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

2021 x-ens-maths__pc

44 maths questions

Q1.1 Proof Proof of Set Membership, Containment, or Structural Property View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that $\|P\|_K$ belongs to $\mathbb{R}$ for all $P \in \mathbb{C}[X]$.

Q1.2 Proof Proof That a Map Has a Specific Property View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Verify that $\|\cdot\|_K$ is a norm on $\mathbb{C}[X]$.

Q1.3 Proof Direct Proof of an Inequality View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Let $Q$ and $R$ be two non-zero polynomials in $\mathbb{C}[X]$. Show that: $$\|Q\|_K \|R\|_K \geq \|QR\|_K.$$

Q1.4 Proof Existence Proof View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that, if the inequality $\|Q\|_K \|R\|_K \geq \|QR\|_K$ is an equality, then there exists $z_0 \in K$ such that: $$\left|Q\left(z_0\right)\right| = \|Q\|_K, \quad \left|R\left(z_0\right)\right| = \|R\|_K \quad \text{and} \quad \left|Q\left(z_0\right)R\left(z_0\right)\right| = \|QR\|_K.$$

Q1.5 Proof Direct Proof of an Inequality View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Show that $C_{n,m}^K > 1$.
To do this, one may choose two distinct elements $a$ and $b$ in $K$ and verify that, for $\rho \in \mathbb{R}$ sufficiently large, we have $\left\|Q_\rho R_\rho\right\|_K < \left\|Q_\rho\right\|_K \left\|R_\rho\right\|_K$ with $Q_\rho(X) = X - (\rho(b-a)+a)$ and $R_\rho(X) = X - (\rho(a-b)+b)$.

Q1.6 Proof Existence Proof View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$. We introduce the $\mathbb{C}$-vector space $V = \mathbb{C}_n[X] \times \mathbb{C}_m[X]$ as well as the set: $$E = \left\{(Q,R) \in V \mid \|Q\|_K = \|R\|_K = 1\right\}.$$ Show that there exists a pair $\left(Q_0, R_0\right) \in E$ such that: $$\left\|Q_0 R_0\right\|_K = \inf\left\{\|QR\|_K \mid (Q,R) \in E\right\}.$$ To do this, one may equip $V$ with the norm defined by $$\|(Q,R)\| = \|Q\|_K + \|R\|_K$$ for $(Q,R) \in V$, then study the application $$\begin{aligned} f : E &\rightarrow \mathbb{R} \\ (Q,R) &\mapsto \|QR\|_K. \end{aligned}$$

Q1.7 Proof Deduction or Consequence from Prior Results View

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce that there exist two monic polynomials $Q_1 \in \mathbb{C}_n[X]$ and $R_1 \in \mathbb{C}_m[X]$ such that: $$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$

Q2.8 Proof Existence Proof View

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. Verify that the integral: $$\int_0^{2\pi} \ln\left|Q\left(e^{i\theta}\right)\right| d\theta$$ converges absolutely in the sense of Definition 1.
One may use the d'Alembert-Gauss theorem.

Q2.9 Proof Direct Proof of an Inequality View

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We set for $p > 0$: $$M_p(Q) = \frac{1}{2\pi} \int_0^{2\pi} \left|Q\left(e^{i\theta}\right)\right|^p d\theta$$ Explain why $M_p(Q)$ is strictly positive for all $p > 0$.

Q2.10 Proof Proof That a Map Has a Specific Property View

Q2.11 Proof Proof That a Map Has a Specific Property View

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show carefully that $\varphi$ is differentiable on $]0,+\infty[$ and compute its derivative on this interval.

