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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 centrale-maths2__mp

25 maths questions

QI.A.1 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.

QI.A.2 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Determine the limits of $f$ and $f^{\prime}$ at $+\infty$.

QI.A.3 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that $$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

QI.A.4 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\left\{ \begin{array}{l} \forall x > 0, \quad f(x) = x \ln(x) - \frac{1}{2} x \ln\left(x^{2} + 1\right) - \arctan(x) + \frac{\pi}{2} \\ f(0) = \frac{\pi}{2} \end{array} \right.$$

QI.A.5 Reduction Formulae Establish an Integral Identity or Representation View

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\forall s \in \mathbb{R}, \quad |s| = \frac{2}{\pi} \int_{0}^{\infty} \frac{1 - \cos(st)}{t^{2}} \mathrm{~d}t$$

QI.B.1 Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Justify the existence of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ and specify the monotonicity of the subsequence $\left(u_{2n}\right)_{n \in \mathbb{N}^{*}}$.

QI.B.2 Reduction Formulae Compute a Base Case or Specific Value of a Parametric Integral View

QI.C.1 Reduction Formulae Perform a Change of Variable or Transformation on a Parametric Integral View

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \frac{\sqrt{n}}{2\sqrt{2}} v_{n} \quad \text{with} \quad v_{n} = \int_{0}^{\infty} \frac{1 - (\cos(\sqrt{2u/n}))^{n}}{u\sqrt{u}} \mathrm{~d}u$$

QI.C.2 Reduction Formulae Bound or Estimate a Parametric Integral View

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\left. \left. \forall (n, u) \in \mathbb{N}^{*} \times \right] 0, 1 \right], \quad \left| 1 - (\cos(\sqrt{2u/n}))^{n} \right| \leqslant u$$

QI.C.3 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ and $v_{n} = \int_{0}^{\infty} \frac{1 - (\cos(\sqrt{2u/n}))^{n}}{u\sqrt{u}} \mathrm{~d}u$.
Show that the sequence $\left(v_{n}\right)_{n \in \mathbb{N}^{*}}$ admits a finite limit $l$ satisfying $$l = \int_{0}^{\infty} \frac{1 - \mathrm{e}^{-u}}{u\sqrt{u}} \mathrm{~d}u$$

QI.C.4 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ We admit the relation $\int_{0}^{\infty} \frac{\mathrm{e}^{-u}}{\sqrt{u}} \mathrm{~d}u = \sqrt{\pi}$.
Conclude that $u_{n} \sim \sqrt{\frac{n\pi}{2}}$.

QII.A.1 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View

QII.A.2 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View

Let $S$ and $T$ be two finite real random variables that are independent and defined on $(\Omega, \mathcal{A}, P)$. We assume that $T$ and $-T$ have the same distribution.
Show that $E(\cos(S + T)) = E(\cos(S)) E(\cos(T))$.

QII.A.3 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$.
We consider the function $\varphi_{n}$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\varphi_{n}(t) = E\left(\cos\left(S_{n} t\right)\right)$ for all real $t$.
Show that $\varphi_{n}(t) = (\cos t)^{n}$ for all integers $n \in \mathbb{N}^{*}$ and all real $t$.

QII.A.4 Discrete Random Variables Integral or Series Representation of Moments View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$.
Show, for all $n \in \mathbb{N}^{*}$, $$E\left(\left|S_{n}\right|\right) = \frac{2}{\pi} u_{n}$$ We will use the integral expression for the absolute value obtained in question I.A.5.

QII.A.5 Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$.
Deduce from the previous question that, for all $n \in \mathbb{N}$, $u_{2n+1} = u_{2n+2}$.

QII.B.1 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View

QII.B.2 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.
Show that, for all $n \in \mathbb{N}^{*}$, $$P\left(U_{n} \geqslant \frac{1}{\sqrt{n}}\right) \leqslant \frac{3}{n^{3/2}}$$

QII.B.3 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and $$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$ Show that $\mathcal{Z}_{n} \in \mathcal{A}$ for all $n \in \mathbb{N}^{*}$ and that $\lim_{n \rightarrow \infty} P\left(\mathcal{Z}_{n}\right) = 0$.

QII.B.4 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and $$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$ By considering $Z = \bigcap_{n \in \mathbb{N}^{*}} \mathcal{Z}_{n}$, show that $\left(\frac{S_{n}}{n}\right)$ converges almost surely to $0$.

QIII.A.1 Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
Show that the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is increasing.

QIII.A.2 Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
Show that if the series $\sum a_{n}^{2}$ is convergent, then the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is convergent.

QIII.A.3 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
We assume $a_{1} \geqslant a_{2} + \cdots + a_{n}$. Show $E\left(\left|T_{n}\right|\right) = E\left(\left|T_{1}\right|\right) = a_{1}$.

QIII.B.1 Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $\left(J_{n}\right)_{n \in \mathbb{N}^{*}}$ is a well-defined sequence and that it is increasing and convergent.
We will set $a_{k} = \frac{1}{2k-1}$ and express the expectation of $\left|T_{n}\right|$ using the method of question II.A.4.

QIII.B.2 Reduction Formulae Bound or Estimate a Parametric Integral View

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.