grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2025 mines-ponts-maths2__pc

23 maths questions

Q1 Arithmetic Sequences and Series Convergence/Divergence Determination of Numerical Series View
Justify that, for all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, the series $\sum u _ { k }$ converges, where $u_k = \dfrac{(-1)^k}{pk+q}$.
Q2 Arithmetic Sequences and Series Evaluation of a Finite or Infinite Sum View
In this question, we set $p = q = 1$. Show that $$\phi _ { 1,1 } ( n ) = \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + t } d t - \int _ { 0 } ^ { 1 } \frac { ( - t ) ^ { n + 1 } } { 1 + t } d t$$ where $\phi_{1,1}(n) = \sum_{k=0}^{n} \dfrac{(-1)^k}{k+1}$.
Q3 Arithmetic Sequences and Series Evaluation of a Finite or Infinite Sum View
Deduce the value of $S _ { 1,1 }$, the sum of the congruent-harmonic series with parameters $p=q=1$.
Q5 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ Prove that the application $I _ { p , q }$ is well-defined and continuous on $\mathbf { R } _ { + }$.
Q6 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ Determine $$\lim _ { n \rightarrow + \infty } I _ { p , q } ( n ) = 0$$
Q7 Arithmetic Sequences and Series Connect a Discrete Sum to an Integral via Reduction Formulae View
We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ For all $x \in [ 0,1 ]$, calculate $\sum _ { k = 0 } ^ { n } \left( - x ^ { \alpha _ { p , q } } \right) ^ { k }$ then deduce that $$\phi _ { p , q } ( n ) = \frac { 1 } { q } \left( \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { \alpha _ { p , q } } } d x + ( - 1 ) ^ { n } I _ { p , q } ( n ) \right)$$
Q8 Integration by Substitution Substitution to Prove an Integral Identity or Equality View
We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ Show that, for all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, $$S _ { p , q } = \int _ { 0 } ^ { 1 } \frac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$$
Q9 Integration by Substitution Definite Integral Evaluation via Substitution or Standard Forms View
We define $E _ { 1 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p = q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
Show that, for all $( p , q ) \in E _ { 1 }$, $$S _ { p , q } = \frac { \ln 2 } { p }$$
Q10 Integration by Substitution Definite Integral Evaluation via Substitution or Standard Forms View
We define $E _ { 2 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p < q , p \mid q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
For all pairs $( p , q ) \in E _ { 2 }$, show that there exists a constant $\lambda := \lambda ( p , q )$ which one will determine, such that $$S _ { p , q } = \frac { ( - 1 ) ^ { \lambda - 1 } } { p } \left( \ln ( 2 ) - \sum _ { k = 1 } ^ { \lambda - 1 } \frac { ( - 1 ) ^ { k - 1 } } { k } \right)$$
Q11 Partial Fractions View
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$.
Show that there exist constants $\left( a _ { 0 } , b _ { 0 } , \ldots , b _ { \lfloor p / 2 \rfloor - 1 } \right) \in \mathbf { C } ^ { \lfloor p / 2 \rfloor + 1 }$ such that $$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { a _ { 0 } } { X + 1 } + \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \left( \frac { b _ { k } } { X - \omega _ { p , k } } + \frac { \overline { b _ { k } } } { X - \overline { \omega _ { p , k } } } \right) ,$$ where the $\omega _ { p , k }$ are constants which one will specify and $F ( X )$ is the rational fraction defined at the beginning of this part.
In the case where $p$ is even, we set $a _ { 0 } = 0$.
Q12 Partial Fractions View
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Calculate $a _ { 0 }$ in the case where $p$ is odd, then show that, for all integers $k \in \llbracket 0 , \lfloor p / 2 \rfloor - 1 \rrbracket$, $b _ { k }$ can be written in the form $$b _ { k } = - \frac { 1 } { p } e ^ { i q \theta _ { k } }$$
Q13 Partial Fractions View
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Deduce the partial fraction decomposition of $F ( X )$ in $\mathbf { R } ( X )$: $$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { ( - 1 ) ^ { q - 1 } } { p } \cdot \frac { 1 } { X + 1 } - \frac { 2 } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } F _ { k } ( X )$$ where, for all $0 \leq k \leq \lfloor p / 2 \rfloor - 1$, $$F _ { k } ( X ) := \frac { \cos \left( q \theta _ { k } \right) X - \cos \left( ( q - 1 ) \theta _ { k } \right) } { X ^ { 2 } - 2 \cos \left( \theta _ { k } \right) X + 1 }$$
Q14 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Show that $$\sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) = \begin{cases} 0 & \text{if } p \text{ is even} \\ \dfrac { ( - 1 ) ^ { q + 1 } } { 2 } & \text{if } p \text{ is odd} \end{cases}$$
Q15 Integration by Substitution Multi-Step Composite Problem Using Identities View
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$. We admit that for all $0 \leq k \leq \lfloor p / 2 \rfloor - 1$, $$\int _ { 0 } ^ { 1 } F _ { k } ( t ) d t = \cos \left( q \theta _ { k } \right) \ln \left( 2 \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) - \frac { \pi } { 2 p } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right)$$
Deduce from the previous questions that, for all $( p , q ) \in E _ { 3 }$, $$S _ { p , q } = \frac { 1 } { p } \left( \frac { \pi } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right) - 2 \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) \ln \left( \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) \right)$$
Q16 Integration by Substitution Multi-Step Composite Problem Using Identities View
Using the formula established for $( p , q ) \in E _ { 3 }$: $$S _ { p , q } = \frac { 1 } { p } \left( \frac { \pi } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right) - 2 \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) \ln \left( \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) \right)$$ where $\theta_k := (2k+1)\dfrac{\pi}{p}$, deduce the exact values of $S _ { 2,1 }$ and $S _ { 3,1 }$.
Q17 Probability Definitions Event Expression and Partition View
We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:
  • $E _ { n }$: "We obtain $( p , q ) \in E _ { 1 } \cup E _ { 2 } \cup E _ { 3 }$".
  • $A _ { n }$: "We obtain $p = q$".
  • $B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".
  • $C _ { n }$: "We obtain $p > q$".
where $E_1 = \{(p,q)\in(\mathbf{N}^*)^2: p=q\}$, $E_2 = \{(p,q)\in(\mathbf{N}^*)^2: pq\}$.
Justify that the set $\left\{ A _ { n } , B _ { n } , C _ { n } \right\}$ forms a partition of $E _ { n }$.
Q18 Probability Definitions Finite Equally-Likely Probability Computation View
We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:
  • $A _ { n }$: "We obtain $p = q$".
  • $C _ { n }$: "We obtain $p > q$".

