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
2022 x-ens-maths__pc_cpge

15 maths questions

Q1.1 Proof Existence Proof View
Let $[ a , b ]$ be a closed bounded interval of $\mathbb { R }$. If $\phi : [ a , b ] \rightarrow [ a , b ]$ is continuous, show that $\phi$ has at least one fixed point.
Q1.2 Proof Existence Proof View
If $\phi : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathcal { C } ^ { 1 }$ and satisfies $$\sup \left\{ \left| \phi ^ { \prime } ( x ) \right| ; x \in \mathbb { R } \right\} < 1$$ show that $\phi$ has at least one fixed point (one may study the sign of $x - \phi ( x )$ for $| x |$ sufficiently large). Show that this fixed point is unique.
Q1.3 Proof True/False Justification View
By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \left| \phi ^ { \prime } ( x ) \right| < 1$$
Q1.4 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $\ell$ be a strictly positive integer. We are given a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ of vectors in $\mathbb { R } ^ { \ell }$ such that the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$ converges.
(a) Show that the sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ is convergent.
(b) Let $v ^ { * }$ denote the limit of this sequence. Bound $\left\| v _ { n } - v ^ { * } \right\|$ by means of a remainder of the sum of the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$.
Q1.5 Proof Existence Proof View
Let $\ell$ be a strictly positive integer. Let $F$ be a closed subset of $\mathbb { R } ^ { \ell }$ and let $\phi : F \rightarrow F$ be a map. We assume that there exists $k \in [ 0,1 [$ such that $$\forall x \in F , \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$
(a) We choose a point $x _ { 0 } \in F$. Show that the formula $x _ { n + 1 } = \phi \left( x _ { n } \right)$ defines a sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ of elements of $F$, and that this sequence is convergent in $F$.
(b) Deduce that $\phi$ has a unique fixed point in $F$.
(c) This fixed point being denoted $x ^ { * }$, bound $\left\| x _ { n } - x ^ { * } \right\|$ as a function of $\left\| x _ { 0 } - x ^ { * } \right\|$.
(d) In what precedes, we assume that $$\phi = \underbrace { \theta \circ \cdots \circ \theta } _ { m \text { times } } ,$$ where $\theta : F \rightarrow F$ is a map and $m \geqslant 2$ is an integer. Show that $\theta$ has a fixed point, and a unique one, in $F$.
Q1.6 Proof Existence Proof View
Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$
Q2.1 Matrices Matrix Power Computation and Application View
For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.
Q2.2 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\epsilon > 0$ be a real number.
(a) Show the existence of a real number $\alpha > 0$ such that for every positive integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$.
(b) Deduce the existence of a real number $\beta > 0$ such that for every positive integer $n$ and all $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$
Q3.1 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $a , b , c , d$ be four real numbers such that $a \leqslant b$ and $c \leqslant d$. Let $U$ be an open set of $\mathbb { R } ^ { 2 }$ containing $[ a , b ] \times [ c , d ]$. Let $h : U \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 2 }$.
(a) Show the identity $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = \int _ { a } ^ { b } \hat { h } \left( s _ { 1 } \right) d s _ { 1 }$$ where $\hat { h }$ is defined by $$\hat { h } \left( s _ { 1 } \right) = \int _ { c } ^ { d } \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( s _ { 1 } , s _ { 2 } \right) d s _ { 2 }$$
(b) Deduce that there exists a point $\left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$ of $[ a , b ] \times [ c , d ]$ such that we have the two equalities $$h ( b , d ) - h ( a , d ) - h ( b , c ) + h ( a , c ) = ( b - a ) \hat { h } \left( \bar { s } _ { 1 } \right) = ( b - a ) ( d - c ) \frac { \partial ^ { 2 } h } { \partial s _ { 1 } \partial s _ { 2 } } \left( \bar { s } _ { 1 } , \bar { s } _ { 2 } \right)$$
Q3.2 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View
Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for all $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$.
We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.
Q3.3 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
We keep, until the end of this third part, the hypotheses and notation of the previous question. For $x , y \in I$ such that $y \neq x$, we set $$H _ { f } ( x , y ) = \frac { x f ( y ) - y f ( x ) } { f ( y ) - f ( x ) }$$
(a) Show that for all $x , y \in I$ such that $y \neq x$ we have $$H _ { f } ( x , y ) = x - f ( x ) \int _ { 0 } ^ { 1 } g ^ { \prime } ( \lambda f ( x ) + ( 1 - \lambda ) f ( y ) ) d \lambda$$
(b) Deduce that $H _ { f }$ admits a unique continuous extension to $I \times I$ as a whole. We still denote this extension by $H _ { f } : I \times I \rightarrow \mathbb { R }$.
(c) Show that $H _ { f }$ is of class $\mathcal { C } ^ { 2 }$ on $I \times I$.
(d) Compute $H _ { f } ( x , x )$.
Q3.4 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
We keep the hypotheses and notation of questions 3.2 and 3.3. We now assume $0 \in f ( I )$ and we denote $x ^ { * } = g ( 0 )$. For $x \in I$ we denote by $I _ { x }$ the closed interval with endpoints $x$ and $x ^ { * }$.
(a) Let $x , y \in I$. Show that there exists $( \bar { x } , \bar { y } ) \in I _ { x } \times I _ { y }$, such that $$H _ { f } ( x , y ) - x ^ { * } = \left( x - x ^ { * } \right) \left( y - x ^ { * } \right) \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( \bar { x } , \bar { y } )$$
(b) Compute $$\frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } \left( x ^ { * } , x ^ { * } \right)$$ as a function of the derivatives of $f$.
Q4.1 Newton-Raphson method View
Illustrate the construction of the secant method by means of a figure. When $f ^ { \prime } > 0$ on $I$, express $x _ { n + 1 }$ as a function of $x _ { n - 1 } , x _ { n }$ by means of the function $H _ { f }$ defined in question 3 of the third part.
(Recall: the secant method initializes with $x_0, x_1 \in I$, and at each step considers the line $L_n$ passing through $(x_{n-1}, f(x_{n-1}))$ and $(x_n, f(x_n))$, defining $x_{n+1}$ as the $x$-intercept of $L_n$.)
Q4.2 Newton-Raphson method View
In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$.
For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$.
(a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$.
(b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$.
(c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$.
(d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that $$x _ { n } - \alpha = O \left( e ^ { s \phi ^ { n } } \right) .$$
Q4.3 Newton-Raphson method View
We return to the general case, $f$ being any function of class $\mathcal { C } ^ { 3 }$. We assume that $f$ vanishes at a point $x ^ { * } \in I$, for which $f ^ { \prime } \left( x ^ { * } \right) > 0$.
(a) Show that there exists $\epsilon > 0$ such that $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] \subset I$ and $f ^ { \prime } > 0$ on the interval $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. We fix such an $\epsilon$ for the rest and we define $$M = \sup _ { ( x , y ) \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] ^ { 2 } } \left| \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( x , y ) \right| .$$
(b) We assume that $x _ { n - 1 } , x _ { n } \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. Show that $$\left| x _ { n + 1 } - x ^ { * } \right| \leqslant M \left| x _ { n - 1 } - x ^ { * } \right| \cdot \left| x _ { n } - x ^ { * } \right| .$$
(c) We fix $\left. \epsilon ^ { \prime } \in \right] 0 , \epsilon ]$ such that $M \epsilon ^ { \prime } < 1$. Show that if $x _ { 0 } , x _ { 1 }$ belong to $\left[ x ^ { * } - \epsilon ^ { \prime } , x ^ { * } + \epsilon ^ { \prime } \right]$ then the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined and converges to $x ^ { * }$.