Found problems: 638
Two cubical dice each have removable numbers $ 1$ through $ 6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $ 7$?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{8} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{2}{11} \qquad
\textbf{(E)}\ \frac{1}{5}$
Solve in the set of $ 2\times 2 $ integer matrices the equation
$$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.
( for example the diffrence of this two row is only in one index
110
100)
[i]Edited by Myth[/i]
Given the following matrix
$$\begin{pmatrix}
11& 17 & 25& 19& 16\\
24 &10 &13 & 15&3\\
12 &5 &14& 2&18\\
23 &4 &1 &8 &22 \\
6&20&7 &21&9
\end{pmatrix},$$
choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.
[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set
$ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $
and construct the following matrices of order $n$:
$ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $
where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$:
$
S= \begin{bmatrix}
E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\
T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\
T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\
\dots & \dots & \dots & \dots & \dots & \dots \\
T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E
\end{bmatrix} $
($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$).
Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.
Let $H$ denote the set of the matrices from $\mathcal{M}_n(\mathbb{N})$ and let $P$ the set of matrices from $H$ for which the sum of the entries from any row or any column is equal to $1$.
a)If $A\in P$, prove that $\det A=\pm 1$.
b)If $A_1,A_2,\ldots,A_p\in H$ and $A_1A_2\cdot \ldots\cdot A_p\in P$, prove that $A_1,A_2,\ldots,A_p\in P$.
Let be three natural numbers $ n,p,q , $ a field $ \mathbb{F} , $ and two matrices $ A,B\in\mathcal{M}_n\left( \mathbb{F} \right) $ such that
$$ A^pB=0=(A+I)^qB. $$
Prove that $ B=0. $
[i]D.M. Bătinețu[/i]
Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers.
Calculate $\det(A)$.
Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that
$$A^5 = I$$ where $I$ is the identity matrix of order $5$
Let $n\in\mathbb{Z}$, $n\geq 3$. A matrix $A\in\mathcal{M}_n(\mathbb{C})$ is said to have property $(\mathcal{P})$ if $\det(A+X_{ij})=\det(A+X_{ji})$, for all $i,j\in\{1,2,\dots ,n\}$, where $X_{ij}\in\mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ on position $(i,j)$ and $0$ otherwise.
[list=a]
[*] Show that if $A\in\mathcal{M}_n(\mathbb{C})$ has property $(\mathcal{P})$ and $\det(A)\neq 0$, then $A=A^T$.
[*] Give an example of a matrix $A\in\mathcal{M}_n(\mathbb{C})$ with property $(\mathcal{P})$ such that $A\neq A^T$.
Suppose $A,B\in M_3(\mathbb C)$, $B\ne0$, and $AB=0$. Prove that there exists $D\in M_3(\mathbb C)$ with $D\ne0$ such that $AD=DA=0$.
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?
If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that
\[AB=BA\]
Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$.
a) Prove that the matrix $A$ is not invertible.
b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$.
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 $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
Let real numbers $ a,b,c$. Prove that $ \sqrt{2(a^2\plus{}b^2)}\plus{}\sqrt{2(b^2\plus{}c^2)}\plus{}\sqrt{2(c^2\plus{}a^2)}\ge \sqrt{3(a\plus{}b)^2\plus{}3(b\plus{}c)^2\plus{}3(c\plus{}a)^2}$.
Let $M \in GL_{2n}(K)$, represented in block form as \[ M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] \]
Show that $\det M.\det H=\det A$.
Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let
$$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.
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}$.
Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$.
(i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices.
(ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.
Prove that $N^2$ arbitrary distinct positive integers ($N>10$) can be arranged in a $N\times N$ table, so that all $2N$ sums in rows and columns are distinct.
[i]Proposed by S. Volchenkov[/i]
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.
Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)