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research_proposal_1 | research_proposal | Critical Lengths and Controllability for the Kawahara Equation | We study the (linearized) Kawahara equation on $(0,L)$:
$$u_t + u_x + u_{xxx} - u_{xxxxx} = 0,$$
with boundary conditions
$$u(t,0)=u(t,L)=u_x(t,0)=u_x(t,L)=0,$$
and boundary control acting at $x=L$ through
$$u_{xx}(t,L)=h(t).$$
A length $L>0$ is called \emph{critical} if there exists a nontrivial solution of the stationary adjoint problem
$$-v^{(5)} + v^{\prime\prime\prime} + v^{\prime} + \lambda v = 0$$
satisfying six homogeneous boundary conditions; such solutions produce ``invisible'' modes and destroy observability. The set of critical lengths is denoted by $\mathcal M$.
Define explicitly the boundary characteristic determinant
$$D(p,L)=\det(\text{boundary matrix built from } e^{\mu_j(p)L})$$
where $\mu_1(p),\dots,\mu_5(p)$ are the five roots of
$$\mu^5-\mu^3-\mu + i p = 0.$$
Show that the critical lengths satisfy the implicit characterization
$$L\in\mathcal M \iff \exists\,p\in\mathbb R\text{ such that } D(p,L)=0.$$ | null | null | https://hal.science/hal-05111010v1/file/ADR-HAL.pdf |
research_proposal_2 | research_proposal | Critical Lengths and Controllability for the Kawahara Equation | For the Kawahara equation, the characteristic equation
$$\mu^5-\mu^3-\mu + i p = 0$$
has five roots, and criticality is determined by transcendental compatibility conditions involving exponentials $e^{\mu_j L}$. Unlike the KdV case, there is no obvious reduction to a simple resonance formula.
Classify possible configurations of the roots of $\mu^5-\mu^3-\mu + i p=0$ (real vs. complex). Determine which configurations can or cannot lead to critical lengths, and try to prove that only certain resonance patterns produce criticality. | null | null | https://hal.science/hal-05111010v1/file/ADR-HAL.pdf |
research_proposal_3 | research_proposal | Stabilizing quantities $\rho_p$ for boundary control of PDE | In the study of stability of nonlinear PDE of the form
$$\partial_t u + A(u)\,\partial_x u + B(u)=0\quad\text{on }[0,L],$$
with boundary conditions
$$\begin{pmatrix}u_+(t,0)\\u_-(t,L)\end{pmatrix}=G\begin{pmatrix}u_+(t,L)\\u_-(t,0)\end{pmatrix},$$
a natural quantity is
$$\rho_p(M)=\inf\{\|\Delta M\Delta^{-1}\|_p\;|\;\Delta=\mathrm{diag}(\delta_1,\dots,\delta_n),\ \delta_i>0\},$$
where $\|\cdot\|_p$ is the matrix $p$-norm.
(Conjecture 1) Prove that for any $M\in\mathbb R^{n\times n}$ one has
$$\rho_2(M)\le \rho_3(M)\le \cdots \le \rho_\infty(M).$$ | null | null | null |
research_proposal_4 | research_proposal | Stabilizing quantities $\rho_p$ for boundary control of PDE | With $\rho_p$ defined by diagonal scaling as
$$\rho_p(M)=\inf_{\Delta=\mathrm{diag}(\delta_1,\dots,\delta_n),\ \delta_i>0}\|\Delta M\Delta^{-1}\|_p,$$
it is known that for any $n\in\mathbb N$ and any $M\in\mathbb R^{n\times n}$,
$$\rho_1(M)\ge \rho_2(M),\qquad \rho_1(M)=\rho_\infty(M),$$
and for $n\ge 2$ there exists $M$ such that $\rho_1(M)>\rho_2(M)$. If $M$ has only non-negative coefficients, then
$$\rho_k(M)=\rho_1(M)\quad\forall k\in[1,+\infty].$$
(Conjecture 3) Show that for any $n>6$ and any integer $p>2$ there exists $M$ such that
$$\rho_p(M)<\rho_{p+1}(M).$$ | null | null | null |
research_proposal_5 | research_proposal | Boundary stability of the viscous Saint-Venant equations | Consider the viscous Saint-Venant system
$$\partial_t H + \partial_x(HV)=0,$$
$$\partial_t V + V\,\partial_x V + g\,\partial_x H + \frac{f(H,V)}{H} -4\mu\,\frac{\partial_x(H\partial_x V)}{H}=0,$$
with boundary conditions
$$V(t,0)=G_1(H(t,0)),\qquad V(t,L)=G_2(H(t,L),\partial_x V(t,L)),\qquad \partial_x V(t,0)=0.$$
(Conjecture 1) Let $(H^*,V^*)$ be a steady state satisfying
$$gH^*>(V^*)^2,$$
and assume in addition that
$$gH^*(0) < (2+\sqrt{2})\,V^*(0)^2.$$ Denote
$$b_0=G_1'(H^*(0)),\quad b_1=\partial_1G_2(H^*(L),V_x^*(L)),\quad c_1=\partial_2G_2(H^*(L),V_x^*(L)).$$
If
$$b_0\in(b_0^-,b_0^+),\qquad b_1\in\mathbb R\setminus[b_1^-,b_1^+],\qquad c_1\in(c_1^-,c_1^+),$$
where
$$b_0^{\pm}=g^{-1}\Bigl(-V^*(0)\pm\sqrt{\frac{1}{V^*(0)^2}-\frac{1}{gH^*(0)}}\Bigr),$$
$$b_1^{\pm}=g^{-1}\Bigl(-V^*(L)\pm\sqrt{\frac{1}{V^*(L)^2}-\frac{1}{gH^*(L)}}\Bigr),$$
$$c_1^{\pm}=\frac{4\Bigl(-V^*(L)-b_1H^*(L)\pm\sqrt{V^*(L)\bigl(H^*(L)V^*(L)b_1^2/g+2H^*(L)b_1+V^*(L)\bigr)}\Bigr)}{gH^*(L)-(V^*(L))^2},$$
then the system is locally exponentially stable for the $H^2$ norm around $(H^*,V^*)$. | null | null | null |
Q764 | RMS | null | a) We consider a set $X = \{x_1 < x_2 < \cdots < x_n\}$ of real numbers, and $k \leq n$. For any subset $F$ of $X$, we denote $\delta(X, F) = \max_{x \in X} \min_{y \in F} |x - y|$. Give an efficient algorithm to calculate the subset $F$ with cardinality $k$ such that $\delta(X, F)$ is minimal.
b) Resume the previous question when $X$ is a subset with cardinality $n$ in a metric space (thus $X$ is unordered). | Discrete Mathematics - Computer Science | difficile | null |
Q825 | RMS | null | Let $\Omega_n$ be the set of binary trees with $n$ leaves.
\begin{enumerate}[a)]
\item Find a recurrence relation for the cardinalities of $\Omega_n$.
