Suppose that $A\in\mathcal M_{n}(\mathbb R)$ with $\text{Rank}(A)=k$. Prove that $A$ is sum of $k$ matrices $X_{1},\dots,X_{k}$ with $\text{Rank}(X_{i})=1$.

Let $A$ be a $n \times n$ matrix of real numbers such that $A^2+\mathbb{I}=A$, where $\mathbb{I}$ is the identity $n \times n$ matrix. Prove that the matrix $A^{3n}$ , where $\nu\in\mathbb{Z}$ takes only two values and find those values.

Let $A=(a_{ij})$ be a real $n\times n$ matrix such that $A^n\ne0$ and $a_{ij}a_{ji}\le0$ for all $i,j$. Prove that there exist two nonreal numbers among eigenvalues of $A$.

Let $A,B$ be $n\times n$ matrices, $n\geq 2$, and $B^2=B$. Prove that: $$\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)$$

Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent. [i]L. Lempert[/i]

Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that $TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal. [i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.[/i]

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]

We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that: [i](i)[/i] in each column there are exactly $k$ terms equal to $0$; [i](ii)[/i] for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$ For what values of $k$ is this possible?

Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$

Consider the multiplicative group $ \left\{ \left.A_k:=\left(\begin{matrix} 2^k& 2^k\\2^k& 2^k\end{matrix}\right)\right| k\in\mathbb{Z} \right\} . $ [b]a)[/b] Prove that $A_xA_y=A_{x+y+1} , $ for all integers $ x,y. $ [b]b)[/b] Show that, for all integers $ t, $ the multiplicative group $ \left\{ A_{jt-1}|j\in\mathbb{Z} \right\} $ is a subgroup of $ G. $ [b]c)[/b] Determine the linear integer polynomials $ P $ for which it exists an isomorphism $ \left( G,\cdot \right)\stackrel{\eta}{\cong}\left( \mathbb{Z} ,+ \right) $ such that $ \eta\left( A_k \right) =P(k). $

Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

Let $A=(a_{ij})_{i, j=1}^n$ be a symmetric $n\times n$ matrix with real entries, and let $\lambda _1, \lambda _2, \dots, \lambda _n$ denote its eigenvalues. Show that $$\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j$$ and determine all matrices for which equality holds. (Proposed by Matrin Niepel, Comenius University, Bratislava)

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.

Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.

Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2, \ldots, n\}$, let $s(i, j)$ be the number of pairs $(a, b)$ of nonnegative integers satisfying $a i+b j=n$. Let $S$ be the $n$-by-n matrix whose $(i, j)$-entry is $s(i, j)$. For example, when $n=5$, we have $S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]$. Compute the determinant of $S$.

In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which: [b](i)[/b] $a_1 < a_2 < \cdots < a_n,$ [b](ii) [/b] $b_1 < b_2 <\cdots < b_n,$ [b](iii) [/b]$a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$

In how many ways can one fill an $n*n$ matrix with $+1$ and $-1$ so that the product of the entries in each row and each column equals $-1$?

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

In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions. [list][*]If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s. [*]If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.[/list] Find all pairs $(a,b)$ for which this is possible.

For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$ entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$.

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

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 $S$.