Q2.12 Proof Computation of a Limit, Value, or Explicit Formula View

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$ and $M(Q) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\ln\left|Q(e^{i\theta})\right|d\theta\right)$. Compute the limit of $\varphi'$ at $0^+$ then deduce that: $$M_p(Q)^{1/p} \underset{p \rightarrow 0^+}{\longrightarrow} M(Q).$$

Q2.13 Proof Direct Proof of a Stated Identity or Equality View

For each complex number $w$, we denote by $\operatorname{Re}(w)$ the real part of $w$. Show that, for all $z \in \stackrel{\circ}{\mathbb{D}}$: $$\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty} \frac{z^n}{n}\right)$$ To do this, one may write $z = re^{i\theta}$ with $0 \leq r < 1$ and $\theta \in \mathbb{R}$, then study the function: $$\begin{aligned} F : [0,1[ &\rightarrow \mathbb{R} \\ \rho &\mapsto \ln\left|1 - \rho e^{i\theta}\right| \end{aligned}$$

Q2.14 Proof Deduction or Consequence from Prior Results View

Let $z \in \stackrel{\circ}{\mathbb{D}}$. Using the result that for all $z \in \stackrel{\circ}{\mathbb{D}}$, $\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty}\frac{z^n}{n}\right)$, deduce that the Mahler measure of the polynomial $X - z$ is 1 and, in the case where $z \neq 0$, that of the polynomial $X - z^{-1}$ is $|z|^{-1}$.

Q2.15 Proof Existence Proof View

Using the result of question 2.14, show that, for all $z \in \partial\mathbb{D}$, the Mahler measure of $X - z$ is 1.
To do this, one may be interested in the function: $$\begin{aligned} g : [0,1[ &\rightarrow \mathbb{R} \\ r &\mapsto M(X - rz) \end{aligned}$$ and note that, for all $r \in [0,1[$ and $\theta, \psi \in \mathbb{R}$, we have the inequality $\left|e^{i\theta} - re^{i\psi}\right| \geq |\sin(\theta - \psi)|$.

Q2.16 Proof Deduction or Consequence from Prior Results View

Let $\lambda$ be the leading coefficient of $Q$ and let $\alpha_1, \ldots, \alpha_n$ be the roots of $Q$ counted with multiplicity. Deduce from the previous questions that: $$M(Q) = |\lambda| \prod_{i=1}^n \max\left\{1, \left|\alpha_i\right|\right\}$$

Q3.17 Complex numbers 2 Contour Integration and Residue Calculus View

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$ and all $r \in \mathbb{R}$, we have: $$P(z) = \frac{1}{2\pi} \int_0^{2\pi} P\left(z + re^{i\theta}\right) d\theta$$

Q3.18 Complex numbers 2 Properties of Analytic/Entire Functions View

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Deduce from question 3.17 that: $$\|P\|_{\mathbb{D}} = \|P\|_{\partial\mathbb{D}}$$ One may apply question 3.17 to an element $z \in \mathbb{D}$ such that $|P(z)| = \|P\|_{\mathbb{D}}$.

Q3.19 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$: $$|P(z)| \leq \|P\|_{\partial\mathbb{D}} \max\{1, |z|\}^d.$$ One may apply question 3.18 to the polynomials $P(X)$ and $Q(X) = X^d P\left(X^{-1}\right)$.

Q3.20 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$

Q3.21 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Deduce from question 3.20 that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq M(S)$$ where $S$ is the polynomial defined by: $$S(X) = (X-1)^{m+n} P\left(\frac{uX - v}{X - 1}\right)$$

Q3.22 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $w = \frac{v}{u}$. Show that: $$M(S) \leq \|P\|_{\mathbb{D}} \exp\left(\frac{n+m}{2\pi} \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} - w\right|\right\}\right) d\theta\right)$$

Q3.23 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Using the previous questions, show that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq C^{n+m} \|P\|_{\mathbb{D}}$$

Q3.24 Sequences and Series Evaluation of a Finite or Infinite Sum View

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Show that: $$I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$$ You may use the result from question 2.13.

Q3.25 Sequences and Series Estimation or Bounding of a Sum View

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with $I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$. The calculator gives: $$\begin{aligned} \exp\left(\frac{2}{\pi}\sum_{k=0}^5 \frac{(-1)^k}{(2k+1)^2}\right) &\approx 1{,}78774486868 \\ \exp\left(\frac{2}{\pi}\sum_{k=0}^6 \frac{(-1)^k}{(2k+1)^2}\right) &\approx 1{,}79449196958 \end{aligned}$$ Can we deduce the rounding of $C$ to $10^{-2}$ precision? If yes, give the value of this rounding. In any case, justify the answer properly.

