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
2023 mines-ponts-maths2__mp

22 maths questions

Q1 Sequences and Series Power Series Expansion and Radius of Convergence View
Determine the domain of definition of $\sigma$ and justify that $\sigma$ is continuous on it, where $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.
Q2 Integration by Parts Definite Integral Evaluation by Parts View
Find two real numbers $\alpha$ and $\beta$ such that: $$\forall n \in \mathbf{N}^*, \int_0^{\pi} (\alpha t^2 + \beta t) \cos(nt) \mathrm{d}t = \frac{1}{n^2}$$ then verify that if $t \in ]0, \pi]$, then: $$\forall n \in \mathbf{N}^*, \sum_{k=1}^n \cos(kt) = \frac{\sin\left(\frac{(2n+1)t}{2}\right)}{2\sin\left(\frac{t}{2}\right)} - \frac{1}{2}$$
Q3 Integration by Parts Prove an Integral Identity or Equality View
Justify that, if $\varphi$ is a $\mathcal{C}^1$ application from $[0, \pi]$ to $\mathbf{R}$, then $$\lim_{x \to +\infty} \int_0^{\pi} \varphi(t) \sin(xt) \mathrm{d}t = 0$$ and conclude that $$\sigma(1) = \frac{\pi^2}{6}$$
Q4 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Determine the domain of definition of $f$ and verify that $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Q5 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Justify that $f$ is of class $\mathcal{C}^2$, decreasing and convex on $I$.
Q6 Taylor series Limit evaluation using series expansion or exponential asymptotics View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Give a simple asymptotic equivalent of $f(x)$ as $x$ tends to $-1$.
Q7 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Show that for every natural number $n$, $$f(n)f(n+1) = \frac{\pi}{2(n+1)}$$ then that: $$f(x) \underset{x \to +\infty}{\sim} \sqrt{\frac{\pi}{2x}}$$
Q8 Curve Sketching Sketching a Curve from Analytical Properties View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Sketch the graph of $f$ by making best use of the previous results.
Q9 Integration by Substitution Substitution to Prove an Integral Identity or Equality View
If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Justify that, if $n \in \mathbf{N}$, the improper integral $D_n$ is convergent, then show that $$D_1 = \int_0^{\pi/2} \ln(\cos(t)) \mathrm{d}t$$
Q10 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Calculate $f'(0)$ and $f'(1)$.
Q11 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View
If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Verify that if $n \in \mathbf{N}^*$, then $$(-1)^n D_n = \int_0^{+\infty} \frac{u^n}{\sqrt{\mathrm{e}^{2u} - 1}} \mathrm{~d}u$$ then that $$D_n \underset{n \to +\infty}{\sim} (-1)^n n!$$
Q12 Taylor series Prove smoothness or power series expandability of a function View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Prove that $f$ is expandable as a power series on $]-1, 1[$.
Q13 Differentiating Transcendental Functions Compute derivative of transcendental function View
We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Show that $\Psi$ is of class $\mathcal{C}^1$ on $\mathbf{R}$, then that for all $x \in \mathbf{R}$, $$\Psi'(x) = 4\sum_{k=1}^{+\infty} \rho^k \sin(2kx)$$
Q14 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Deduce that for all $x \in \mathbf{R}$, $$\Psi(x) = 2\ln\left(\frac{a+b}{2}\right) - 2\sum_{k=1}^{+\infty} \frac{\cos(2kx)}{k}\rho^k$$
Q15 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ and $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.
Conclude that $$\int_0^{\pi} \Psi(x)^2 \mathrm{d}x = 4\pi\left(\ln\left(\frac{a+b}{2}\right)\right)^2 + 2\pi\sigma(\rho^2)$$
Q16 Sequences and series, recurrence and convergence Sequence of functions convergence View
We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$.
Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by: $$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$ Deduce $f''(0)$.
Q17 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
Verify that $f$ is an application from $I$ to $\mathbf{R}$ that is log-convex.
Q18 Sequences and Series Functional Equations and Identities via Series View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. We call $\tilde{f}$ the application from $\mathbf{R}_+$ to $\mathbf{R}$, defined by: $$\forall x \in \mathbf{R}^+, \tilde{f}(x) = \ln(f(2x))$$
Show that $$\forall p \in \mathbf{N}^*, \forall x \in \mathbf{R}_+, \tilde{f}(x+p) = \tilde{f}(x) + \sum_{k=0}^{p-1} \ln\left(\frac{2x+2k+1}{2x+2k+2}\right)$$
Q19 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. We call $\tilde{f}$ the application from $\mathbf{R}_+$ to $\mathbf{R}$, defined by: $$\forall x \in \mathbf{R}^+, \tilde{f}(x) = \ln(f(2x))$$
Suppose here that $x \in \mathbf{R}_+^*$, $(n,p) \in (\mathbf{N}^*)^2$ and $x \leq p$. Verify that $$\tilde{f}(n) - \tilde{f}(n-1) \leq \frac{\tilde{f}(n+x) - \tilde{f}(n)}{x} \leq \frac{\tilde{f}(n+p) - \tilde{f}(n)}{p}$$ and that $(\tilde{f}(n+x) - \tilde{f}(n))$ has a limit as $n$ tends to $+\infty$.
Q20 Sequences and Series Recurrence Relations and Sequence Properties View
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$. The functional equation referred to as (1) is: $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Conclude that $f$ is the unique application from $I$ to $\mathbf{R}$, which is log-convex, which satisfies (1) and such that $$f(0) = \frac{\pi}{2}$$
Q21 Sequences and Series Limit Evaluation Involving Sequences View
An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
More generally, determine, if $T \in \mathbf{R}_+^*$, all applications $g$ from $]-T, +\infty[$ to $\mathbf{R}$, log-convex and satisfying $$\forall t \in ]-T, +\infty[, (t+T)g(t) = (t+2T)g(t+2T).$$
Q22 Sequences and Series Limit Evaluation Involving Sequences View
An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
Does there exist an application $h$, from $\mathbf{R}$ to $\mathbf{R}$ and log-convex, satisfying $$\forall t \in \mathbf{R}, (t+T)h(t) = (t+2T)h(t+2T)?$$