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
2018 centrale-maths2__pc

35 maths questions

Q1 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Determine $\mathcal{D}_{\zeta}$, the domain of definition of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.
Q2 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Show that $\zeta$ is continuous on $\mathcal{D}_{\zeta}$, where $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.
Q3 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Study the monotonicity of $\zeta$, where $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.
Q4 Sequences and Series Limit Evaluation Involving Sequences View
Justify that $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$ admits a limit at $+\infty$.
Q5 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $x \in \mathcal{D}_{\zeta}$ and let $n \in \mathbb{N}$ such that $n \geqslant 2$. Show: $$\int_{n}^{n+1} \frac{\mathrm{~d}t}{t^{x}} \leqslant \frac{1}{n^{x}} \leqslant \int_{n-1}^{n} \frac{\mathrm{~d}t}{t^{x}}$$
Q6 Sequences and Series Estimation or Bounding of a Sum View
Using the result of Q5, deduce that for all $x \in \mathcal{D}_{\zeta}$, $$1 + \frac{1}{(x-1)2^{x-1}} \leqslant \zeta(x) \leqslant 1 + \frac{1}{x-1}$$
Q7 Sequences and Series Limit Evaluation Involving Sequences View
Determine the limit of $\zeta(x)$ as $x$ tends to 1 from above.
Q8 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Determine the limit of $\zeta(x)$ as $x$ tends to $+\infty$.
Q9 Curve Sketching Sketching a Curve from Analytical Properties View
Give the shape of the representative curve of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.
Q10 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Determine $\mathcal{D}_{f}$, the domain of definition of $f$.
Q11 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is continuous on $\mathcal{D}_{f}$ and study its variations.
Q12 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Calculate $f(k)$.
Q13 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q12, deduce an asymptotic equivalent of $f$ at $+\infty$.
Q14 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.
Q15 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Using the result of Q14, deduce an asymptotic equivalent of $f$ at $-k$. What are the right and left limits of $f$ at $-k$?
Q16 Sequences and Series Power Series Expansion and Radius of Convergence View
We consider the power series in the real variable $x$ given by $\sum_{k \in \mathbb{N}^{*}} (-1)^{k} \zeta(k+1) x^{k}$.
Determine the radius of convergence $R$ of this power series. Is there convergence at $x = \pm R$?
Q17 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}_{f}$ and calculate $f^{(k)}(x)$ for all $x \in \mathcal{D}_{f}$ and all $k \in \mathbb{N}^{*}$.
Q18 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that there exists $A \in \mathbb{R}_{+}^{*}$ such that $$\forall k \in \mathbb{N}^{*}, \forall x \in {]-1,1[}, \quad \left|f^{(k)}(x)\right| \leqslant k! \left(A + \frac{1}{(x+1)^{k+1}}\right)$$
Q19 Sequences and Series Power Series Expansion and Radius of Convergence View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q18, deduce that $f$ is expandable as a power series on $]-1,1[$ and that $$\forall x \in {]-1,1[}, \quad f(x) = \sum_{k=1}^{+\infty} (-1)^{k} \zeta(k+1) x^{k}$$
Q20 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Determine for which $x \in \mathbb{R}$ the integral below is convergent $$\int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$
Q21 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ By noting that, for all $t \in [0,1[$, $\frac{1}{1-t} = \sum_{n=0}^{\infty} t^{n}$, show that $$\forall x \in {]-1,+\infty[}, \quad f(x) = \int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$
Q22 Sequences and Series Evaluation of a Finite or Infinite Sum View
Using the results of the previous questions, deduce an integral expression of $\zeta(k+1)$ for all $k \in \mathbb{N}^{*}$.
Q23 Sequences and Series Evaluation of a Finite or Infinite Sum View
Show that $$\forall k \in \mathbb{N}^{*}, \quad \zeta(k+1) = \frac{1}{k!} \int_{0}^{+\infty} \frac{u^{k}}{\mathrm{e}^{u} - 1} \mathrm{~d}u$$
Q24 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
Let $x \in \mathbb{R}$ such that $x > 1$. Show that we define the probability distribution of a random variable $X$ taking values in $\mathbb{N}^{*}$ by setting $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$
Q25 Discrete Random Variables Existence of Expectation or Moments View
Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Give a necessary and sufficient condition on $x$ for $X$ to have a finite expectation. Express this expectation using $\zeta$.
Q26 Discrete Random Variables Existence of Expectation or Moments View
Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $k \in \mathbb{N}$, give a necessary and sufficient condition on $x$ for $X^{k}$ to have a finite expectation. Express this expectation using $\zeta$.
Q27 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Using the result of Q26, deduce the variance of $X$.
Q28 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Show that, for all $a \in \mathbb{N}^{*}$, $$\mathbb{P}\left(X \in a\mathbb{N}^{*}\right) = \frac{1}{a^{x}}$$
Q29 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.
Q30 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order.
Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that $$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$
Q31 Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event $$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$ Express the event $A$ using the events $C_{n}$. Deduce that $$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$
Q32 Discrete Probability Distributions Combinatorial Counting in Probabilistic Context View
Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).
For all $k \in \mathbb{N}^{*}$, show that $$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$
Q33 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$.
Show that $$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$
Q34 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.
Q35 Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution.
Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?