Q3.26 Complex numbers 2 Algebraic Number Theory over C View

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with $I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta}-1\right|, \left|e^{i\theta}+1\right|\right\}\right)d\theta$. For each natural integer $k \geq 2$, we set: $$\begin{aligned} & Q_k(X) = \prod_{\zeta \in U}(X - \zeta), \\ & R_k(X) = \prod_{\zeta \in V}(X - \zeta), \end{aligned}$$ where $U$ denotes the set of $k$-th roots of unity $\zeta$ such that $|\zeta - 1| \leq |\zeta + 1|$ and $V$ the set of $k$-th roots of unity that are not in $U$. By bounding below the quotient: $$\frac{\left\|Q_k\right\|_{\mathbb{D}} \left\|R_k\right\|_{\mathbb{D}}}{\left\|Q_k R_k\right\|_{\mathbb{D}}},$$ show that: $$C = \inf\left\{\lambda \in \mathbb{R} \left\lvert\, \begin{array}{c} \forall Q \in \mathbb{C}[X]\backslash\{0\},\quad \forall R \in \mathbb{C}[X]\backslash\{0\}, \\ \|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq \lambda^{\operatorname{deg}(QR)} \|QR\|_{\mathbb{D}} \end{array} \right.\right\},$$ where $\operatorname{deg}(QR)$ denotes the degree of $QR$.

Q4.27 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set: $$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$ Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties: $$\begin{gathered} \|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\ A(a) = C(c), \quad B(b) = D(d). \end{gathered}$$

Q4.28 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall: $$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.

Q4.29 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$, i.e., $Q_0$ and $R_0$ are monic and $\frac{\|Q_0\|_I \|R_0\|_I}{\|Q_0 R_0\|_I} = C_{n,m}$.
Let $J$ be a segment contained in $I$ such that $\left\|Q_0\right\|_J = \left\|Q_0\right\|_I$ and $\left\|R_0\right\|_J = \left\|R_0\right\|_I$. Show that: $$\left\|Q_0 R_0\right\|_J = \left\|Q_0 R_0\right\|_I$$

Q4.30 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$. Deduce from questions 4.27 and 4.29 that there exists an extremal pair $(Q_1, R_1)$ such that: $$\left\|Q_1\right\|_I = \left|Q_1(-1)\right| \quad \text{and} \quad \left\|R_1\right\|_I = \left|R_1(1)\right|$$

Q4.31 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_1, R_1)$ be an extremal pair such that $\left\|Q_1\right\|_I = \left|Q_1(-1)\right|$ and $\left\|R_1\right\|_I = \left|R_1(1)\right|$. Let $n_1$ and $m_1$ be the degrees of $Q_1$ and $R_1$ respectively. We set $Q_2 = X^{n-n_1} Q_1$ and $R_2 = X^{m-m_1} R_1$.
Show that $(Q_2, R_2)$ is a good extremal pair, i.e., $Q_2$ and $R_2$ are monic of degrees $n$ and $m$ respectively, $\frac{\|Q_2\|_I \|R_2\|_I}{\|Q_2 R_2\|_I} = C_{n,m}$, and $\|Q_2\|_I = |Q_2(-1)|$, $\|R_2\|_I = |R_2(1)|$.

Q4.32 Proof Existence Proof View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_2, R_2)$ be a good extremal pair. Let $w$ be a root of $Q_2$ and let $S \in \mathbb{C}[X]$ be such that: $$Q_2(X) = (X - w)S(X)$$ By setting: $$S_2(X) = (X + 1 - |w+1|)S(X)$$ show that $(S_2, R_2)$ is a good extremal pair.

Q4.33 Proof Deduction or Consequence from Prior Results View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Deduce from question 4.32 that there exists a polynomial $Q_3$ whose roots are all in $[-1, +\infty[$ and such that the pair $(Q_3, R_2)$ forms a good extremal pair.

