Found problems: 638
Two real square matrices $A$ and $B$ satisfy the conditions $A^{2002}=B^{2003}=I$ and $AB=BA$. Prove that $A+B+I$ is invertible. (The symbol $I$ denotes the identity matrix.)
Consider the sytem of equations
\[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:
a) $a_{11}, a_{22}, a_{33}$ are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution $x_1=x_2=x_3=0$.
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}
$
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.
Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$.
(Proposed by Gerhard Woeginger, Austria)
Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.
Let be three natural numbers $ k,m,n $ an $ m\times n $ matrix $ A, $ an $ n\times m $ matrix $ B, $ and $ k $ complex numbers $ a_0,a_1,\ldots ,a_k $ such that the following conditions hold.
$ \text{(i)}\quad m\ge n\ge 2 $
$ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $
$ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $
Prove that $ a_0=0. $
Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$. (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$).
For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix
$$\begin{bmatrix}
1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\
0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\
1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\
0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\
0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1
\end{bmatrix}$$
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.
a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1;
b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.
[i]After an idea of Harazi[/i]
Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that
$$ AB^2=A-B. $$
[b]a)[/b] Show that the matrix $ I_n+B $ is inversable.
[b]b)[/b] Show that $ AB=BA. $
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy
$$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.
Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$
Let $A=[a_{ij}]_{n \times n}$ be a $n \times n$ matrix whose elements are all numbers which belong to set $\{1,2,\cdots ,n\}$. Prove that by swapping the columns of $A$ with each other we can produce matrix $B=[b_{ij}]_{n \times n}$ such that $K(B) \le n$ where $K(B)$ is the number of elements of set $\{(i,j) ; b_{ij} =j\}$.
Given such positive real numbers $a, b$ and $c$, that the system of equations:
$ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $
has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.
Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
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 $1$.
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$
Let $n\geq 2$ and $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $$\{\text{rank}(A^k)\mid k\geq 1\}=\{\text{rank}(B^k)\mid k\geq 1\}.$$ Prove that $\text{rank}(A^k)=\text{rank}(B^k)$ for all $k\geq 1$.
[i]Cristi Săvescu[/i]
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$. Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$. Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$´s for eigenvalues.