Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

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. $

Let $n \geq 1$. Find a set of distincts real numbers $\left(x_j\right)_{1\leq j\leq n}$ such that for any bijections $f : \{1, 2,\cdots ,n\}^2 \to \{1, 2,\cdots ,n\}^2$ the matrix $\left(x_{f(i,j)}\right)_{1\leq i,j\leq n}$ is invertible.

Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.

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$? $ \setlength{\unitlength}{6mm} \begin{picture}(9,9)(0,0) \multiput(0,0)(1,0){10}{\line(0,1){9}} \multiput(0,0)(0,1){10}{\line(1,0){9}} \linethickness{1.2pt} \multiput(0,0)(3,0){4}{\line(0,1){9}} \multiput(0,0)(0,3){4}{\line(1,0){9}} \put(0,8){\makebox(1,1){1}} \put(1,7){\makebox(1,1){2}} \put(3,6){\makebox(1,1){?}} \end{picture}$

Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide. find the number of all possible codes(in terms of $n$).

Let $n$ be a positive integer. For any positive integer $k$, let $0_k=diag\{\underbrace{0, ...,0}_{k}\}$ be a $k \times k$ zero matrix. Let $Y=\begin{pmatrix} 0_n & A \\ A^t & 0_{n+1} \end{pmatrix}$ be a $(2n+1) \times (2n+1)$ where $A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}$ is a $n \times (n+1)$ real matrix. Let $A^T$ be transpose matrix of $A$ i.e. $(n+1) \times n$ matrix, the element of $(j, i)$ is $x_{i, j}$. (a) Let complex number $\lambda$ be an eigenvalue of $k \times k$ matrix $X$. If there exists nonzero column vectors $v=(x_1, ..., x_k)^t$ such that $Xv=\lambda v$. Prove that 0 is the eigenvalue of $Y$ and the other eigenvalues of $Y$ can be expressed as a form of $\pm \sqrt{\lambda}$ where nonnegative real number $\lambda$ is the eigenvalue of $AA^t$. (b) Let $n=3$ and $a_1$, $a_2$, $a_3$, $a_4$ are $4$ distinct positive real numbers. Let $a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}$ and $x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j$ where $1\leq i \leq 3, 1\leq j \leq 4$, $\delta_{i, j}= \begin{cases} 1 \text{ if } i=j\\ 0 \text{ if } i\neq j\\ \end{cases}\,$. Prove that $Y$ has 7 distinct eigenvalue.

Let $n\geq 2$ and $1 \leq r \leq n$. Consider the set $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$. Compute the sum $\sum_{X \in S_r}X$

a)Prove that any matrix $A\in \mathcal{M}_4(\mathbb{C})$ can be written as a sum of four matrices $B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C})$ with the rank equal to $1$. b)$I_4$ can't be written as a sum of less than four matrices with the rank equal to $1$. [i]Manuela Prajea & Ion Savu[/i]

What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?

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 $j=1, \ldots ,q.$

Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that \[A^2B-2ABA+BA^2=0.\] Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.

Let $ \mathbb{K} $ be a field and let be a matrix $ M\in\mathcal{M}_3(\mathbb{K} ) $ having the property that $ \text{tr} (A) =\text{tr} (A^2) =0 . $ Show that there is a $ \mu\in \mathbb{K} $ such that $ A^3=\mu A $ or $ A^3=\mu I. $ [i]Cristinel Mortici[/i]

Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?

In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n\plus{}1,n\plus{}2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that: \[ \frac{1^2}{a_1}\plus{}\frac{2^2}{a_2}\plus{}\ldots \plus{}\frac{n^2}{a_n}\geq \frac{n\plus{}2}{2}\minus{}\frac{1}{n^2\plus{}1}\]

Let $ A,B,S $ be three $ 3\times 3 $ complex matrices with $ B=S^{-1}AS , $ and $ S $ nonsingular. Show: $$ \text{tr} \left( B^2\right) +2\text{tr}(C(B)) = \left(\text{tr} (A)\right)^2 , $$ where $ C(B) $ is the cofactor of $ B. $ [i]Mihai Haivas[/i]

$ 15 $ minors of order $ 3 $ of a $ 4\times 4 $ real matrix whose determinant is a nonzero rational number, are rational. Prove that this matrix is rational.

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)

Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity matrix of the same size as $S$. Find $x^T(I-S)^{-1}x$ Where $x$ is a real unit vector.

Consider the matrix $X\in\mathcal{M}_2(\mathbb{C})$ which satisfies $X^{2022}=X^{2023}.$ Prove that $X^2=X^3.$

Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by \[ f(x)=\det(A+xB)\] has a multiple root, then $ \det(A+B)=\det B$.

Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following: \begin{align*} &(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\ &(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\ &(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\ &(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5}) \end{align*} Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.