\item Describe an algorithm to simulate a random binary tree with a uniform distribution on $\Omega_n$.
\item If $X$ is a random variable with a uniform distribution on $\Omega_n$, what is the expected height of $X$ (the height is the maximum length of a branch in the tree descending from the root)?
\end{enumerate} | Discrete Mathematics - Computer Science | moyen | null |
Q1038 | RMS | null | Following the corrected oral exercise 227 (RMS 111-9) and the response R712 published in RMS 132-2. A matrix $A = (a_{i,j})$ is said to satisfy $P(n, h, k)$ if
\begin{itemize}
\item $A$ is square of order $n$;
\item $a_{i,j} \in \{0, 1\}$ for all $i, j$ ;
\item there exists, for each $j$, exactly $h + k$ indices $i$ such that $a_{i,j} = 1$;
\item there exists, for all $j < j'$, exactly $k$ indices $i$ such that $a_{i,j} = a_{i,j'} = 1$;
\end{itemize}
In the solution to the exercise, it is proven that if $A$ satisfies $P(n, h, k)$ then $\transpose{A}$ also does and:
\begin{equation}
nh = (h + k)^2 - k.
\end{equation}
We consider three natural numbers $n, h, k$ satisfying (1) such that $n \geq 2$ and $k \geq 1$.
Give sufficient conditions for the existence of a matrix satisfying $P(n, h, k)$.
A counterexample was given in $R712$:
$n \geq 4$, $h = n - 2$, $k = 1$. | Discrete Mathematics - Computer Science | moyen | null |
Q1062 | RMS | null | In the response R1001 present in this journal, the solutions of the diophantine equation $(E_n)$ are studied:
\[ x_1 x_2 \cdots x_n = 2(x_1 - 2)(x_2 - 2) \cdots (x_n - 2), \]
where $5 \leq x_1 \leq x_2 \leq \cdots \leq x_n$. Here are three unresolved questions.
a) It is established at the end of the response that the sequence $(x_1, \ldots, x_n)$ given by $x_1 = 5$, then $x_{i+1} = x_i^2 - 3x_i + 3$ for $i < n - 1$ and $x_n = x_{n-1}^2 - 3x_{n-1} + 2$ is a solution of $(E_n)$ and there is strong reason to believe that it maximizes the product $x_1 x_2 \cdots x_n$ among all the solutions of $(E_n)$. Is this conjecture true?
b) Can we determine the solution of $(E_n)$ minimizing the product $x_1 x_2 \cdots x_n$?
c) It is established that the number $s_n$ of solutions of $(E_n)$ is at least $2^n$. Improve this; is the growth of $(s_n)$ geometric? | Discrete Mathematics - Computer Science | moyen | null |
Q604 | RMS | null | Let $A$, $B$ be polynomials in $K[X]$, coprime, and $U$ and $V$ be polynomials in $K[X]$ such that $\deg U < \deg B$ and $AU + BV = 1$. Let $m$ and $n$ be non-zero natural integers. Explicitly find the pairs $(U_{m,n}, V_{m,n})$ of polynomials in $K[X]$ such that $\deg U_{m,n} < \deg B^n$ and $A^m U_{m,n} + B^n V_{m,n} = 1$. | General Algebra - Arithmetic | facile | null |
Q831 | RMS | null | We denote $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$ as the elementary symmetric polynomials of $X_1, X_2, \ldots, X_n$. For $m = 2, 3, \ldots$, can we make explicit the $\Sigma_i(X_1^m, X_2^m, \ldots, X_n^m)$? | General Algebra - Arithmetic | difficile | null |
Q834 | RMS | null | In oral exam exercise 5 (118-4) it is shown that there exists a real sequence $(a_n)_{n \in \mathbb{N}}$ such that for every natural number $n \in \mathbb{N}$, the polynomial $a_n X^n + \cdots + a_1 X + a_0$ has exactly $n$ distinct real roots, but without providing an explicit formula. Does there exist a sequence of non-zero rational numbers $(a_n)_{n \in \mathbb{N}}$ such that for every $n$ in $\mathbb{N}^*$, the polynomial $a_0 + a_1 X + \cdots + a_n X^n$ is split and has simple roots over $\mathbb{Q}$? If YES, can such a sequence be made explicit? | General Algebra - Arithmetic | difficile | null |
Q938 | RMS | null | Let $n$ be an integer at least equal to two. In the group $S_n$ of permutations of order $n$, we denote by $f(n, k)$ the number of elements that are powers $k^e$. Let $p$ and $q$ be prime numbers divisors of $n!$ such that $p < q$. Is it true that $f(n, p) < f(n, q)$? | General Algebra - Arithmetic | difficile | null |
Q947 | RMS | null | Characterize the pairs $(G, f)$ where $G$ is a finite group of odd order and $f$, an involutive automorphism of $G$. | General Algebra - Arithmetic | facile | null |
Q995 | RMS | null | a) Can we characterize the monic polynomials $A$ with integer coefficients such that for all prime $p$ there exists an integer $a$ such that $p$ divides $P(a)$? b) Can we characterize the polynomials $A$ with integer coefficients such that for all integer $n$ there exists an integer $a$ such that $n$ divides $P(a)$? | General Algebra - Arithmetic | facile | null |
Q1003 | RMS | null | One classically shows that a polynomial $P$ with real coefficients taking only positive values can be written in the form $A^2 + B^2$ where $A$ and $B$ are in $\mathbb{R}[X]$. Find necessary and sufficient conditions for a polynomial $Q$ with rational coefficients to be written in the form $A^2 + B^2$ where $A$ and $B$ are in $\mathbb{Q}[X]$. | General Algebra - Arithmetic | difficile | null |
Q1021 | RMS | null | Let $G$ be a non-abelian finite group. What are the elements of $G$ that can be written as products of all elements of $G$ in any order, each element appearing exactly once? In particular, handle the case $G = S_n$.
