The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $ Prove that $ A=0. $ [i]Marius Ghergu[/i]

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

Given a natural number $ n\ge 2 $ and an $ n\times n $ matrix with integer entries, consider the multiplicative monoid $$ M=\{ M_k=I+kA| k\in \mathbb{Z} \} . $$ [b]a)[/b] Prove that $ M $ is a commutative group if the [url=https://en.wikipedia.org/wiki/Nilpotent_matrix]index[/url] of $ A $ is $ 2. $ [b]b)[/b] Prove that all elements of $ M $ are units if $ M_1,M_2,\ldots M_{2n} $ are all units.

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.

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}$

[i][b]a)[/b][/i] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$. [b][i]b)[/i][/b] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$. [b]Note[/b]: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$.

Let $d$ be a positive integer and let $A$ be a $d\times d$ matrix with integer entries. Suppose $I+A+A_2+\ldots+A_{100}=0$ (where $I$ denotes the identity $d\times d$ matrix, and $0$ denotes the zero matrix, which has all entries $0$). Determine the positive integers $n\le100$ for which $A_n+A_{n+1}+\ldots+A_{100}$ has determinant $\pm1$.

Let $M$ be a set of real $n \times n$ matrices such that i) $I_{n} \in M$, where $I_n$ is the identity matrix. ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$. iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$. Prove that $M$ contains at most $n^2 $ matrices.

Mientka Publishing Company prices its bestseller [i]Where's Walter?[/i] as follows: \[C(n) \equal{} \begin{cases} 12n, &\text{if } 1 \le n \le 24\\ 11n, &\text{if } 25 \le n \le 48\\ 10n, &\text{if } 49 \le n \end{cases}\] where $ n$ is the number of books ordered, and $ C(n)$ is the cost in dollars of $ n$ books. Notice that $ 25$ books cost less than $ 24$ books. For how many values of $ n$ is it cheaper to buy more than $ n$ books than to buy exactly $ n$ books? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

Let $A{}$ and $B$ be invertible $n\times n$ matrices with real entries. Suppose that the inverse of $A+B^{-1}$ is $A^{-1}+B$. Prove that $\det(AB)=1$. Does this property hold for $2\times 2$ matrices with complex entries?

Let be a natural number $ n, $ a number $ t\in (0,1) $ and $ n+1 $ numbers $ a_0\ge a_1\ge a_2\ge\cdots\ge a_n\ge 0. $ Prove the following matrix inequality: $$ \begin{vmatrix}\frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0& 0 & \cdots & 0 & 0 \\ 0 & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 \\ a_0 & a_1 & a_2 & a_3 & \cdots & a_{n-1} & a_n \end{vmatrix}^2\le a_0^2\left( 1+\frac{1}{t^2} \right) $$