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
2019 x-ens-maths1__mp

24 maths questions

Q1 Number Theory Algebraic Number Theory and Minimal Polynomials View
Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.
Q2 Number Theory Algebraic Number Theory and Minimal Polynomials View
Show that $\alpha$ has degree 1 if and only if $\alpha \in \mathbb { Q }$, where the degree of $\alpha$ is the degree of its minimal polynomial $\Pi_{\alpha}$.
Q3 Number Theory Algebraic Number Theory and Minimal Polynomials View
(a) Show that $\Pi _ { \alpha }$ is irreducible in $\mathbb { Q } [ X ]$.
(b) Let $P \in \mathbb { Q } [ X ]$ be a monic polynomial, irreducible in $\mathbb { Q } [ X ]$. Show that if $z$ is a complex root of $P$, then $P$ is the minimal polynomial of $z$.
Q4 Roots of polynomials Divisibility and minimal polynomial arguments View
(a) Let $A , B \in \mathbb { Q } [ X ]$ be two polynomials that have a common root in $\mathbb { C }$. Show that $A$ and $B$ are not coprime in $\mathbb { Q } [ X ]$.
(b) Show that the roots of $\Pi _ { \alpha }$ in $\mathbb { C }$ are simple.
Q5 Number Theory Algebraic Number Theory and Minimal Polynomials View
(a) Show that if $\alpha \in \mathbb { Q }$ is an algebraic integer, then $\alpha \in \mathbb { Z }$.
(b) Show that if $\alpha \in \mathbb { C }$ is an algebraic integer then $\Pi _ { \alpha } \in \mathbb { Z } [ X ]$.
Hint: use the theorem admitted in the introduction (the set of algebraic integers is a subring of $\mathbb{C}$) as well as question 5a.
Q6 Number Theory Algebraic Number Theory and Minimal Polynomials View
(a) Let $\alpha \in \mathbb { C }$ be an algebraic integer of degree 2 and of modulus 1. Show that $\alpha$ is a root of unity.
(b) Show that $\frac { 3 + 4 i } { 5 }$ is an algebraic number of degree 2 and of modulus 1 but is not a root of unity.
Q7 Number Theory Algebraic Number Theory and Minimal Polynomials View
Show that for all $n \geq 1$ we have $$X ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d }$$ the product being taken over the set of positive integers $d$ dividing $n$, where $\Phi_n = \prod_{z \in \mathbb{P}_n}(X - z)$ and $\mathbb{P}_n$ is the set of primitive $n$-th roots of unity.
Q8 Number Theory Algebraic Number Theory and Minimal Polynomials View
(a) Show that if $p$ is a prime number and $k \geq 1$ is an integer, then $$\Phi _ { p ^ { k } } = X ^ { ( p - 1 ) p ^ { k - 1 } } + X ^ { ( p - 2 ) p ^ { k - 1 } } + \cdots + X ^ { p ^ { k - 1 } } + 1$$ (b) Calculate $\Phi _ { n }$ for $n = 1,2,3,4,5,6$.
Q9 Number Theory Algebraic Number Theory and Minimal Polynomials View
We fix an integer $n \geq 2$.
(a) Calculate $\Phi _ { n } ( 0 )$.
(b) Calculate $\Phi _ { n } ( 1 )$ as a function of the prime factorization of $n$. Hint: reason by induction on $n$, using question 7.
Q10 Number Theory Algebraic Number Theory and Minimal Polynomials View
Show that $\Phi _ { n } \in \mathbb { Z } [ X ]$.
Q11 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities, so that $P = \prod _ { i = 1 } ^ { n } \left( X - z _ { i } \right)$. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$.
(a) Show that the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$ converges for all $z \in \mathbb { C }$ such that $| z | < 1$.
(b) Let $z \in \mathbb { C }$ be non-zero such that $| z | < 1$ and let $f ( z )$ be the sum of the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$. Show that $$z f ( z ) P \left( \frac { 1 } { z } \right) = P ^ { \prime } \left( \frac { 1 } { z } \right)$$ (c) Deduce that $a _ { k } \in \mathbb { Z }$ for all $k \geq 0$.
Q12 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$.
(a) Show that there exist two integers $0 \leq k < l$ such that $a _ { k + i } = a _ { l + i }$ for all $i \in \{ 0,1 , \ldots , n \}$. We fix two such integers $k , l$ in questions 12b and 12c.
(b) Show that $\sum _ { i = 1 } ^ { n } F \left( z _ { i } \right) \left( z _ { i } ^ { l } - z _ { i } ^ { k } \right) = 0$ for every polynomial $F \in \mathbb { C } [ X ]$ of degree at most $n$.
(c) Show that $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are pairwise distinct. Deduce that $z _ { i } ^ { l - k } = 1$ for all $i \in \{ 1,2 , \ldots , n \}$ and conclude.