Oral exercise 6, the solution of which appears in RMS 131-3, deals with the case $G = S_3$. | General Algebra - Arithmetic | moyen | null |
Q1037 | RMS | null | Let $K$ be a field. We denote by $(K^*)[2]$ the set of squares of the non-zero elements of $K$. A square root on $K^*$ is a group homomorphism $\varphi$ from $(K^*)[2]$ to $K^*$ such that $\varphi(x)^2 = x$ for all $x \in (K^*)[2]$. In the corrected oral exam problem 13 (RMS 128) we study for $K = \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}/p\mathbb{Z}$ the existence and uniqueness of a square root function. In fact, we prove that there exists one and only one if $K = \mathbb{R}$, $K$ has characteristic $2$, $K$ is finite with a cardinal congruent to $-1$ modulo $4$. There exists none if $K = \mathbb{C}$, $K$ is finite with a cardinal congruent to $1$ modulo $4$, $K$ has characteristic $p$ congruent to $1$ modulo $4$, and more generally, $K$ has an element whose square is $-1$. Here are some additional questions. a) A subfield $K$ of $\mathbb{R}$ has at least one square root: $x \mapsto \sqrt{x}$ defined on $(K^*)[2]$. Does there exist a strict subfield of $\mathbb{R}$ having no other square root? b) Let $K$ be a quadratic extension of $\mathbb{Q}$, real or not, and not containing $i$. Construct the square roots on $K$. c) Does there exist a field containing no element with a square $-1$ and possessing no square root? | General Algebra - Arithmetic | facile | null |
Q1043 | RMS | null | An integer $n > 1$ is said to be a powerful number if, for every prime divisor $p$ of $n$, $p^2$ also divides $n$.
\begin{enumerate}
\item Show that there exists a sequence of $n$ consecutive natural integers containing no powerful numbers.
\item Even better: show that there exists a sequence of $n$ consecutive natural integers containing no number that is the sum of two powerful numbers.
\item Can the result of b) be further improved?
\end{enumerate} | General Algebra - Arithmetic | difficile | null |
Q1070 | RMS | null | Here is problem 3, day 1 of the IMC 2023 Bulgaria International Mathematics Competition.
Determine the real polynomials $P(x, y)$ satisfying for all real numbers $x, y, z, t$ :
\[
P(x, y) P(z, t) = P(xz - yt, xt + yz).
\]
Now here is a proposition with four variables. Determine all real polynomials $P(x_1, x_2, x_3, x_4)$ satisfying for all real numbers $x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4$ :
\[
P(x_1, x_2, x_3, x_4) P(y_1, y_2, y_3, y_4) = P(z_1, z_2, z_3, z_4)
\]
where :
\[
z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4,\\
z_2 = x_1 y_2 + x_2 y_1 + x_3 y_4 - x_4 y_3, \\
z_3 = x_1 y_3 - x_2 y_4 + x_3 y_1 + x_4 y_2 \\
z_4 = x_1 y_4 + x_2 y_3 - x_3 y_2 + x_4 y_1.
\] | General Algebra - Arithmetic | facile | null |
Q655 | RMS | null | Following R626. Let $K$ be any field and $E$ a $K$-vector space of finite dimension $n \geq 2$. Let $C(f)$ denote the commutant of $f \in L(E)$. When $f$ and $g$ are not homotheties, what is the maximum dimension of $C(f)+C(g)$ and which pairs $(f, g)$ achieve the equality? | Linear and Bilinear Algebra | difficile | null |
Q706 | RMS | null | We denote $S_n([a, b])$ as the set of real symmetric matrices whose coefficients are all in $[a, b]$. For any real symmetric matrix $A$, we denote $\lambda_k(A)$ as the k extsuperscript{th} eigenvalue of $A$ in decreasing order. Study as a function of $n$, $a$, $b$ the bounds of the set of $\lambda_k(A)$ where $A$ runs through $S_n([a, b])$. | Linear and Bilinear Algebra | facile | null |
Q857 | RMS | null | The corrected oral exercise 73 (102-10) involved studying, for $A$ in $M_n(\mathbb{C})$, the infimum of $\|A - M\|$, where $M$ ranges over the set of matrices $M$ such that $\det M = 0$, $\|\cdot\|$ being the norm subordinate to the Hermitian norm.
Study in $M_n(\mathbb{C})$ (resp. $M_n(\mathbb{R})$) the infimum of $\|A - M\|$, where $M$ ranges over the set of matrices $M$ such that $\det M = 1$, $\|\cdot\|$ being the norm subordinate to the Hermitian (resp. Euclidean) norm. | Linear and Bilinear Algebra | difficile | null |
Q942 | RMS | null | This question follows on from R499 and the article “What is the maximum dimension of a vector subspace of \( M(n, \mathbb{R}) \) whose every non-zero element is invertible?”, (see RMS 127-4).
Let \( G(n) \) be the set consisting of \( \mathrm{GL}(n, \mathbb{R}) \) and the zero matrix of \( M(n, \mathbb{R}) \).
\begin{enumerate}
\item[a)] As a consequence of the Adams's theorem and the aforementioned article in this issue, \( M(6, \mathbb{R}) \) does not contain a vector subspace of dimension three that is included in \( G(6) \). Prove this result elementarily.
\item[b)] Two vector subspaces \( F \) and \( G \) of \( M(n, \mathbb{R}) \) are said to be equivalent if there exist \( P \) and \( Q \) in \( \mathrm{GL}(n, \mathbb{R}) \) such that \( M \mapsto PMQ \) maps \( F \) to \( G \).
Let \( H \) be the algebra of quaternions, i.e., the vector subspace of \( M(4, \mathbb{R}) \) generated by the matrices
\[
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix},
\begin{pmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0
\end{pmatrix}.
\]
Is there in \( M(4, \mathbb{R}) \) a four-dimensional vector subspace included in \( G(4) \) and not equivalent to \( H \)?
\end{enumerate} | Linear and Bilinear Algebra | difficile | null |
Q949 | RMS | null | Continuation of the corrected oral exam exercise 516 (124 4).
