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
2016 x-ens-maths__psi

20 maths questions

Q1 Proof Direct Proof of an Inequality View
Let $x , y$ be strictly positive vectors of $\mathbb { R } ^ { n }$ and let $S , R$ be two sign diagonal matrices.
(a) Show that $$( S x \mid R y ) \leq ( x \mid y )$$ with equality if and only if $R = S$.
(b) Prove the uniqueness of $S$ in Broyden's theorem.
(c) Show that $$\| S x + R y \| \leq \| x + y \|$$ with equality if and only if $R = S$.
Q2 Matrices Linear System and Inverse Existence View
Let $O$ be an orthogonal matrix of $M _ { n } ( \mathbb { R } )$ and $S$ a sign diagonal matrix. Show that the equality $O x = S x$ with $x \in \mathbb { R } ^ { n }$ strictly positive is equivalent to $$( * ) \left\{ \begin{array} { c } \left( I _ { n } + O \right) x \geq 0 \\ \left( I _ { n } - O \right) x \geq 0 \\ x > 0 \end{array} \right.$$
Q3 Linear transformations View
We assume $n = 2$. We identify elements $x = \binom { x _ { 1 } } { x _ { 2 } } \in \mathbb { R } ^ { 2 }$ with vectors $\vec { x } = x _ { 1 } \vec { i } + x _ { 2 } \vec { j }$ of the Euclidean plane relative to an orthonormal frame $(\Omega , \vec { i } , \vec { j })$.
Let $O$ be the matrix of a reflection relative to a line passing through $\Omega$ and directed by a vector $\vec { v } _ { + }$. Determine a vector $x \in \mathbb { R } ^ { 2 }$ strictly positive and a sign diagonal matrix $S \in M _ { 2 } ( \mathbb { R } )$ such that $O x = S x$. Hint: begin by treating the case where $\vec { v } _ { + } \in \{ \vec { i } , \vec { j } \}$.
Q4 Linear transformations View
We assume $n = 2$. We identify elements $x = \binom { x _ { 1 } } { x _ { 2 } } \in \mathbb { R } ^ { 2 }$ with vectors $\vec { x } = x _ { 1 } \vec { i } + x _ { 2 } \vec { j }$ of the Euclidean plane relative to an orthonormal frame $(\Omega , \vec { i } , \vec { j })$.
Let $O$ be the matrix of a rotation with center $\Omega$ and angle $\theta \in ] - \pi , \pi ]$ nonzero. Using a drawing, find two vectors $x _ { + }$ and $x _ { - }$ such that $$O x _ { + } = \operatorname { diag } ( 1 , - 1 ) x _ { + } \text { and } O x _ { - } = \operatorname { diag } ( - 1,1 ) x _ { - }$$ Then discuss according to the sign of $\theta$, which of $x _ { + }$ and $x _ { - }$ is strictly positive.
Q5 Matrices Linear System and Inverse Existence View
With the notations of Broyden's theorem, we denote by $M \in M _ { 3 n } ( \mathbb { R } )$ the following block matrix $$M = \left( \begin{array} { c c c } 0 & 0 & I _ { n } + O \\ 0 & 0 & I _ { n } - O \\ - \left( I _ { n } + { } ^ { t } O \right) & - \left( I _ { n } - { } ^ { t } O \right) & 0 \end{array} \right)$$ Using Tucker's theorem, show that there exist positive vectors $x , z _ { 1 } , z _ { 2 } \in \mathbb { R } ^ { n }$ such that $$\left\{ \begin{array} { l } \left( I _ { n } + O \right) x \geq 0 \\ \left( I _ { n } - O \right) x \geq 0 \\ - \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } \geq 0 \\ z _ { 1 } + \left( I _ { n } + O \right) x > 0 \\ z _ { 2 } + \left( I _ { n } - O \right) x > 0 \\ x - \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } > 0 \end{array} \right.$$
Q6 Matrices Matrix Algebra and Product Properties View
Show that $\left\| z _ { 1 } - z _ { 2 } \right\| = \left\| z _ { 1 } + z _ { 2 } \right\|$ and that $- \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } = 0$.
Q7 Proof Deduction or Consequence from Prior Results View
Deduce from question 6 that $x > 0$ and $x + O x \geq 0$ as well as $x - O x \geq 0$. Conclude.
Q8 Matrices Linear System and Inverse Existence View
Show that if $M \in M _ { n } ( \mathbb { R } )$ is antisymmetric (that is ${ } ^ { t } M = -M$) then $I _ { n } + M$ is an invertible matrix.
Q9 Matrices Matrix Group and Subgroup Structure View
Show that if $M \in M _ { n } ( \mathbb { R } )$ is antisymmetric, the matrix $$O = \left( I _ { n } + M \right) ^ { - 1 } \left( I _ { n } - M \right)$$ is orthogonal.
Q10 Proof Deduction or Consequence from Prior Results View
Deduce from Broyden's theorem that there exist a strictly positive vector $x$ and a sign diagonal matrix $S$ such that $O x = S x$ and deduce that $u = x + S x$ is the positive vector of Tucker's theorem.
Q11 Matrices Matrix Norm, Convergence, and Inequality View
We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Show that $| \alpha | \leq 1$ with equality if and only if $q = r = 0$.
Q12 Proof by induction Prove a general algebraic or analytic statement by induction View
We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Treat the case $| \alpha | = 1$.
