ^ { \top }$ and $A = Q Q ^ { \top }$. What can be said about the diagonal elements of $D$ if $B \in S _ { n } ^ { + + } ( \mathbf { R } )$ ?
Hint: You may use question 3.
Q9 Applied differentiation Convexity and inflection point analysis View
Study the convexity of the function $t \mapsto \ln \left( 1 + \mathrm { e } ^ { t } \right)$.
Q10 Proof Direct Proof of an Inequality View
Show the inequality
$$\forall ( A , B ) \in S _ { n } ^ { + + } ( \mathrm { R } ) ^ { 2 } , \quad \operatorname { det } ^ { 1 / n } ( A + B ) \geq \operatorname { det } ^ { 1 / n } ( A ) + \operatorname { det } ^ { 1 / n } ( B )$$
Q11 3x3 Matrices Determinant of Parametric or Structured Matrix View
Show that, if $A$ and $B$ belong to $S _ { n } ^ { + + } ( \mathrm { R } )$, then:
$$\forall t \in [ 0,1 ] , \quad \operatorname { det } ( ( 1 - t ) A + t B ) \geq \operatorname { det } ( A ) ^ { 1 - t } \operatorname { det } ( B ) ^ { t }$$
Justify that this inequality remains valid for $A$ and $B$ only in $S _ { n } ^ { + } ( \mathbf { R } )$.
Q12 3x3 Matrices Determinant of Parametric or Structured Matrix View
What can we deduce about the function $\ln \circ \det$ on $S _ { n } ^ { + + } ( \mathbf { R } )$ ?
Q13 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $g : t \in \mathbf { R } \mapsto \operatorname { det } \left( I _ { n } + t A \right)$. Express, for all $t \in \mathbf { R } , g ( t )$ using the eigenvalues of $A$. Deduce that $g$ is of class $C ^ { \infty }$ on $\mathbf { R }$.
Q14 Differentiating Transcendental Functions Prove inequality or sign of transcendental expression View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $f : t \mapsto \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right)$. Show that
$$\forall t \in \mathbf { R } _ { + } , \quad \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right) \leq \operatorname { Tr } ( A ) t$$
Q15 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.
Q16 Proof Existence Proof View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that there exists $\varepsilon _ { 0 } > 0$ such that, for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$.
Q17 Proof Computation of a Limit, Value, or Explicit Formula View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A } ( t ) \underset { t \rightarrow 0 } { = } \operatorname { det } ( A ) + \operatorname { det } ( A ) \operatorname { Tr } \left( A ^ { - 1 } M \right) t + o ( t )$.
Hint: You may begin by treating the case where $A = I _ { n }$.
Q18 Matrices Determinant and Rank Computation View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Determine $f _ { A } ^ { \prime } ( t )$ for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$.
Q19 Matrices Linear System and Inverse Existence View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. We admit that the function $\Phi : t \mapsto ( A + t M ) ^ { - 1 }$ is of class $C ^ { 1 }$ on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$. By noting that $\Phi ( t ) \times ( A + t M ) = I _ { n }$, show that
$$\Phi ( t ) \underset { t \rightarrow 0 } { = } A ^ { - 1 } - A ^ { - 1 } M A ^ { - 1 } t + o ( t )$$
Q20 Matrices Determinant and Rank Computation View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. We define the application $\varphi _ { \alpha }$ by
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \right.$$
Show that $\varphi _ { \alpha }$ is differentiable on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$ and that
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \quad \varphi _ { \alpha } ^ { \prime } ( t ) = - \operatorname { Tr } \left( ( A + t M ) ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A + t M ) . \right.$$
Q21 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$ and $\varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M )$. Show that $\varphi _ { \alpha }$ is twice differentiable at 0 and that
$$\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right) .$$
Q22 Matrices Diagonalizability and Similarity View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Show that $A ^ { - 1 } M$ is similar to a real symmetric matrix.
Hint: You may use question 3.
Q23 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. With $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right)$, deduce that $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) \geq 0$.
Q24 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. Show that, if $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) > 0$, then there exists $\eta > 0$, such that for all $t \in ] - \eta , \eta [$,
$$\frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \geq \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A ) - \operatorname { Tr } \left( A ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A ) t$$