Calculate $\mathbf { P } \left( A _ { n } \right)$ then $\mathbf { P } \left( C _ { n } \right)$.
Q19 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:
  • $A _ { n }$: "We obtain $p = q$".
  • $B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".

Show that $$\mathbf { P } \left( B _ { n } \right) = \frac { 1 } { n ^ { 2 } } \sum _ { p = 1 } ^ { n } \left\lfloor \frac { n } { p } \right\rfloor - \frac { 1 } { n } ,$$ and deduce $\mathbf { P } \left( A _ { n } \cup B _ { n } \right)$.
Q20 Sequences and series, recurrence and convergence Asymptotic Equivalents and Growth Estimates for Sequences/Series View
By denoting $H _ { n } := \sum _ { k = 1 } ^ { n } \dfrac { 1 } { k }$ the harmonic series, show that $$H _ { n } \sim \ln n \quad ( n \rightarrow + \infty )$$
Q21 Probability Definitions Proof of a General Conditional Expectation or Independence Property View
We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. Using the result $H_n \sim \ln n$ as $n \to +\infty$, show that $$\mathbf { P } \left( A _ { n } \cup B _ { n } \right) \sim \frac { \ln n } { n } \quad ( n \rightarrow + \infty )$$
Q22 Probability Definitions Probability Using Set/Event Algebra View
We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. The event $E_n$ is defined as "We obtain $(p,q) \in E_1 \cup E_2 \cup E_3$".
Using the result $\mathbf{P}(A_n \cup B_n) \sim \dfrac{\ln n}{n}$ as $n \to +\infty$, deduce $$\lim _ { n \rightarrow + \infty } \mathbf { P } \left( E _ { n } \right) .$$
Q23 Integration by Substitution Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, define $R _ { p , q } := \dfrac { 1 } { q } I _ { p , q }$ where $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x, \quad \alpha_{p,q} = \frac{p}{q}.$$
Using the change of variables $s = x ^ { n + 1 }$ in $I _ { p , q } ( n )$, prove that $$R _ { p , q } ( n ) \sim \frac { 1 } { 2 p n } \quad ( n \rightarrow + \infty )$$
Q24 Integration by Substitution Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, define $R _ { p , q } := \dfrac { 1 } { q } I _ { p , q }$ where $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x, \quad \alpha_{p,q} = \frac{p}{q},$$ and recall that $$\phi _ { p , q } ( n ) = \frac { 1 } { q } \left( \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { \alpha _ { p , q } } } d x + ( - 1 ) ^ { n } I _ { p , q } ( n ) \right).$$
Using the result $R _ { p , q } ( n ) \sim \dfrac { 1 } { 2 p n }$ as $n \to +\infty$, deduce the convergence rate of the alternating congruent-harmonic series $\sum u _ { k }$, that is, that of the sequence of partial sums $\left( \phi _ { p , q } ( n ) \right) _ { n \in \mathbf { N } }$.