Q13 Matrices Matrix Algebra and Product Properties View
We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that ${ } ^ { t } P P + q { } ^ { t } q = I _ { n - 1 }$, ${ } ^ { t } P r + \alpha q = 0$ and ${ } ^ { t } r r + \alpha ^ { 2 } = 1$.
Q14 Matrices Matrix Group and Subgroup Structure View
We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that the matrices $Q _ { + }$ and $Q _ { - }$ are orthogonal.
Q15 Matrices Matrix Algebra and Product Properties View
We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that $${ } ^ { t } Q _ { + } Q _ { - } = I _ { n - 1 } - \frac { 2 } { 1 - \alpha ^ { 2 } } q { } ^ { t } q$$ and deduce that $$Q _ { - } = Q _ { + } - \frac { 2 } { 1 - \alpha ^ { 2 } } Q _ { + } q { } ^ { t } q$$
Q16 Matrices Matrix Algebra and Product Properties View
We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ Using the induction hypothesis for $Q _ { + }$ (resp. for $Q _ { - }$), we denote by $x _ { + } > 0$ (resp. $x _ { - } > 0$) a vector of $\mathbb { R } ^ { n - 1 }$ and $S _ { + }$ (resp. $S _ { - }$) the sign diagonal matrix, such that $$Q _ { + } x _ { + } = S _ { + } x _ { + } , \quad \text { resp. } \quad Q _ { - } x _ { - } = S _ { - } x _ { - }$$
Show that $$\left( S _ { + } x _ { + } \mid S _ { - } x _ { - } \right) = \left( x _ { + } \mid x _ { - } \right) - \frac { 2 } { 1 - \alpha ^ { 2 } } \left( x _ { + } \mid q \right) \left( x _ { - } \mid q \right)$$
Q17 Proof Deduction or Consequence from Prior Results View
We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$. Using the induction hypothesis for $Q _ { + }$ (resp. for $Q _ { - }$), we denote by $x _ { + } > 0$ (resp. $x _ { - } > 0$) a vector of $\mathbb { R } ^ { n - 1 }$ and $S _ { + }$ (resp. $S _ { - }$) the sign diagonal matrix, such that $Q _ { + } x _ { + } = S _ { + } x _ { + }$, resp. $Q _ { - } x _ { - } = S _ { - } x _ { - }$. We set
  • $\eta _ { + } = - \frac { \left( x _ { + } \mid q \right) } { \alpha + 1 } , \quad \eta _ { - } = - \frac { \left( x _ { - } \mid q \right) } { \alpha - 1 }$
  • $z _ { + } = \binom { x _ { + } } { \eta _ { + } } , \quad z _ { - } = \binom { x _ { - } } { \eta _ { - } }$
  • $S ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , \quad S ^ { - } = \left( \begin{array} { c c } S _ { - } & 0 \\ 0 & - 1 \end{array} \right)$
Show using question 1(a) that in the case where $S _ { + } \neq S _ { - }$ then one of the pairs $(z _ { + } , S ^ { + })$ or $(z _ { - } , S ^ { - })$ satisfies Broyden's theorem.
Q18 Proof Existence Proof View
We now assume that $S _ { + } = S _ { - }$ and we assume that $\left( x _ { + } \mid q \right) = 0$. We denote by $z = \binom { x _ { + } } { 0 }$, $R ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , R ^ { - } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & - 1 \end{array} \right)$.
(a) Show that $O z = R^{+} z = R^{-} z$.
(b) We now write $$O = \left( \begin{array} { c c } \alpha ^ { \prime } & { } ^ { t } q ^ { \prime } \\ r ^ { \prime } & P ^ { \prime } \end{array} \right)$$ where $P ^ { \prime } \in M _ { n - 1 } ( \mathbb { R } )$. Construct then $z ^ { \prime } = \binom { \eta ^ { \prime } } { x ^ { \prime } } \in \mathbb { R } ^ { n }$ with $x ^ { \prime } \in \mathbb { R } ^ { n - 1 }$ strictly positive and $\eta ^ { \prime } \geq 0$ such that there exists a sign diagonal matrix $R ^ { \prime }$ satisfying $O z ^ { \prime } = R ^ { \prime } z ^ { \prime }$.
(c) In the case where $\eta ^ { \prime } = 0$, and using question 1(c), show that there exists a sign diagonal matrix $S$ such that $O \left( z + z ^ { \prime } \right) = S \left( z + z ^ { \prime } \right)$ and conclude.
Q19 Proof Deduction or Consequence from Prior Results View
For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set $$B = \left( \begin{array} { c c c c } 0 & 0 & A & - b \\ 0 & 0 & - A & b \\ - { } ^ { t } A & { } ^ { t } A & 0 & 0 \\ { } ^ { t } b & - { } ^ { t } b & 0 & 0 \end{array} \right)$$ Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.
Show that if $t > 0$ then for $z = z _ { 1 } - z _ { 2 }$, we have $- { } ^ { t } A z \geq 0$ and $( b \mid z ) > 0$.
Q20 Proof Deduction or Consequence from Prior Results View
For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set $$B = \left( \begin{array} { c c c c } 0 & 0 & A & - b \\ 0 & 0 & - A & b \\ - { } ^ { t } A & { } ^ { t } A & 0 & 0 \\ { } ^ { t } b & - { } ^ { t } b & 0 & 0 \end{array} \right)$$ Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.
If $t > 0$ show that $A x = t b$ and conclude.