For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set \[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\] By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

Given the determinant of order $n$ $$\begin{vmatrix} 8 & 3 & 3 & \dots & 3 \\ 3 & 8 & 3 & \dots & 3 \\ 3 & 3 & 8 & \dots & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & 3 & 3 & \dots & 8 \end{vmatrix}$$ Calculate its value and determine for which values of $n$ this value is a multiple of $10$.

Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle. Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of the lamp; if $i$ is turned on, a [i]signal[/i] of range $i$ is sent by this lamp, and it will be received only by the next $i$ lamps which follow $i$, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed. The example in attachment, for $n=4$, ilustrates a configuration where lamps $2$ and $4$ are initially turned on. Lamp $2$ sends signal only for the lamps $3$ e $4$, while lamp $4$ sends signal for lamps $1$, $2$, $3$ e $4$. Therefore, we verify that lamps $1$ e $2$ received only one signal, while lamps $3$ e $4$ received two signals. Therefore, in the next configuration, lamps $1$ e $4$ will be turned on, while lamps $2$ e $3$ will be turned off. Let $\Psi$ to be the set of all $2^n$ possible configurations, where $0 \le \ell \le n$ random lamps are turned on. We define a function $f: \Psi \rightarrow \Psi$ where, if $\xi$ is a configuration of lamps, then $f(\xi)$ is the configurations obtained after we perform the [i]cool procedure[/i] described above. Determine all values of $n$ for which $f$ is bijective.

Let $A=a_{ij}$ is simetrical real matrix. Prove that : $\sum_i e^{a_{ii}} \leq tr (e^A)$

Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that: (i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and (ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$ Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$

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 $A=\left( \begin{array}{ccc} 1 & 1& 0 \\ 0 & 1& 0 \\ 0 &0 & 2 \end{array} \right),\ B=\left( \begin{array}{ccc} a & 1& 0 \\ b & 2& c \\ 0 &0 & a+1 \end{array} \right)\ (a,\ b,\ c\in{\mathbb{C}}).$ (1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$ (2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.

Let $k$ and $n$ be positive integers. A sequence $\left( A_1, \dots , A_k \right)$ of $n\times n$ real matrices is [i]preferred[/i] by Ivan the Confessor if $A_i^2\neq 0$ for $1\le i\le k$, but $A_iA_j=0$ for $1\le i$, $j\le k$ with $i\neq j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$. (Proposed by Fedor Petrov, St. Petersburg State University)

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.

[b]Problem 1.[/b]Find all $ (x,y)$ such that: \[ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\ \sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\; \]

Let $A \in \mathcal{M}_n(\mathbb{R})$ be an invertible matrix. a) Prove that the eigenvalues of $AA^T$ are positive real numbers. b) We assume that there are two distinct positive integers, $p$ and $q$, such that $(AA^T)^p=(A^TA)^q.$ Prove that $A^T=A^{-1}.$

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 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?

Let $V$ be a $n-$dimensional vector space over a field $F$ with a basis $\{e_1,e_2, \cdots ,e_n\}$.Prove that for any $m-$dimensional linear subspace $W$ of $V$, the number of elements of the set $W \cap P$ is less than or equal to $2^m$ where $P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}$.

Let $A$ be a $n\times n$ matrix such that $A_{ij} = i+j$. Find the rank of $A$. [hide="Remark"]Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.[/hide]

Let $v_1,v_2,\ldots,v_n$ vectors in $\mathbb{R}^2$ such that $|v_i|\leq 1$ for $1 \leq i \leq n$ and $\sum_{i=1}^n v_i=0$. Prove that there exists a permutation $\sigma$ of $(1,2,\ldots,n)$ such that $\left|\sum_{j=1}^k v_{\sigma(j)}\right| \leq\sqrt 5$ for every $k$, $1\leq k \leq n$. [i]Remark[/i]: If $v = (x,y)\in \mathbb{R}^2$, $|v| = \sqrt{x^2 + y^2}$.

Let $A$ be an $n\times n$ matrix with strictly positive elements and two vectors $u,v\in\mathbb{R}^n$, also with strictly positive elements, such that $$Au=v\text{ and }Av=u.$$ Prove that $u=v$.

Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$ \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\ &w(a,b)&-2&-1&0&1&2\\ \hline &-2&-1&-2&2&-2&-1\\ &-1&-2&4&-4&4&-2\\ a&0&2&-4&12&-4&2\\ &1&-2&4&-4&4&-2\\ &2&-1&-2&2&-2&-1\\ \hline\end{array}\] For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)

Given a $19 \times 19$ matrix where each component is either $1$ or $-1$. Let $b_i$ be the product of all components in the $i$-th row, and $k_i$ be the product of all components in the $i$-th column, for all $1 \le i \le 19$. Prove that for any such matrix, $b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \neq 0$.

Find a matrix $A_{(2 \times 2)}$ for which \[ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.\]

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form \[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\] where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

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 \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with lamp $k+1$,..., lamp $n-1$ with lamp $n$, lamp $n$ with lamp $1$. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.