$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$, is a square matrix of order four. Given that: [list] [*] [b](i)[/b] for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$, \[a_{i,k} = a_{i,k-4};\]\[P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};\]\[S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};\]\[L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};\]\[C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},\] [*][b](ii)[/b] for each $i, j = 1, 2, 3, 4$, $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$, and [*][b](iii)[/b] $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$, and $a_{3,3} = 15$.[/list] find the matrix M.

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$. 1.Prove that $\det(AB-BA)=0$. 2.If $rank(A-B)=1$, then prove that $AB=BA$.

Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that $$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$ Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle. [i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]

The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: [list][*]Every row consists of distinct numbers; [*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; [*]$x_{17}=x_{28}=x_{39}=0$; [*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. [*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list] Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.

Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

Let $A\in M_2(\mathbb Q)$ such that there is $n\in\mathbb N,n\ne0$, with $A^n=-I_2$. Prove that either $A^2=-I_2$ or $A^3=-I_2$.

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left( \begin{array}{cc} 1-n & 1 \\ -n(n+1) & n+2 \end{array} \right)$ on the $xy$ plane. Answer the following questions: (1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines. (2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$. (3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$ [i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/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 $A$ be a real $n\times n$ matrix satisfying $$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.

Let $ T \in \textsl{SL}(n,\mathbb{Z})$, let $ G$ be a nonsingular $ n \times n$ matrix with integer elements, and put $ S\equal{}G^{\minus{}1}TG$. Prove that there is a natural number $ k$ such that $ S^k \in \textsl{SL}(n,\mathbb{Z})$. [i]Gy. Szekeres[/i]

$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value. \[\sum_{A\in T} \text{det}(A)\]

Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal. [i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$?

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

On a chess table $n\times m$ we call a [i]move [/i] the following succesion of operations (i) choosing some unmarked squares, any two not lying on the same row or column; (ii) marking them with 1; (iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move). We call a [i]game [/i]a succession of moves that end in the moment that we cannot make any more moves. What is the maximum possible sum of the numbers on the table at the end of a game?

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.