A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$: $$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$ Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define: $$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$ Investigate a) whether $M$ is a semigroup; b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.

For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$ ,$M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$. Find the smallest integer $n\geq 2 $ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$. ($M^T$ means the transpose of a matrix $M$ . A square matrix is called normal if $M^T M=M M^T$ holds). [i]Proposed by Martin Niepel (Comenius University, Bratislava)..[/i]

Let $n \ge 2$ be a positive integer and let $A \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $A^2=-I_n$. If $B \in \mathcal{M}_n(\mathbb{R})$ and $AB = BA$, prove that $\det B \ge 0$.

Let $ n\in{N^*}$,$ n\ge{3}$ and $ a_1,a_2,...,a_n\in{R^*}$, so that $ |a_i|\neq{|a_j|}$, for every $ i,j\in{\{1,2,...,n\}}, i\neq{j}$. Find $ p\in{S_n}$ with the property: $ a_ia_j < \equal{} a_{p(i)}a_{p(j)}$, for every $ i,j\in{\{1,2,....n\}}$,$ i\neq{j}$ (Teodor Radu)

Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)

Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that: [b]a)[/b] $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $ [b]b)[/b] $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$

Let be two $ 3\times 3 $ real matrices that have the property that $$ AX=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies BX=\begin{pmatrix}0\\0\\0\end{pmatrix} , $$ for any three-dimensional vectors $ X. $ Prove that there exists a $ 3\times 3 $ real matrix $ C $ such that $ B=CA. $

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by $m_{ij}=x$ if $i=j$, $m_{ij}=a$ if $i \not= j$ and $i+j$ is even, $m_{ij}=b$ if $i \not= j$ and $i+j$ is odd. For example $ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}$. Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ . P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)

A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

Let $$A=\left( \begin{array}{cc} 4 & -\sqrt{5} \\ 2\sqrt{5} & -3 \end{array} \right) $$ Find all pairs of integers \(m,n\) with \(n \geq 1\) and \(|m| \leq n\) such as all entries of \(A^n-(m+n^2)A\) are integer.

Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.

Let $\alpha\in\mathbb R\backslash \{ 0 \}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb R^n$ into $\mathbb R^n$ satisfying $F\circ G - G\circ F=\alpha F$. a) Show that for all $k\in\mathbb N$ one has $F^k\circ G-G\circ F^k=\alpha kF^k$. b) Show that there exists $k\geq 1$ such that $F^k=0$.

Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that $$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$ for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $ [i]Nelu Chichirim[/i]

Denote by $ H_n$ the linear space of $ n\times n$ self-adjoint complex matrices, and by $ P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $ H_n$ \[ \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)\] and its derived metric. Show that every $ \phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as \[ \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)\] or \[ \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)\] where $ U$ is an $ n\times n$ unitary matrix, $ X$ is a positive-semidefinite matrix, and $ ^T$ and $ ^*$ denote taking the transpose and the adjoint, respectively.

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

Let $A=[a_{ij}]_{n\times n}$ be a matrix with nonnegative entries such that $$\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.$$ (a) Prove that $|\det A|\le1$. (b) If $|\det A|=1$ and $\lambda\in\mathbb C$ is an arbitrary eigenvalue of $A$, show that $|\lambda|=1$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the question from a) still true if $AB\neq BA$ ?

$A,B \in M_{2\times 2}(C)$ Prove that: $Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$

Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that: \[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\] $ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$

Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$ let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$ Prove that the following statements are equivalent: $\textbf{i) }$ $|\det A | = 1$ $\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$ [i]Romeo Raicu[/i]