Let $E$ be a real normed vector space of finite dimension at least 2. In exercise 516, it is shown that for any non-zero element $a$ of $E$ and any element $b$ of $E$, the infimum of $\|\|\|f\|\|\|$, where $f$ is an endomorphism of $E$ such that $f(a) = b$, is $\frac{\|b\|}{\|a\|}$. Study the infimum of $\|\|\|f\|\|\|$, where $f$ is an endomorphism of $E$ such that $f(a_1) = b_1$ and $f(a_2) = b_2$, $(a_1, a_2)$ being a free system of $E$ and $(b_1, b_2)$ an element of $E^2$. | Linear and Bilinear Algebra | facile | null |
Q1013 | RMS | null | Let $M \in \mathrm{M}_n(\mathbb{C})$. We say that $M$ satisfies (P) if the set of subalgebras of $\mathrm{M}_n(\mathbb{C})$ containing $M$ is finite. In the oral exercise 15 of the year 2014, corrected in RMS 125-4, it is proven that if $M$ has $n$ distinct eigenvalues then $M$ satisfies (P).
a) Show that, for $n = 2$, the matrices satisfying (P) are the cyclic matrices $M$.
b) Is this result true for any $n$? | Linear and Bilinear Algebra | facile | null |
Q1026 | RMS | null | Let $E$ be a finite-dimensional vector space; for each endomorphism $u$ of $E$, denote by $z(u)$ the algebra of endomorphisms commuting with $u$.
\begin{enumerate}[a)]
\item If $u = [v, w]$, where $v$ and $w$ are in $z(u)$, show that $u$ is nilpotent.
\item Conversely, if $E$ is of dimension 3 and if $u$ is a nilpotent of rank 1, show that $u$ is of the form indicated in a).
\item Show that if $E$ is of dimension $pq$, where $p \geq 2$, and if $u$, nilpotent, has a Jordan form consisting of $q$ blocks of size $(p, p)$, then $u$ is not of the form indicated in a).
\item Do there exist nilpotent endomorphisms $u$ having at least two Jordan blocks of unequal sizes that are not of the form indicated in a)?
\end{enumerate} | Linear and Bilinear Algebra | difficile | null |
Q1034 | RMS | null | In this question, all matrices involved are complex square matrices of size $n$.
a) What are the real matrices $B$ such that for any real symmetric positive definite matrix $A$, $AB$ is diagonalizable in $M_n(\mathbb{R})$?
b) What are the complex matrices $B$ such that for any real symmetric positive definite matrix $A$, $AB$ is diagonalizable in $M_n(\mathbb{C})$?
c) What are the complex matrices $B$ such that for any Hermitian positive definite matrix $A$, $AB$ is diagonalizable in $M_n(\mathbb{C})$?
In oral exam 58 (stated in RMS 131-2 and corrected in RMS 131-3), one is asked to show that if $B$ is a real anti-symmetric matrix, it satisfies the property of b). | Linear and Bilinear Algebra | facile | null |
Q1051 | RMS | null | What is the maximum $a_n$ of the dimension of a subalgebra of $M_n(K)$ generated by two projectors? $K$ is an arbitrary field and $n$ is an integer at least equal to two. In the response R1020 published in the RMS 133-1, it is proven that $a_n = n^2$ if and only if $n = 2$ and $K$ has at least three elements. | Linear and Bilinear Algebra | facile | null |
Q1069 | RMS | null | We consider the space $S$ of real sequences, the subspace $B$ of bounded sequences, and we focus on operators of the form:
\[
\Phi : U \in S \mapsto V \in S \quad \text{where} \quad \forall n, \ v_n = a_1 u_{f_1(n)} + \ldots + a_d u_{f_d(n)}
\]
where the $a_i$ are strictly positive real numbers whose sum is $1$, and the $f_i : \mathbb{N} \to \mathbb{N}$ are functions such that $\forall i, \forall n, f_i(n) \geq n$.
\begin{enumerate}
\item[(a)] We assume that the $f_i$ commute pairwise. Show that $U \in B$ is invariant by $\Phi$ if and only if for every $i$ and every $n$, we have $u_n = u_{f_i(n)}$.
\item[(b)] We assume $\forall n, f_1(n) = n + 1$ and that for every $i$, $f_i - \text{Id}$ is bounded on $\mathbb{N}$. Show that $U \in B$ is invariant by $\Phi$ if and only if it is constant.
\end{enumerate}
We now take $\Phi(U) = V$ where $\forall n, v_n = \frac{1}{2}(u_{n+1} + u_{2n})$. We denote by $I$ the space of invariants of $\Phi$.
\begin{enumerate}
\item[(c)] Show that $\Theta : U \in I \mapsto (\Phi(U)_{2n+1})_{n \in \mathbb{N}} \in S$ is an isomorphism and that the image of $I \cap B$ by $\Theta$ is of infinite codimension in $B$.
\item[(d)] Can we characterize the bounded invariants of $\Phi$?
\end{enumerate} | Linear and Bilinear Algebra | facile | null |
Q1072 | RMS | null | a) Let $A \in M_n(\mathbb{C})$ and $A = UH$ be its polar decomposition ($U$ unitary, $H$ Hermitian positive). Show that for all $k \in \mathbb{N}$, $| \operatorname{tr}(A^k)| \leq \operatorname{tr}(H^k)$.
b) More generally, if $U_1, \ldots, U_k$ are unitary matrices and $H$ a positive Hermitian matrix, is it true that
$| \operatorname{tr}(U_1HU_2H \cdots U_kH)| \leq \operatorname{tr}(H^k)$ ?
The inequality in the first question is proved in Horn, Johnson, Topics in Matrix Analysis, as a corollary of a long succession of theorems (much more general). Is there a reasonably short direct proof? | Linear and Bilinear Algebra | facile | null |
Q756 | RMS | null | Following the corrected exercise 126 (121-4). Let $\beta$ be a real number and $\lambda$ a real number such that $|\lambda| > 1$. We denote by $F(\beta, \lambda)$ the set of functions $g$ in $C^1(\mathbb{R}, \mathbb{R})$ such that
\[ \forall t \in \mathbb{R}, \quad g'(t) = g(\lambda t) - \beta g(t). \]
There exists a non-zero polynomial function of degree $p$ belonging to $F(\lambda^p, \lambda)$ for every natural number $p$.
