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
2022 x-ens-maths-b__mp

19 maths questions

Q1a Sequences and series, recurrence and convergence Convergence/Divergence Determination of Numerical Series View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent.
Q1b Proof Proof That a Map Has a Specific Property View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are odd.
Q1c Proof Proof That a Map Has a Specific Property View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are periodic with period 1.
Q1d Proof Proof That a Map Has a Specific Property View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.
Q2a Trig Proofs Trigonometric Identity Simplification View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$
Q2b Proof Functional Equations and Identities via Series View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x).$$
Q3a Proof Existence Proof View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.
Q3b Proof Existence Proof View
Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M.$$
Q5a Taylor series Construct series for a composite or related function View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Show that: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{x}{2} \operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}.$$
Q5b Taylor series Construct series for a composite or related function View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Deduce: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}.$$
Q6 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1} \zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right).$$
Q7a Taylor series Extract derivative values from a given series View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n).$$
Q7b Taylor series Functional Equations and Identities via Series View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}.$$
Q7c Sequences and series, recurrence and convergence Evaluation of a Finite or Infinite Sum View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Calculate $b_2$, $b_4$ and $b_6$ then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.
Q18 Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:
  1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
  2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} P(r \mid X_n) = P(r \mid X)$.
Show that then the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.
Q19 Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$.
We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$.
For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $P(r \mid X_n^{(i)})$ and show that $P(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} P\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}.$$
Q20a Discrete Random Variables Properties of Named Discrete Distributions (Non-Binomial) View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $P(k \mid Z)$ for $k \in \mathbb{N}^*$.
Q20b Discrete Random Variables Limit and Convergence of Probabilistic Quantities View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $s \geqslant 2$ be an integer. Let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$, and let $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ be their gcd.
Deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.
Q21 Discrete Random Variables GCD, LCM, and Coprimality View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that $$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$ and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.