We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$. Moreover,\begin{align*} 53 & \equiv 1 \pmod{2} \\ 53 & \equiv 2 \pmod{3} \\ 53 & \equiv 3 \pmod{5} \\ 53 & \equiv 4 \pmod{7}. \end{align*} Let \[ M = \left( \begin{array}{ccc} 53 & 158 & 53 \\ 23 & 93 & 53 \\ 50 & 170 & 53 \end{array} \right). \] Based on the above, find $\overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})}$.

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 $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$. 1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$ 2) Prove that $f$ is linear.

Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.

Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 14$

$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow: $L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$ [i]a)[/i] For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points) [i]b)[/i] Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant. Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.$ . (7 points) [i]c)[/i] Prove that $a+b+c = -3$. (4 points) [i]d)[/i] Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points) [i]e)[/i] Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points) (${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)

Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations: $ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$ Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)

In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4\times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2\times 2$ squares.)

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).

For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as: $P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$ Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $$A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.$$ Prove that $A^6=I$.

[list=a] [*]Let $A$ be a matrix from $\mathcal{M}_{2}(\mathbb{C})$, $A\neq aI_{2}$, for any $a\in\mathbb{C}$. Prove that the matrix $X$ from $\mathcal{M} _{2}(\mathbb{C})$ commutes with $A$, that is, $AX=XA$, if and only if there exist two complex numbers $\alpha$ and $\alpha^{\prime}$, such that $X=\alpha A+\alpha^{\prime}I_{2}$. [*]Let $A$, $B$ and $C$ be matrices from $\mathcal{M}_{2}(\mathbb{C})$, such that $AB\neq BA$, $AC=CA$ and $BC=CB$. Prove that $C$ commutes with all matrices from $\mathcal{M}_{2}(\mathbb{C})$.[/list]

Consider $A\in \mathcal{M}_{2020}(\mathbb{C})$ such that $$ (1)\begin{cases} A+A^{\times} =I_{2020},\\ A\cdot A^{\times} =I_{2020},\\ \end{cases} $$ where $A^{\times}$ is the adjugate matrix of $A$, i.e., the matrix whose elements are $a_{ij}=(-1)^{i+j}d_{ji}$, where $d_{ji}$ is the determinant obtained from $A$, eliminating the line $j$ and the column $i$. Find the maximum number of matrices verifying $(1)$ such that any two of them are not similar.

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

Let $M_2(\mathbb{Z})$ be the set of $2 \times 2$ matrices with integer entries. Let $A \in M_2(\mathbb{Z})$ such that $$A^2+5I=0,$$ where $I \in M_2(\mathbb{Z})$ and $0 \in M_2(\mathbb{Z})$ denote the identity and null matrices, respectively. Prove that there exists an invertible matrix $C \in M_2(\mathbb{Z})$ with $C^{-1} \in M_2(\mathbb{Z})$ such that $$CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.$$

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $ that don’t commute. [b]a)[/b] If $ A^3=B^3, $ then $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n. $ [b]b)[/b] If $ A^n\neq B^n $ and $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n, $ then find some of the matrices $ A,B. $

Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$ a) Prove that $(AB - BA)^2 = O_{n}.$ b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$

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 $\text{rank } A$.

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

Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)

Let $n$ and $k$ be positive integers, where $n > 1$ is odd. Suppose $n$ voters are to elect one of the $k$ cadidates from a set $A$ according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a $k$ x $n$ voting matrix. We denote the number of ccurences of $a \in A$ in the $i$-th row of the voting matrix by $a_{i}$ . Let $l_{a}$ stand for the minimum integer $l$ for which $\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}$. Setting $l'= min \{l_{a} | a \in A\}$, we will regard the voting matrices which make the set $\{a \in A | l_{a} = l' \}$ as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if $w_{1} \geq w_{2} \geq ... \geq
w_{k} \geq 0$ are given, for each admissible voting matrix, $\sum^{k}_{i=1}{w_{i}a_{i}}$ is called the total weighted score of $a \in A$. We will say that the system $(w_{1},w_{2}, . . . , w_{k})$ of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates. (a) Determine whether there is a system of weights representing majoritarian compromise if $k = 3$. (b) Show that such a system of weights does not exist for $k > 3$.

Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.

Let $A_1, A_2, \ldots , A_m$ be subsets of the set $\{ 1,2, \ldots , n \}$, such that the cardinal of each subset $A_i$, such $1 \le i \le m$ is not divisible by $30$, while the cardinal of each of the subsets $A_i \cap A_j$ for $1 \le i,j \le m$, $i \neq j$ is divisible by $30$. Prove \begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor \le 3n \end{align*}