Study, for all $(\beta, \lambda)$, the existence of non-polynomial functions belonging to $F(\beta, \lambda)$. | Real and Functional Analysis | moyen | null |
Q758 | RMS | null | Let $v$ be a continuous function from $[0, 1]$ to $[0, 1]$. We denote by $E$ the vector space of continuous real functions on $[0, 1]$. For $f$ in $E$, we set :
$$Tf(x) = \int_0^1 f(xv(t))\,dt.$$
Find necessary and sufficient conditions for $T$ to be injective. | Real and Functional Analysis | difficile | null |
Q850 | RMS | null | It is known that any irrational algebraic number $\alpha \in \mathbb{R}$ satisfies a Diophantine inequality of the type
\[ \exists d \geq 2, \exists C > 0, \forall (p, q) \in \mathbb{Z} \times \mathbb{N}, \quad \left| \alpha - \frac{p}{q} \right| \geq \frac{C}{q^d} \cdot \]
A typical example of a number that does not satisfy this inequality is the Liouville number $x = \sum_{n \geq 1} 10^{-n!}$. For this real $x$, we are asked to find explicit examples of decreasing functions
$\psi : \mathbb{N}^* \rightarrow ]0, 1[$ such that
\[ \forall (p, q) \in \mathbb{Z} \times \mathbb{N}, \quad \left| x - \frac{p}{q} \right| \geq \psi(q). \] | Real and Functional Analysis | facile | null |
Q909 | RMS | null | Let $\varphi : [0, 1] \rightarrow \mathbb{R}$ be a convex function and $I$ the interval $]0, +\infty[$. Determine a necessary and sufficient condition on $\varphi$ so that there exists a function $f \in L^1(I) \cap L^\infty(I)$ such that $\ln \|f\|_{L^p(0,+\infty)} = \varphi\left(\frac{1}{p}\right)$ for all $p \geq 1$. | Real and Functional Analysis | difficile | null |
Q912 | RMS | null | According to exercise 112 of the RMS 124-2. We denote by $E$ the set of $C^1$ functions, concave on $[-1, 1]$, vanishing at $-1$ and $1$, and not identically zero.
a) Show the existence of $c > 0$ such that for every $f$ in $E$,
\[
\int_{-1}^{1} f(t)\,dt \geq c \int_{-1}^{1} |f'(t)|\,dt.
\]
What is the best possible constant $c$?
b) Show the existence of $c' > 0$ such that $\forall f \in E$,
\[
\int_{-1}^{1} f(t)\,dt \geq c' \int_{-1}^{1} \frac{f'(t)^2}{\sqrt{1 + f'(t)^2}}\,dt.
\]
What is the best possible constant $c'$?
c) For every $f$ in $E$ and every real number $\lambda$ such that $0 < \lambda \leq c'$, we define
\[
J_\lambda(f) = \int_{-1}^{1} f(t)\,dt - \lambda \int_{-1}^{1} \sqrt{1 + f'(t)^2}\,dt.
\]
Calculate the infimum of $f \mapsto J_\lambda(f)$. | Real and Functional Analysis | difficile | null |
Q941 | RMS | null | Let $S$ denote the Schwarz space on $\mathbb{R}$, which is the vector space of functions $f \in C^{\infty}(\mathbb{R}, \mathbb{R})$ such that $\sup_{x \in \mathbb{R}^n} |x^n f^{(m)}(x)| < +\infty$ for all $(m, n) \in \mathbb{N}^2$. For every integer $p \geq 1$, we define
\[ J_p := \inf_{f \in S} \int_{\mathbb{R}} f'(x)^2 + x^{2p}f(x)^2 \, dx, \ \text{and} \ \int_{\mathbb{R}} f(x)^2 \, dx = 1. \]
It can be shown that $J_1 = 1$ (and is attained for $f(x) = \pm \frac{1}{\sqrt[4]{\pi}} e^{-x^2/2}$ as the first "eigenfunction of the harmonic oscillator $-\frac{d^2}{dx^2} + x^2$ ").
a) Is it true that $(J_p)_{p \geq 1}$ is monotonic from a certain rank onward?
b) Is it true that
\[ \lim_{p \to +\infty} J_p = \frac{\pi^2}{4} ? \] | Real and Functional Analysis | difficile | null |
Q955 | RMS | null | Following oral exercise 93 corrected in RMS 128-3. For any natural integer $q$ and any real $x \in ]0, \pi[$, we consider the expression:
\[ S(q, x) = (q + 1) \sum_{k=q+1}^{\infty} \frac{\sin kx}{k} \cdot \frac{\sin x}{2}. \]
Determine its supremum. | Real and Functional Analysis | moyen | null |
Q983 | RMS | null | In the oral exercise 33 corrected in RMS 129-3, we are asked to study the limit values of the sequence $\{ \ln(n!) \}$ where $\{ x \}$ is the fractional part of $x$. For any integer $k > 0$, we define $u_k(n) = \{ \ln(\ln(\ldots(\ln(n!)) \ldots )) \}$ where the natural logarithm is iterated $k$ times.
\begin{itemize}
\item[a)] Show that the sequence $u_1(n)$ i.e. $\{ \ln(n!) \}$ is equidistributed modulo 1.
\item[b)] Show that the sequence $u_2(n)$ is dense in $[0, 1]$ but not equidistributed modulo 1.
\item[c)] What can be said about $u_k(n)$ for $k \geq 3$?
\end{itemize} | Real and Functional Analysis | difficile | null |
Q986 | RMS | null | Exercise 291 from RMS 119-2 asked if there exists a subset $A$ of $\mathbb{N}$ such that $f_A : x \mapsto \sum_{n \in A} \frac{x^n}{n!} \sim_{x \rightarrow +\infty} \frac{e^x}{x^2}$. A negative and fairly simple answer was given in RMS 119-4. This answer shows that the exponent 2 is not exceptional but leaves room for the following question. Does there exist $A \subset \mathbb{N}$, $\alpha \geq \frac{1}{2}$, and $C > 0$ such that
\[ f_A(x) \sim_{x \rightarrow +\infty} \frac{C \cdot e^x}{x^\alpha} \]? | Real and Functional Analysis | difficile | null |
Q1000 | RMS | null | Following R827, published in RMS 130-3.