Q13 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Let $F , G \in \mathbb { Z } [ X ]$. Show that there exists $H \in \mathbb { Z } [ X ]$ such that $$( F + G ) ^ { p } = F ^ { p } + G ^ { p } + p H$$ (b) Show that $\Pi _ { z } \in \mathbb { Z } [ X ]$ and deduce the existence of a polynomial $F \in \mathbb { Z } [ X ]$ such that $$\Pi _ { z } \left( X ^ { p } \right) = \Pi _ { z } ( X ) ^ { p } + p F ( X )$$ (c) Show that $\frac { \Pi _ { z } \left( z ^ { p } \right) } { p }$ is an algebraic integer.
Q14 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Express as a function of $n$ the number $\prod _ { 1 \leq i < j \leq n } \left( z _ { i } - z _ { j } \right) ^ { 2 }$, where $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are the roots of the polynomial $P = X ^ { n } - 1$. Hint: One may consider the numbers $P ^ { \prime } \left( z _ { i } \right)$.
(b) Show that $\Pi _ { z } \left( z ^ { p } \right) = 0$. Hint: show that if $\Pi _ { z } \left( z ^ { p } \right) \neq 0$, then there exists an algebraic integer $u$ such that $n ^ { n } = u \cdot \Pi _ { z } \left( z ^ { p } \right)$.
(c) Conclude that $\Phi _ { n } = \Pi _ { z }$.
Q15 Roots of polynomials Reciprocal and antireciprocal polynomial properties View
A monic polynomial of degree $d \geq 1$ $$P = \sum _ { i = 0 } ^ { d } a _ { i } X ^ { i } \in \mathbb { C } [ X ]$$ is called reciprocal if $a _ { i } = a _ { d - i }$ for $0 \leq i \leq d$.
(a) Show that a monic polynomial $P \in \mathbb { C } [ X ]$ of degree $d$ is reciprocal if and only if $X ^ { d } P \left( \frac { 1 } { X } \right) = P$.
(b) Let $P \in \mathbb { C } [ X ]$ be a monic reciprocal polynomial. Show that if $x \in \mathbb { C }$ is a root of $P$, then $x \neq 0$ and $\frac { 1 } { x }$ is also a root of $P$, with the same multiplicity.
Q16 Roots of polynomials Reciprocal and antireciprocal polynomial properties View
If $\alpha$ is an algebraic number with minimal polynomial $\Pi _ { \alpha }$, the complex roots of $\Pi _ { \alpha }$ different from $\alpha$ are called the conjugates of $\alpha$. We denote by $C ( \alpha )$ the set of conjugates of $\alpha$.
Let $x$ be an algebraic number of modulus 1 and such that $x \notin \{ - 1,1 \}$. Show that $\frac { 1 } { x }$ is a conjugate of $x$. Deduce that $\Pi _ { x }$ is reciprocal.
Q17 Number Theory Algebraic Number Theory and Minimal Polynomials View
We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Let $\alpha$ be an element of $\mathcal { S }$ and let $\gamma \in C ( \alpha )$ have modulus 1.
(a) Show that the minimal polynomial of $\alpha$ is reciprocal and that $\frac { 1 } { \alpha }$ is a conjugate of $\alpha$.
(b) Show that $\gamma$ is not a root of unity.
(c) Show that all conjugates of $\alpha$ other than $\frac { 1 } { \alpha }$ have modulus 1.
Q18 Number Theory Algebraic Number Theory and Minimal Polynomials View
We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Show that the degree of every element of $\mathcal { S }$ is an even integer, greater than or equal to 4.
Q19 Number Theory Algebraic Number Theory and Minimal Polynomials View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Verify that $P _ { n }$ has no root in $\mathbb { Q }$ and that $P _ { n }$ has at least one real root strictly greater than 1. We fix such a root $\alpha _ { n }$ in the sequel.
Q20 Roots of polynomials Reciprocal and antireciprocal polynomial properties View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Show that if $x \in \mathbb { C }$ is a root of $P _ { n }$, then $\frac { 1 } { x }$ is also a root of $P _ { n }$, with the same multiplicity.
Q21 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.
Q22 Roots of polynomials Location and bounds on roots View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $s _ { n }$ is real and that $0 < s _ { n } < 2$. Deduce that $\gamma _ { n }$ is not real and that $\gamma _ { n }$ has modulus 1.
Q23 Roots of polynomials Divisibility and minimal polynomial arguments View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
(a) Show that $t _ { n }$ and $s _ { n }$ are irrational.
(b) Deduce that $P _ { n }$ is irreducible in $\mathbb { Q } [ X ]$ and that $\alpha _ { n } \in \mathcal { S }$.
(c) Show that $\lim _ { n \rightarrow + \infty } \alpha _ { n } = + \infty$.
Q24 Roots of polynomials Location and bounds on roots View
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$. Let $\mathcal { T }$ be the set of $\alpha \in \mathcal { S }$ of degree 4. Show that $\mathcal { T }$ has a smallest element and calculate this number.