$A, B$ are $n \times n$ matrices such that $\text{rank}(AB-BA+I) = 1.$ Prove that $\text{tr}(ABAB)-\text{tr}(A^2 B^2) = \frac{1}{2}n(n-1).$

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.

Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$. Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as $$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$, and $N_i\ne N_j$ for $i\ne j$.

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

Let $A{}$ and $B{}$ be $3\times 3{}$ matrices with complex entries, satisfying $A^2=B^2=O_3$. Prove that if $A{}$ and $B{}$ commute, then $AB=O_3$. Is the converse true?

Do either $(1)$ or $(2)$ $(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$. $(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$

Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties: 1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$ 2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\] Show that in fact \[\mathbf M=\mathbb R^2.\]

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative: $ \begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix} $

The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.

Let $A$ be a $3 \times 9$ matrix. All elements of $A$ are positive integers. We call an $m\times n$ submatrix of $A$ "ox" if the sum of its elements is divisible by $10$, and we call an element of $A$ "carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in $A$.

Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.

[list=a] [*]Let $A$ be a matrix from $\mathcal{M}_{2}(\mathbb{C})$, $A\neq aI_{2}$, for any $a\in\mathbb{C}$. Prove that the matrix $X$ from $\mathcal{M} _{2}(\mathbb{C})$ commutes with $A$, that is, $AX=XA$, if and only if there exist two complex numbers $\alpha$ and $\alpha^{\prime}$, such that $X=\alpha A+\alpha^{\prime}I_{2}$. [*]Let $A$, $B$ and $C$ be matrices from $\mathcal{M}_{2}(\mathbb{C})$, such that $AB\neq BA$, $AC=CA$ and $BC=CB$. Prove that $C$ commutes with all matrices from $\mathcal{M}_{2}(\mathbb{C})$.[/list]

Let $n$ and $k$ be positive integers, where $n > 1$ is odd. Suppose $n$ voters are to elect one of the $k$ cadidates from a set $A$ according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a $k$ x $n$ voting matrix. We denote the number of ccurences of $a \in A$ in the $i$-th row of the voting matrix by $a_{i}$ . Let $l_{a}$ stand for the minimum integer $l$ for which $\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}$. Setting $l'= min \{l_{a} | a \in A\}$, we will regard the voting matrices which make the set $\{a \in A | l_{a} = l' \}$ as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if $w_{1} \geq w_{2} \geq ... \geq
w_{k} \geq 0$ are given, for each admissible voting matrix, $\sum^{k}_{i=1}{w_{i}a_{i}}$ is called the total weighted score of $a \in A$. We will say that the system $(w_{1},w_{2}, . . . , w_{k})$ of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates. (a) Determine whether there is a system of weights representing majoritarian compromise if $k = 3$. (b) Show that such a system of weights does not exist for $k > 3$.

Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.

Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

Let $A,B\in\mathcal{M}_n(\mathbb{Z})$ and $p$ a prime number. Prove that $$\text{Tr}((A+B)^p)\equiv\text{Tr}(A^p+B^p)\pmod p.$$

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties: (1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s. (2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.

We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.

Let be a $ 2\times 2 $ real matrix $ A $ whose primary diagonal has positive elements and whose secondary diagonal has negative elements. If $ \det A>0, $ show that [b]a)[/b] for any $ 2\times 2 $ matrix $ X $ of positive real numbers there exists a $ 2\times 2 $ matrix of positive real numbers such that $ AY=X. $ [b]b)[/b] there is a $ 2\times 2 $ matrix $ Z $ of positive real numbers having the property that all elements of $ AZ $ are positive. [i]Vasile Pop[/i]

Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$ a) Prove that $(AB - BA)^2 = O_{n}.$ b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$

Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left( \begin{array}{c} 1 \\ -1 \\ -1 \\ 1 \end{array} \right),\left( \begin{array}{c} 1 \\-1 \\ 1 \\ -1 \end{array} \right).$ Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$