For all integers $p$ and $q$ such that $0 < p < q$, we define $M(p, q) = \max_{x \in \mathbb{R}} |\sin(px) - \sin(qx)|$.
a) Show that the sequences
\[M(1, 4n)\]_{n \geq 1},
\[M(1, 4n + 1)\]_{n \geq 0},
\[M(1, 4n + 2)\]_{n \geq 0}
are strictly increasing.
b) Show that, for all $n \geq 1$, $M(1, 4n + 1) < M(1, 4n) < M(1, 4n + 2)$. | Real and Functional Analysis | moyen | null |
Q1007 | RMS | null | Consider the functional equation: (E) $f'(t) = f(t - t^2)$. We know that the set of functions defined on $[0, 1]$, differentiable and solutions of (E) is a vector space of dimension 1. Let $f$ be the unique solution on $[0, 1]$ such that $f(0) = 1$.
\begin{enumerate}
\item Is there a solution $g$ of $E$ defined on an interval strictly including $[0, 1]$ and extending $f$?
\item Is $f$ expandable in a power series at $0$?
\end{enumerate} | Real and Functional Analysis | difficile | null |
Q1023 | RMS | null | It is well known that the series with general term $\sin(\pi e n!)$ converges. Study the set of real $x$ such that the series with general term $\sin(x n!)$ converges. | Real and Functional Analysis | difficile | null |
Q1032 | RMS | null | n X k=0 a_n X^n \in \mathbb{R}[X] of degree n, has a mode at M \leq n if we have a_0 \leq a_1 \leq \cdots \leq a_M \geq a_{M+1} \geq \cdots \geq a_n.
Let $P$ be a polynomial with strictly positive coefficients; we are interested in its powers. If $P$ represents the generating function of a random variable $X$, then $P^N$ is the generating function of the sum of $N$ independent copies of $X$.
\begin{enumerate}
\item Is there a rank $N$ for which $P^N$ has a mode?
\item Does $P^N$ always have a mode for sufficiently large $N$?
\item Does this mode appear at an index $M_N$ equivalent to $\frac{P'(1)}{P(1)} N$?
\end{enumerate} | Real and Functional Analysis | facile | null |
Q565 | RMS | null | Let $T(n)$ be the Bell number of index $n : T(n) = e^{-1} \sum_{k=0}^{\infty} \frac{k^n}{k!}$. In \cite{R506} (116-4) it is established that $T(n) \sim f(n)$ where
\[ f(x) := \frac{1}{\sqrt{\ln x}} \frac{w(x)^x}{x} e^{w(x)-x-1}, \]
and $w : [0, +\infty[\to[1, +\infty[$ is the inverse function of $y \mapsto y \ln y$.
Estimate the next term in the asymptotic expansion. | Asymptotics | difficile | null |
Q776 | RMS | null | Let the entire series $f_m(x) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^m}$ for $m = 2, 3, \ldots$. Find a simple equivalent in $-\infty$ for $f_m$.
Note. In the article by Jean-Christophe Feauveau “some simple equivalents of sums of entire series” (107-5), we find the following equivalent of $f_m$ for $+\infty$, as a particular case of a more general formula: \[f_m(x) \sim (2\pi)^{(1-m)/2}m^{-1/2}x^{(1-m)(2m)}e^{mx^{1/m}}.\] | Asymptotics | facile | null |
Q839 | RMS | null | Let $q$ be a real number strictly between $-1$ and $1$. Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that is a solution of the functional equation
\[ f(x) = (1 - qx)f(qx). \]
In oral exam 21 (103-5), it is asked to prove the existence of a power series expansion of $f$ on $\mathbb{R}$.
Here, it is asked to study the asymptotic behavior of $f$ as $x$ approaches $+\infty$. | Asymptotics | difficile | null |
Q855 | RMS | null | Consider the real sequences $(u_n)$ given by the second-order recurrence relation:
\[ u_{n+2} = u_{n+1} + \lambda u_n \]
In the answer R773 (124-1) it has been proven that the solution space has a basis
$\{(v_n), (w_n)\}$, where $(v_n)$ is equivalent to $n^\lambda$ and $(w_n)$ to
\[ (-1)^n \frac{n^\lambda}{n! n^{\lambda + 1}}. \]
Continue the development of the dominant sequence $(v_n)$. More precisely, establish the existence of a real sequence $(\alpha_n)_{n \in \mathbb{N}^*}$ such that for every natural number $N$
\[ v_n = n^\lambda + \alpha_1 n^{\lambda-1} + \alpha_2 n^{\lambda-2} + \cdots + \alpha_N n^{\lambda-N} + o(n^{\lambda-N}). \] | Asymptotics | facile | null |
Q865 | RMS | null | This question followed the corrected oral exercise 2 (113-4) and then the answer R469 (116-2).
For $n \geq 2$, we consider the polynomial $P_n = X^n - \sum_{k=0}^{n-1} X^k$.
In R469, we showed the following results.
(i) The polynomial $P_n$ has a unique positive real root denoted $\rho_n$. The sequence $(\rho_n)$ is increasing and tends towards $2$. It admits for all $N$ an asymptotic expansion of the type:
\[
\rho_n = 2 \left(1 - \sum_{k=0}^{N} A_k(n) u_n^{k+1} + o(u_n^{N+1}) \right)
\]
with $u_n = 2^{-n-1}$, where the $A_k$ are polynomials.
(ii) We denote $r_n$ the smallest of the absolute values of the roots of $P_n$. The sequence $(r_n)$ tends towards $1$.
(iii) Furthermore, $\lim (2^n (1 - r_{2n})) = \ln 3$.
a) Can we provide a global formula for each polynomial $A_k$?
b) Can we affirm: $\rho_n = 2 \left(1 - \sum_{k=0}^{\infty} A_k(n) u_n^{k+1} \right)$?
