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 x-ens-maths-c__mp

27 maths questions

Q1 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Let $f$ be a power series and $z$ a complex number such that $\hat{f}(|z|) < \infty$. Show then that the series $f$ converges at $z$ and that $|f(z)| \leqslant \hat{f}(|z|)$. Give an example where this inequality is strict.
Q3 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show, for $r > 0$, that $$r < \rho(f) \Rightarrow \exists a > 0 \text{ such that } f \prec \frac{a}{r - z} \Rightarrow r \leqslant \rho(\hat{f})$$ deduce in particular that $\rho(\hat{f}) = \rho(f)$.
Q4 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that $\widehat{f \cdot g} \prec \hat{f} \cdot \hat{g}$, deduce that $\rho(f \cdot g) \geqslant \min(\rho(f), \rho(g))$.
Q6 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $f$ and $g$ be power series, with $g \in O_1$. Show that $\widehat{f \circ g} \prec \hat{f} \circ \hat{g}$. Deduce that, if $f$ and $g$ have strictly positive radius of convergence, then $\rho(f \circ g) > 0$.
Q7 Proof Direct Proof of an Inequality View
If $f, g$ have non-negative real coefficients, $h, g \in O_1$, show that $h \prec g \Rightarrow f \circ h \prec f \circ g$.
Q8 Proof Direct Proof of a Stated Identity or Equality View
Show, if $f$ and $g \in O_1$ have non-negative real coefficients and if $r \in [0, \infty]$, that $f \circ g(r) = f(g(r))$.
Q9 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $f$ and $g$ be power series, with $g \in O_1$. For all $z$ satisfying $|z| < \rho(\hat{f} \circ \hat{g})$, show that the series $f$ converges at $g(z)$ and that $f \circ g(z) = f(g(z))$.
Q10 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $f, g$ and $h$ be power series, with $g, h \in O_1$, show that $(f \circ g) \circ h = f \circ (g \circ h)$.
Q11 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ Show that there exist $r > 0$ and a function $h: ]-r, r[ \longrightarrow \mathbb{R}$, expandable as a power series at 0, satisfying $h(0) = 0$ and such that $$h(x) = a\left(x + \frac{h(x)^2}{b - h(x)}\right)$$ for all $x \in ]-r, r[$. We also denote by $h$ the element of $O_1$ associated with the function $h$.
Q12 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ and $h$ is the function from question (11). Show, by induction on $k$, that $(g)_k \leqslant (h)_k$ for all $k \in \mathbb{N}$, conclude.
Q13 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $h \in O_1$ such that $h \circ f = I$, and that $(h)_1 = 1/\lambda$.
Q14 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $g \in O_1$ such that $f \circ g = I$.
Q15 Proof Direct Proof of a Stated Identity or Equality View
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $h$ the unique series in $O_1$ such that $h \circ f = I$ and $g$ the unique series in $O_1$ such that $f \circ g = I$. Show that $g = h$.
Q16 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $g = f^\dagger$ the reciprocal series. Show that $\hat{g} \prec (1/\lambda)(I + \hat{F} \circ \hat{g})$, conclude using part C that $\rho(g) > 0$ if $\rho(f) > 0$.
Q17 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Show that $[G]_{d+1} + F \circ (I + [G]_d) \in O_{d+2}$ for all $d \geqslant 1$ (the notation $[f]_d$ is defined in the introduction to the subject).
Q18 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Assume that there exist $s > 0$ and $\alpha \in ]0,1[$ such that $\hat{F}(s) \leqslant \alpha s$. Show then that for all $d \geqslant 2, \widehat{[G]_d}((1-\alpha)s) \leqslant \alpha s$. Conclude that $$\hat{G}((1-\alpha)s) \leqslant \alpha s.$$
Q19 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity, that is, $\lambda^n \neq 1$ for all integer $n \geqslant 1$. We propose to show that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$. Show that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$.
Q20 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity. We have shown in question (19) that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$. Conclude that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$.
Q21 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Show that there exists $\omega > 0$ such that $|\lambda^m - \lambda| \geqslant \omega$ for all integer $m \geqslant 2$.
