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-c__mp

17 maths questions

Q3 Taylor 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 Taylor 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 $\rho(f \cdot g) \geqslant \min(\rho(f), \rho(g))$.
Q5 Taylor series Taylor's formula with integral remainder or asymptotic expansion View
If $f \in O_n, n \geqslant 0, g \in O_1, h \in O_l, l \geqslant 1$ and $r \geqslant 1$, show that $h^r \in O_{rl}$, that $f \circ h \in O_{nl}$ and $f \circ (g + h) - f \circ g \in O_{n+l-1}$.
Q6 Taylor 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 Taylor series 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 Taylor series 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 Taylor 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 Taylor 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 Taylor 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.
Q23 Taylor 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 Taylor series 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 Taylor series 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 Taylor series 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 $\rho(h) \geqslant r_\infty$.