c) Can we affirm: $\lim ((2n + 1) (1 - r_{2n+1})) = \ln 3$? | Asymptotics | facile | null |
Q1039 | RMS | null | If $P = \sum_{k=0}^{n} a_k X^n$ is a real or complex polynomial, we denote
\[ \|P\|_\infty = \sup_{t \in [0,1]} |P(t)| \]
and
\[ N(P) = \max_{0 \leq k \leq n} |a_k|; \]
and, for $n \in \mathbb{N}$, we denote
\[ C_n = \sup_{P \in \mathbb{R}_n[X], \|P\|_\infty = 1} N(P). \]
What estimates of $C_n$ can we establish? By "estimate," we mean: lower bound, upper bound; simple equivalent; predominance over and under other simple sequences. | Asymptotics | moyen | null |
Q1046 | RMS | null | In oral exam 116 from the RMS list 132 2, we ask to study certain properties of the solutions of the differential equation, defined on $\mathbb{R}^+$ :
\[ y'' = \frac{8x^n}{(1 + x^2)^2} y. \]
Study, according to the natural integer $n$, the asymptotic behavior of the solutions when $x$ tends towards $+\infty$. | Asymptotics | difficile | null |
Q1066 | RMS | null | Voici l’exercice d’oral 83 corrigé dans RMS 133-3. For $x \in \mathbb{R}$, let $\lceil x\rceil$ be the smallest integer greater than or equal to $x$. We define a sequence $(u_n)_{n \in \mathbb{N}^*}$ by $u_0 = 1$, $u_n = 2u_{n-1}$ for every integer $n \geq 2$ which is a power of $2$, $u_n = \lceil \frac{u_{n-1}}{3} \rceil$ for every integer $n \geq 3$ which is a power of $3$, and finally $u_n = u_{n-1}$ for every other natural number $n \geq 1$. Show that $u$ tends towards $+\infty$. The statement is supplemented with the following questions. a) Show that $u_n$ is negligible compared to any positive power of $n$. b) Provide a lower bound for $u_n$ with a simple sequence. c) Specify the asymptotic behavior of $u_n$. | Asymptotics | difficile | null |
Q701 | RMS | null | Let $a \in ]0, 1[$; we consider the set $C$ of functions from $[0, 1]$ to $\mathbb{R}$ which vanish at $0$ and are Hölder continuous of order $a$ with exponent $1$. Equipped with the uniform convergence on $[0, 1]$, $C$ is a convex compact set. Can we exhibit simple properties of the extreme points of $C$ and can we even characterize them? | Topology | facile | null |
Q748 | RMS | null | Let $A$ be a subset of the Euclidean space $\mathbb{R}^n$, arcwise connected and $C^1$ piecewise. If $x$ and $y$ are any two points $x$ and $y$ of $A$, we define their geodesic distance $d(x, y)$ as the infimum of the set of lengths of arcs connecting them. We identify $\mathrm{M}_n(\mathbb{R})$ with the Euclidean space $\mathbb{R}^{n^2}$. Determine the geodesic distance between two points in $\mathrm{GL}_n^+(\mathbb{R})$, then in the set of non-regular matrices. | Topology | difficile | null |
Q803 | RMS | null | Let $(a_n)$ be a real sequence, with strictly positive terms, decreasing and summable. We denote $\Sigma$ as the set of sums of extracted series (finite or infinite) from this sequence. Can we characterize the sequences of this type such that $\Sigma$ has an empty interior? | Topology | facile | null |
Q805 | RMS | null | Consider \( \mathbb{R}^2 \) with the usual dot product. For any norm \( N \) on \( \mathbb{R}^2 \), the dual norm \( N^\star \) is defined by: \[ \forall x \in \mathbb{R}^2, \quad N^\star(x) = \sup_{y \in \mathbb{R}^2\setminus\{0\}} \frac{\langle x, y \rangle}{N(y)}. \] Can we characterize the norms \( N \) such that there exists \( u \in \mathrm{GL}(\mathbb{R}^2) \) with \( N^\star = N \circ u \) ? | Topology | facile | null |
Q820 | RMS | null | \text{Following the corrected oral exercise 39 (113-4).} \\ \text{Let } C \text{ be a convex compact in } \mathbb{R}^2 \text{ with non-empty interior. We denote } \delta(C) \text{ the diameter of } C. \\ \text{For any point } P, \text{ we denote } \mu(P, C) \text{ the average of the distances from } P \text{ to the points of } C. \text{ We then set } \\ \rho(C) = \inf_{P} \mu(P, C). \\ \text{Study the bounds of } \frac{\rho(C)}{\delta(C)} \text{ where } C \text{ ranges over the set of convex compacts in } \mathbb{R}^2 \text{ with non-empty interior.} \\ \text{Exercise 39 shows that the lower bound is at least } \frac{1}{12}. | Topology | difficile | null |
Q963 | RMS | null | For each $\|\cdot\|$ on $\mathbb{R}^3$ we denote by $|||\cdot|||$ the associated subordinate norm. For any real $p$, we denote $A(p)$ the matrix
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & \frac{1}{2} & p \\
0 & 0 & \frac{1}{2}
\end{pmatrix}
\]. In oral exam 997 of RMS 128-2, $\|\cdot\|$ being one of the three standard norms on $\mathbb{R}^3$, we study the set of $p$ such that $|||A(p)||| \leq 1$.
a) Explicit a norm $\|\cdot\|$ such that $|||A(p)||| > 1$ for all $p$.
b) Can such norms be characterized? | Topology | difficile | null |
Q978 | RMS | null | Let $n \geq 1$. For all $P \in \mathbb{R}_n[X]$, we denote $\|P\|_1 = \int_0^1 |P(t)| \, dt$ and $\|P\|_\infty = \max_{0 \leq x \leq 1} |P(x)|$. Find $\inf_{P \in \mathbb{R}_n[X]} \frac{\|P\|_1}{\|P\|_\infty}$. | Topology | moyen | null |
Q1068 | RMS | null | Let $f : \mathbb{R}^n \to \mathbb{R}$ and $g : \mathbb{R}^n \to \mathbb{R}$ be two convex, differentiable, and lower-bounded functions. We equip $\mathbb{R}^n$ with the Euclidean norm $\| . \|$. Let $(\nabla f)(x)$ (resp. $(\nabla g)(x)$) denote the gradient vector of $f$ (resp. $g$) at $x$. Show that if $\| (\nabla f)(x) \| = \| (\nabla g)(x) \|$ for all $x$, then the function $f - g$ is constant. Is the convexity assumption on $f$ and $g$ necessary? | Topology | facile | null |
Q793 | RMS | null | Following the corrected oral exercise 24 (121-4). We denote $E_n$ the set of square matrices of order $n$ where all coefficients are 1 or -1. In the mentioned exercise, it is shown that the average of the determinants of the elements of $E_n$ is zero and that the average of the squares of these determinants is $n!$. Study the average of the absolute values of these determinants. | Probability | moyen | null |
Q975 | RMS | null | This question is from the oral exercise 169 corrected in the present journal.
