Found problems: 638
suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$.
SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.
the exam time was 5 hours.
Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$
Find $\text{rank } A$.
Prove that a field $K$ can be ordered if and only if every $A\in M_n(K)$ symmetric matrix can be diagonalized over the algebraic closure of $K$. (In other words, for all $n\in\mathbb{N}$ and all $A\in M_n(K)$, there exists an $S\in GL_n(\overline{K})$ for which $S^{-1}AS$ is diagonal.)
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.
Calculate the minimum value of $ \text{tr} (A^tA) , $ where $ A $ in the cases where is a matrix of pairwise distinct nonnegative integers and:
[b]a)[/b] $ \det A\equiv 1\pmod 2 $
[b]b)[/b] $ \det A=0 $
[i]Vlad Mihaly[/i]
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?
Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$
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 $A_n$.
Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that
\[
A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O.
\]
Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.
[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers.
[b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $
[i]Cristinel Mortici[/i]
In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$
Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$
If $ n$ is a real number, then the simultaneous system
$ nx \plus{} y \equal{} 1$
$ ny \plus{} z \equal{} 1$
$ x \plus{} nz \equal{} 1$
has no solution if and only if $ n$ is equal to
$ \textbf{(A)}\ \minus{}1 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 0 \text{ or } 1 \qquad
\textbf{(E)}\ \frac12$
A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$?
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For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$.
Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
Let $A$ be a symmetric $m\times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.
The point $ P \equal{} (1,2,3)$ is reflected in the $ xy$-plane, then its image $ Q$ is rotated by $ 180^\circ$ about the $ x$-axis to produce $ R$, and finally, $ R$ is translated by 5 units in the positive-$ y$ direction to produce $ S$. What are the coordinates of $ S$?
$ \textbf{(A)}\ (1,7, \minus{} 3) \qquad \textbf{(B)}\ ( \minus{} 1,7, \minus{} 3) \qquad \textbf{(C)}\ ( \minus{} 1, \minus{} 2,8) \qquad \textbf{(D)}\ ( \minus{} 1,3,3) \qquad \textbf{(E)}\ (1,3,3)$
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i]
Solve with full solution:
\[\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y}
\\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right. \]
Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then
$$|\mathcal F|\leq \binom{n-1}{k-1}^2$$
[Gy. Katona]
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?