Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

Let $ (x,y)$ be a pair of real numbers satisfying \[ 56x \plus{} 33y \equal{} \frac{\minus{}y}{x^2\plus{}y^2}, \qquad \text{and} \qquad 33x\minus{}56y \equal{} \frac{x}{x^2\plus{}y^2}. \]Determine the value of $ |x| \plus{} |y|$.

Let $(a_i)_{1\le i\le2015}$ be a sequence consisting of $2015$ integers, and let $(k_i)_{1\le i\le2015}$ be a sequence of $2015$ positive integers (positive integer excludes $0$). Let $$A=\begin{pmatrix}a_1^{k_1}&a_1^{k_2}&\cdots&a_1^{k_{2015}}\\a_2^{k_1}&a_2^{k_2}&\cdots&a_2^{k_{2015}}\\\vdots&\vdots&\ddots&\vdots\\a_{2015}^{k_1}&a_{2015}^{k_2}&\cdots&a_{2015}^{k_{2015}}\end{pmatrix}.$$Prove that $2015!$ divides $\det A$.

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?

For any positive integer $n$, $S_{n}$ be the set of all permutations of $\{1,2,3,\dots,n\}$. For each permutation $\pi \in S_n$, let $f(\pi)$ be the number of ordered pairs $(j,k)$ for which $\pi(j)>\pi(k)$ and $1\leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k)\equiv k \pm 1 \pmod{n}$. Compute \[ \sum_{\pi \in S_{999}} (-1)^{f(\pi)+g(\pi)}. \]

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

Calculate $ \begin{pmatrix}1&0&0& \ldots &0\\\binom{1}{0} &\binom{1}{1} &0& \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \binom{n}{0} &\binom{n}{1} & \binom{n}{2} & \ldots & \binom{n}{n}\end{pmatrix}^{-1} . $

Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $ Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $ [i]Vasile Pop[/i]

Let $A\in\mathcal{M}_n(\mathbb{C})$ be an antisymmetric matrix, i.e. $A=-A^t.$[list=a] [*]Prove that if $A\in\mathcal{M}_n(\mathbb{R})$ and $A^2=O_n$ then $A=O_n.$ [*]Assume that $n{}$ is odd. Prove that if $A{}$ is the adjoint of another matrix $B\in\mathcal{M}_n(\mathbb{C})$ then $A^2=O_n.$ [/list]

Let be an odd integer $ n\ge 3 $ and an $ n\times n $ real matrix $ A $ whose determinant is positive and such that $ A+\text{adj} A=2A^{-1} . $ Prove that $ A^{2010} +\text{adj}^{2010} A =2A^{-2010} . $ [i]Lucian Petrescu[/i]

Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

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}$.

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\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 $n$ be a positive integer. Given a sequence $\varepsilon_1$, $\dots$, $\varepsilon_{n - 1}$ with $\varepsilon_i = 0$ or $\varepsilon_i = 1$ for each $i = 1$, $\dots$, $n - 1$, the sequences $a_0$, $\dots$, $a_n$ and $b_0$, $\dots$, $b_n$ are constructed by the following rules: \[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\] \[\begin{array}{lll} a_{i+1} = \begin{cases} 2a_{i-1} + 3a_i, \\ 3a_{i-1} + a_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_i = 0, \\ \text{if } \varepsilon_i = 1, \end{array} & \text{for each } i = 1, \dots, n - 1, \\[15pt] b_{i+1}= \begin{cases} 2b_{i-1} + 3b_i, \\ 3b_{i-1} + b_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_{n-i} = 0, \\ \text{if } \varepsilon_{n-i} = 1, \end{array} & \text{for each } i = 1, \dots, n - 1. \end{array}\] Prove that $a_n = b_n$. [i]Proposed by Ilya Bogdanov, Russia[/i]

There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.

Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?

[i][b]a)[/b][/i] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$. [b][i]b)[/i][/b] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$. [b]Note[/b]: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$.

The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.

Suppose $A\in{M_2(\mathbb{C})}$ is not a scalar matrix. Let $S=\{B\in{M_2(\mathbb{C})}|\ AB=BA\}$. If $X,\ Y\in{S}$, then prove that $XY=YX$.

Let $GL(n, K)$ be a linear group over the field K with a topology induced by a non-Archimedean absolute value of the field K. Prove that if the matrix $M \in GL (n, K)$ is contained by some compact subgroup of $GL(n, K)$, then all eigenvalues of M have absolute value 1.

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.