Let $A = (0, 0, 0)$ in 3D space. Define the [i]weight[/i] of a point as the sum of the absolute values of the coordinates. Call a point a [i]primitive lattice point[/i] if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an [i]unbalanced primitive integer square[/i] if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.

Let $ A \equal{} (a_{ik})$ be an $ n\times n$ matrix with nonnegative elements such that $ \sum_{k \equal{} 1}^n a_{ik} \equal{} 1$ for $ i \equal{} 1,...,n$. Show that, for every eigenvalue $ \lambda$ of $ A$, either $ |\lambda| < 1$ or there exists a positive integer $ k$ such that $ \lambda^k \equal{} 1$

Let $A$ be a $n\times n$ diagonal matrix with characteristic polynomial $$(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k}$$ where $c_1, c_2, \ldots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1+d_2+\ldots + d_k=n$). Let $V$ be the space of all $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is $$d_1^2+d_2^2+\cdots + d_k^2$$

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

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

An elementary change in a 0-1 matrix is ​​a change in an element and with it all its horizontal, vertical, and diagonal neighbors (0 to 1 or 1 to 0). Can any 1791 x 1791 0-1 matrix be transformed into a zero matrix with elementary changes?

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 $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and \[(A^2+B^2)(A^4+B^4)=A^5+B^5.\] Find all possible values of $\det(AB)$ for the given $n$.

Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that \[ \mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array} \right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n \end{array} \right). \] (Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)

Let $n$ be a positive integer. For any positive integer $k$, let $0_k=diag\{\underbrace{0, ...,0}_{k}\}$ be a $k \times k$ zero matrix. Let $Y=\begin{pmatrix} 0_n & A \\ A^t & 0_{n+1} \end{pmatrix}$ be a $(2n+1) \times (2n+1)$ where $A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}$ is a $n \times (n+1)$ real matrix. Let $A^T$ be transpose matrix of $A$ i.e. $(n+1) \times n$ matrix, the element of $(j, i)$ is $x_{i, j}$. (a) Let complex number $\lambda$ be an eigenvalue of $k \times k$ matrix $X$. If there exists nonzero column vectors $v=(x_1, ..., x_k)^t$ such that $Xv=\lambda v$. Prove that 0 is the eigenvalue of $Y$ and the other eigenvalues of $Y$ can be expressed as a form of $\pm \sqrt{\lambda}$ where nonnegative real number $\lambda$ is the eigenvalue of $AA^t$. (b) Let $n=3$ and $a_1$, $a_2$, $a_3$, $a_4$ are $4$ distinct positive real numbers. Let $a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}$ and $x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j$ where $1\leq i \leq 3, 1\leq j \leq 4$, $\delta_{i, j}= \begin{cases} 1 \text{ if } i=j\\ 0 \text{ if } i\neq j\\ \end{cases}\,$. Prove that $Y$ has 7 distinct eigenvalue.

Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible. Prove that $ A^5\neq I_5. $

We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.

Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$. Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.

We consider the nonzero matrices $A_0, A_1, \ldots, A_n \in \mathcal{M}_2(\mathbb{R}),$ $n \ge 2,$ with the properties: $A_0 \neq aI_2$ for any $a \in \mathbb{R}$ and $A_0A_k=A_kA_0$ for $k= \overline{1,n}.$ Prove that a) $\det \left(\sum\limits_{k=1}^n A_k^2 \right) \ge 0$; b) If $\det \left(\sum\limits_{k=1}^n A_k^2 \right) = 0$ and $A_2 \ne aA_1$ for any $a \in \mathbb{R},$ then $\sum\limits_{k=1}^n A_k^2=O_2.$

Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$. [i]Marian Ionescu, Pitesti[/i]

Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements.

Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$. [i]J. Szucs[/i]

Let $ A\in M_2\left( \mathbb{R}\right) $ be a matrix having (strictly) positive numbers as its elements. Show that there is no natural number $ n $ such that $ A^n=I_2. $

Let $(a_{i,j})^n_{i,j=1}$ be a real matrix such that $a_{i,i}=0$ for $i=1,2,\ldots,n$. Prove that there exists a set $\mathcal J\subset\{1,2,\ldots,n\}$ of indices such that $$\sum_{\begin{smallmatrix}i\in\mathcal J\\j\notin\mathcal J\end{smallmatrix}}a_{i,j}+\sum_{\begin{smallmatrix}i\notin\mathcal J\\j\in\mathcal J\end{smallmatrix}}a_{i,j}\ge\frac12\sum_{i,j=1}^na_{i,j}.$$

Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$. Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

If $A\in \mathcal{M}_2(\mathbb{R})$, prove that: \[\det(A^2+A+I_2)\ge \frac{3}{4}(1-\det A)^2\]

Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: [list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. [/list] (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. [i](Paolo Leonetti)[/i]

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]