Q4.34 Proof Existence Proof View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Show that there exists a polynomial $Q_4$ whose roots are all in $I$ and such that the pair $(Q_4, R_2)$ forms a good extremal pair.
To do this, given a root $w$ of $Q_3$ that is not in $I$, one may introduce the polynomial: $$S_3(X) = \frac{X-1}{X-w} Q_3(X)$$ then one may follow the method used in the two previous questions.

Q4.35 Proof Existence Proof View

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Briefly explain why there exists a polynomial $R_4$ such that the pair $(Q_4, R_4)$ forms a very good extremal pair, i.e., a good extremal pair in which all complex roots of $Q_4$ and $R_4$ are contained in $I$.

Q4.36 Proof Direct Proof of a Stated Identity or Equality View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. Show that: $$Q = \prod_{k=m+1}^{n+m}\left(X - x_k\right) \quad \text{and} \quad R = \prod_{k=1}^{m}\left(X - x_k\right)$$

Q4.37 Proof Direct Proof of an Inequality View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity, with $Q = \prod_{k=m+1}^{n+m}(X-x_k)$ and $R = \prod_{k=1}^{m}(X-x_k)$.
Verify that for all $x \in ]-\infty, -1[$, we have $|Q(x)| > |Q(-1)|$.

Q4.38 Proof Direct Proof of a Stated Identity or Equality View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By contradiction, show that $|P(-1)| = \|P\|_I$.
To do this, one may choose a real number $\epsilon > 0$, introduce the segment $I_\epsilon = [-1-\epsilon, 1]$ and bound the quantity: $$\frac{\|Q\|_{I_\epsilon} \|R\|_{I_\epsilon}}{\|P\|_{I_\epsilon}}$$ using question 4.37.

Q4.39 Proof Existence Proof View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. We are given an integer $k \in \{m+1, m+2, \ldots, m+n-1\}$ and we set: $$S(X) = \left(X - x_k\right)\left(X - x_{k+1}\right).$$ Show that for all $\epsilon > 0$, there exists a polynomial $T \in \mathbb{R}[X]$ such that $S - T$ has degree 1 and: $$\begin{gathered} \|S - T\|_I \leq \epsilon \\ |T(-1)| = |S(-1)| \\ \forall x \in ]-1,1] \backslash ]x_k - \epsilon, x_{k+1} + \epsilon[, |T(x)| < |S(x)|. \end{gathered}$$ Deduce that there exists $y \in ]x_k, x_{k+1}[$ such that $|P(y)| = \|P\|_I$.
To handle this last point, one may proceed by contradiction, write $Q$ in the form $SU$ for a certain polynomial $U$, then verify that if $\epsilon$ is chosen appropriately, the pair $(TU, R)$ forms a very good extremal pair.

Q4.40 Proof Existence Proof View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. Following the method used in question 4.39, show that there exists an element $y \in ]x_m, x_{m+1}[$ such that $|P(y)| = \|P\|_I$.

Q4.41 Proof Direct Proof of a Stated Identity or Equality View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Show that $P$ satisfies the differential equation: $$\|P\|_I^2 - P^2 = \frac{1}{(n+m)^2}\left(1 - X^2\right)P'^2$$

Q4.42 Proof Deduction or Consequence from Prior Results View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Deduce from question 4.41 that: $$\left(1 - X^2\right)P'' - XP' + (n+m)^2 P = 0$$

Q4.43 Proof Direct Proof of a Stated Identity or Equality View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By considering the function: $$\begin{aligned} f : \mathbb{R} &\rightarrow \mathbb{R} \\ y &\mapsto P(\cos y), \end{aligned}$$ verify that for all $x \in [-1,1]$, $$P(x) = \|P\|_I \cos\left((n+m)\operatorname{Arccos} x\right).$$

Q4.44 Proof Deduction or Consequence from Prior Results View

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Deduce from question 4.43 that: $$C_{n,m} = 2^{n+m-1} \cdot \left[\prod_{k=1}^n \left(1 + \cos\left(\frac{2k-1}{2(n+m)}\pi\right)\right)\right] \cdot \left[\prod_{k=1}^m \left(1 + \cos\left(\frac{2k-1}{2(n+m)}\pi\right)\right)\right].$$