Let $Z$ be a random variable with values in $\mathbb{N}^*$.
\begin{itemize}
\item[a)] Give a necessary and sufficient condition on $Z$ for there to exist $X$ and $Y$, independent random variables that are not almost surely constant, such that $Z = XY$.
\item[b)] Give a necessary and sufficient condition on $Z$ for there to exist $X$ and $Y$, independent random variables with the same distribution, such that $Z = XY$.
\end{itemize} | Probability | moyen | null |
Q1048 | RMS | null | This question follows the answer R859 published in RMS 133-1.
Let $(X_n)$ be a sequence of independent random variables with the same Bernoulli distribution, that is, uniformly distributed over \{0, 1\}. We consider the random version of the Fibonacci sequence $(u_n)$ defined by the recurrence relation
\[ u_{n+1} = u_n + X_n u_{n-1}, \quad u_0 = u_1 > 0 \]
It has been shown that: $(u_n)^{\frac{1}{n}}$ almost surely converges to a universal constant $K \simeq 1.33 \ldots$ while \[ \frac{u_{n+1}}{u_n} \]
lawfully converges to a discrete distribution supported on $[1, 2]$. We can thus naturally ask if for some constant $A > 0$ (possibly $A = 1$) and a positive sequence $(\omega_n)$ there would be a kind of central limit theorem, that is, a remarkable convergence in law of the random variable \[ \frac{u_n - A K^n}{\omega_n} \] ? (we naturally wish to avoid the case where $(\omega_n)$ grows too fast so that the limiting distribution could only be a Dirac mass). | Probability | facile | null |
Q1060 | RMS | null | In exercise 28 corrected in RMS 117-4 it is proven that any function $f : \mathbb{Z}^2 \to \mathbb{R}$ which is positive and harmonic on $\mathbb{Z}^2$, that is, satisfying
\[ f(m, n) = \frac{1}{4} \left(f(m, n -1) + f(m, n + 1) + f(m -1, n) + f(m + 1, n) \right) \]
for all integers $m$ and $n$, is constant.
The proof uses the recurrent nature of a random walk on $\mathbb{Z}^2$.
The same question is posed here regarding other lattices.
\begin{itemize}
\item[a)] The lattice $\mathbb{Z}[j]$ in $\mathbb{C}$. Here
\[ f(z) = \frac{1}{6} \sum_{u \in U_6} f(z + u), \]
where $U_6$ is the set of sixth roots of $1$, for all $z \in \mathbb{Z}[j]$.
\item[b)] The lattice $\mathbb{Z}^3$ in $\mathbb{R}^3$. Here
\[ f(x) = \frac{1}{6} \sum_{e \in E_3} f(x + e), \]
where $E_3$ is the set of vectors of the canonical basis of $\mathbb{R}^3$ and their opposites, for all $x$ in $\mathbb{R}^3$.
\end{itemize} | Probability | moyen | null |
Q1063 | RMS | null | Let $X$ be a real random variable such that $E(|X|)$ is finite. We assume that $X$ is symmetric, i.e., $X$ and $-X$ have the same distribution.
A point is placed on the real line according to the distribution of $X$. I am initially placed at $0$ and I set off blindly in search of the point, making a series of journeys alternately to the right and to the left. As soon as I have landed on the point, I return to the origin. Let $T$ be the random variable indicating the total length of the journey I have made.
a) Does there exist a strategy such that $E(T)$ is finite?
b) Does there exist an optimal strategy, i.e., one that minimizes $E(T)$?
c) Specify an optimal strategy, if it exists, in particular cases, for example: $X$ follows the uniform distribution on $[-1, 1]$, $X$ follows a Gaussian distribution. | Probability | difficile | null |
Q804 | RMS | null | We denote by $S$ the unit sphere of $\mathbb{R}^3$. For every finite subset $X$ of $S$ with at least two elements, we denote $\delta(X)$ as the minimum distance between two points of $X$; we denote $d(n)$ as the maximal value of $\delta(X)$, where $X$ consists of $n$ points of $S$.
a) Suppose there exists a regular polyhedron with $n$ vertices inscribed in $S$. Is it true that $d(n)$ is the length of the side of this regular polyhedron?
b) Study the limit of $\sqrt{n} \, d(n)$. | Geometry | facile | null |
Q832 | RMS | null | Let us consider a regular dodecagon $P$ and form the twelve squares 'directed' towards the center of $P$ with sides that are sides of $P$. The twelve squares thus obtained verify the following two properties:
\begin{itemize}
\item two different squares do not share a common side;
\item a vertex belongs to exactly two squares.
\end{itemize}
What are the natural numbers $n$ such that one can arrange $n$ isometric squares in the Euclidean plane with the two aforementioned properties? | Geometry | difficile | null |
Q896 | RMS | null | In the oriented real affine plane, equipped with a direct coordinate system $(O, \vec{ı}, \vec{ȷ})$, we consider $n$ distinct rays all having $O$ as origin. They divide the plane into $n$ zones $Z_1, Z_2, \ldots, Z_n$ which are the connected components of the complement of the union of these rays. It is assumed that none of these zones include an open half-plane. A direct convex polygonal line with $p$ sides is called any sequence $(A_t)_{t \in \mathbb{Z}}$ with period $p$, such that $\det(\overrightarrow{A_kA_{k+1}}, \overrightarrow{A_kA_\ell}) > 0$ for all $k, \ell$ between $0$ and $p -1$ such that $\ell$ differs from $k$ and from $k + 1$. We denote by $M_k$ the point such that $\overrightarrow{OM_k} = \overrightarrow{A_kA_{k+1}}$ and it is assumed that for every $k$, there is no $j$ such that $M_{k+1}$ and $M_k$ are both adherent to $Z_j$. Show that $p$ is at most equal to $n$. | Geometry | moyen | null |
Q971 | RMS | null | Construct the sets of four distinct, non-collinear points in the Euclidean plane such that every distance between two of these points is an integer. In exercise 163 of RMS 125-2, solved in RMS 125-3, it is shown that these distances cannot all be odd integers. Moreover, the book "Key Problems for Advanced Mathematics" by H. Gianella, R. Krust, F. Taïeb, N. Tosel at Calvage & Mounet provides a construction method for establishing solutions that are inscriptible quadrilaterals or sets consisting of triangles and their circumcenter.
The task is to complete the list. | Geometry | facile | null |
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