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 mines-ponts-maths1__psi

24 maths questions

Q1 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Let $z \in D$. Show the convergence of the series $\sum_{n \geq 1} \frac{z^n}{n}$. Specify the value of its sum when $z \in ]-1,1[$. We denote $$L(z) := \sum_{n=1}^{+\infty} \frac{z^n}{n}$$
Q3 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $z \in D$. Show that the function $\Psi : t \mapsto (1-tz)e^{L(tz)}$ is constant on $[0,1]$, and deduce that $$\exp(L(z)) = \frac{1}{1-z}$$
Q4 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Show that $|L(z)| \leq -\ln(1-|z|)$ for all $z$ in $D$. Deduce that the series $\sum_{n \geq 1} L(z^n)$ is convergent for all $z$ in $D$.
Q6 Proof Proof That a Map Has a Specific Property View
In this part, we introduce the function $q$ which associates to any real $x$ the real number $q(x) = x - \lfloor x \rfloor - \frac{1}{2}$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf{R}$, that it is 1-periodic and that the function $|q|$ is even.
Q7 Proof Existence Proof View
Show that $\int_{1}^{+\infty} \frac{q(u)}{e^{tu}-1} \mathrm{~d}u$ is well defined for all real $t > 0$.
Q8 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Show that for all integer $n > 1$, $$\int_{1}^{n} \frac{q(u)}{u} \mathrm{~d}u = \ln(n!) + (n-1) - n\ln(n) - \frac{1}{2}\ln(n) = \ln\left(\frac{n! e^n}{n^n \sqrt{n}}\right) - 1.$$
Q9 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Show that $\int_{\lfloor x \rfloor}^{x} \frac{q(u)}{u} \mathrm{~d}u$ tends to 0 when $x$ tends to $+\infty$, and deduce the convergence of the integral $\int_{1}^{+\infty} \frac{q(u)}{u} \mathrm{~d}u$, as well as the equality $$\int_{1}^{+\infty} \frac{q(u)}{u} \mathrm{~d}u = \frac{\ln(2\pi)}{2} - 1$$
Q10 Sequences and Series Evaluation of a Finite or Infinite Sum View
Using a series expansion under the integral, show that $$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$
Q11 Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View
Show that $$\int_{0}^{1} \ln\left(\frac{1-e^{-tu}}{t}\right) \mathrm{d}u \underset{t \rightarrow 0^+}{\longrightarrow} -1.$$ One may begin by establishing that $x \mapsto \frac{1-e^{-x}}{x}$ is decreasing on $\mathbf{R}_+$.
Q12 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Show that $u_k$ is continuous on $\mathbf{R}$, for all $k \in \mathbf{N}^*$.
Q13 Reduction Formulae Bound or Estimate a Parametric Integral View
For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Let $t \in \mathbf{R}_+$. Show successively that $|u_k(t)| = \int_{k/2}^{(k+1)/2} \frac{t|q(u)|}{e^{tu}-1} du$ then $u_k(t) = (-1)^k |u_k(t)|$ for all integer $k \geq 1$, and finally establish that $$\forall n \in \mathbf{N}^*, \left|\sum_{k=n}^{+\infty} u_k(t)\right| \leq \frac{1}{2n}.$$
Q14 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View
For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ We admit that the bound $\left|\sum_{k=n}^{+\infty} u_k(t)\right| \leq \frac{1}{2n}$ holds for $t = 0$.
Deduce that $$\int_{1}^{+\infty} \frac{tq(u)}{e^{tu}-1} \mathrm{~d}u \underset{t \rightarrow 0^+}{\longrightarrow} \frac{\ln(2\pi)}{2} - 1$$
Q15 Proof Direct Proof of a Stated Identity or Equality View
Show, for all real $t > 0$, the identity $$\int_{1}^{+\infty} \frac{tq(u)}{e^{tu}-1} \mathrm{~d}u = -\frac{1}{2}\ln(1-e^{-t}) - \ln P(e^{-t}) - \int_{1}^{+\infty} \ln(1-e^{-tu}) \mathrm{d}u$$
Q16 Taylor series Taylor's formula with integral remainder or asymptotic expansion View
Conclude that $$\ln P(e^{-t}) = \frac{\pi^2}{6t} + \frac{\ln(t)}{2} - \frac{\ln(2\pi)}{2} + o(1) \text{ when } t \text{ tends to } 0^+.$$
Q17 Sequences and Series Recurrence Relations and Sequence Properties View
For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality.