Q22 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. We consider the series $H \in O_2$ from part E. Show that the series $H$ satisfies $\hat{H} \prec \frac{1}{\omega} \hat{F} \circ (I + \hat{H})$.
Q23 Sequences and Series Power Series Expansion and Radius of Convergence View
We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Using the result of question (22), conclude that the power series $h$ and $H$ of part E have strictly positive radius of convergence.
Q24 Taylor series Determine radius or interval of convergence View
We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$.
We are given a series $F \in O_{m+1}, m \geqslant 1$ such that $\rho(F) > 0$. Show that there exists $r_0 \in ]0,1[$ such that $\hat{F}(r) \leqslant r$ for all $r \in [0, r_0]$. Show then, for $\gamma \in ]0,1[$, that $$\hat{F}(r) \leqslant \gamma^m r$$ for all $r \in [0, \gamma r_0]$.
Q25 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $r_0 > 0$ is as given by question (24).
Still for $F \in O_{m+1}, m \geqslant 1$, we set $$P := \sum_{k=m+1}^{2m} \frac{(F)_k}{\lambda^k - \lambda} z^k \in O_{m+1} \quad , \quad R := (I + P)^\dagger - I.$$ Show that $P \circ (\lambda I) - \lambda P - F \in O_{2m+1}$ and that $R + P \in O_{2m+1}$. Show that $\hat{P}(r) \leqslant \alpha_m r$ for all $r \in [0, \gamma_m r_0]$, and that $$\hat{R}(r) \leqslant \frac{\alpha_m}{1 - \alpha_m} r$$ for all $r \in [0, (1-\alpha_m)\gamma_m r_0]$.
Q26 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $P, R$ as defined in question (25).
For $F \in O_{m+1}, m \geqslant 1$, show that $$G := (I + P)^\dagger \circ (\lambda I + F) \circ (I + P) - \lambda I = (I + R) \circ (\lambda I + F) \circ (I + P) - \lambda I$$ satisfies $G \in O_{2m+1}$.
Q27 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $G$ as defined in question (26).
Show that $$\hat{G}(r) \leqslant \left(\alpha_m + (1 + \alpha_m)\alpha_m^2 + \frac{\alpha_m(1 + \alpha_m)(1 + \alpha_m^2)}{1 - \alpha_m}\right) r \leqslant r$$ for all $r$ such that $$0 \leqslant r \leqslant \frac{1 - \alpha_m}{(1 + \alpha_m)(1 + \alpha_m^2)} \gamma_m r_0$$
Q28 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We consider a power series $f = \lambda I + F$ with $F \in O_2, \rho(F) > 0$. We still assume that $\lambda$ has modulus 1 and is not a root of unity. We consider the real $r_0 > 0$ given by question (24) (applied for $m = 1$) and the sequence $r_k$ defined by recursion from $r_0$ by the relation $$r_{k+1} = (1 - \alpha_{2^k})(1 + \alpha_{2^k}^2)^{-1}(1 + \alpha_{2^k})^{-1} \gamma_{2^k} r_k.$$ Show that there exist sequences $F_k$ and $P_k$ of elements of $O_2$, defined for $k \geqslant 0$, such that $F_0 = F$ and, for all $k \geqslant 0$, $$\begin{aligned} & \lambda I + F_{k+1} = (I + P_k)^\dagger \circ (\lambda I + F_k) \circ (I + P_k), \\ & F_k \in O_{1+2^k}, \quad P_k \in O_{1+2^k}, \\ & \widehat{F_k}(r_k) \leqslant r_k, \quad \widehat{P_k}(r_{k+1}) \leqslant r_k - r_{k+1}. \end{aligned}$$
Q29 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We set $r_\infty := \lim r_k$ and $$h_k := (I + P_0) \circ (I + P_1) \circ \cdots \circ (I + P_{k-1}).$$ Explain why $r_\infty$ is well defined, and show that $\hat{h}_k(r_k) \leqslant r_0$ for all $k \geqslant 1$. Deduce that the series $h$ of question E satisfies $\hat{h}(r_\infty) \leqslant r_0$, thus that $\rho(h) \geqslant r_\infty$.