Let $n \in \mathbf{N}$. Show that $P_{n,N}$ is included in $[0,n]^N$ and non-empty for all $N \in \mathbf{N}^*$, that the sequence $(p_{n,N})_{N \geq 1}$ is increasing and that it is constant from rank $\max(n,1)$ onwards.
Q18 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
Let $N \in \mathbf{N}^*$. Give a sequence $(a_{n,N})_{n \in \mathbf{N}}$ such that $$\forall z \in D, \frac{1}{1-z^N} = \sum_{n=0}^{+\infty} a_{n,N} z^n$$ Deduce, by induction, the formula $$\forall N \in \mathbf{N}^*, \forall z \in D, \prod_{k=1}^{N} \frac{1}{1-z^k} = \sum_{n=0}^{+\infty} p_{n,N} z^n$$
Q19 Sequences and series, recurrence and convergence Series convergence and power series analysis View
For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
We fix $\ell \in \mathbf{N}$ and $x \in [0,1[$. Using the result of the previous question, establish the bound $\sum_{n=0}^{\ell} p_n x^n \leq P(x)$. Deduce the radius of convergence of the power series $\sum_n p_n z^n$.
Q20 Sequences and series, recurrence and convergence Series convergence and power series analysis View
For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
Let $z \in D$. By examining the difference $\sum_{n=0}^{+\infty} p_n z^n - \sum_{n=0}^{+\infty} p_{n,N} z^n$, prove that $$P(z) = \sum_{n=0}^{+\infty} p_n z^n$$
Q21 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $n \in \mathbf{N}$. Show that for all real $t > 0$, $$p_n = \frac{e^{nt} P(e^{-t})}{2\pi} \int_{-\pi}^{\pi} e^{-in\theta} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta \tag{1}$$
Q22 Taylor series Lagrange error bound application View
Let $x \in [0,1[$ and $\theta \in \mathbf{R}$. Using the function $L$, show that $$\left|\frac{1-x}{1-xe^{i\theta}}\right| \leq \exp(-(1-\cos\theta)x)$$ Deduce that for all $x \in [0,1]$ and all real $\theta$, $$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{1-x} + \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right)\right)$$
Q23 Taylor series Lagrange error bound application View
Let $x \in [0,1[$ and $\theta$ a real. Show that $$\frac{1}{1-x} - \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right) \geq \frac{x(1-\cos\theta)}{(1-x)\left((1-x)^2 + 2x(1-\cos\theta)\right)}.$$ Deduce that if $x \geq \frac{1}{2}$ then $$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1-\cos\theta}{6(1-x)^3}\right) \quad \text{or} \quad \left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{3(1-x)}\right).$$ For this last result, distinguish two cases according to the relative values of $x(1-\cos\theta)$ and $(1-x)^2$.
Q24 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Show that there exists a real $a > 0$ such that $$\forall \theta \in [-\pi,\pi], 1-\cos\theta \geq a\theta^2.$$ Deduce that there exist three reals $t_0 > 0$, $\beta > 0$ and $\gamma > 0$ such that, for all $t \in ]0,t_0]$ and all $\theta \in [-\pi,\pi]$, $$\left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\beta(t^{-3/2}\theta)^2} \quad \text{or} \quad \left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\gamma(t^{-3/2}|\theta|)^{2/3}}.$$
Q25 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Deduce that $$\int_{-\pi}^{\pi} e^{-i\frac{\theta^2}{6t^2}} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta = O(t^{3/2}) \text{ when } t \text{ tends to } 0^+.$$
Q26 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
By taking $t = \frac{\pi}{\sqrt{6n}}$ in formula $$p_n = \frac{e^{nt} P(e^{-t})}{2\pi} \int_{-\pi}^{\pi} e^{-in\theta} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta,$$ conclude that $$p_n = O\left(\frac{\exp\left(\pi\sqrt{\frac{2n}{3}}\right)}{n}\right) \quad \text{when } n \text{ tends to } +\infty.$$