For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

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 $\sum_{X \in S_r}X$

Given the linear fractional transformation of $x$ into $f_1(x) = \tfrac{2x-1}{x+1},$ define $f_{n+1}(x) = f_1(f_n(x))$ for $n=1,2,3,\ldots.$ It can be shown that $f_{35} = f_5.$ Determine $A,B,C,D$ so that $f_{28}(x) = \tfrac{Ax+B}{Cx+D}.$

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

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]

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? $ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $

Denote $\textrm{SL}_2 (\mathbb{Z})$ and $\textrm{SL}_3 (\mathbb{Z})$ the sets of matrices with $2$ rows and $2$ columns, respectively with $3$ rows and $3$ columns, with integer entries and their determinant equal to $1$. $\textbf{(a)}$ Let $N$ be a positive integer and let $g$ be a matrix with $3$ rows and $3$ columns, with rational entries. Suppose that for each positive divisor $M$ of $N$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $N$ and a matrix $\gamma_M \in \textrm{SL}_3 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . \] Moreover, if $q_1 = 1$, prove that $\det (g) = N$ and $g$ has the following shape: \[ g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), \] where $a_{ij}$ are all integers, $i, j \in \{ 1, 2, 3 \} .$ $\textbf{(b)}$ Provide an example of a matrix $g$ with $2$ rows and $2$ columns which satisfies the following properties: $\bullet$ For each positive divisor $M$ of $6$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $6$ and a matrix $\gamma_M \in \textrm{SL}_2 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) \] and $q_1 = 1$. $\bullet$ $g$ does not have its determinant equal to $6$ and is not of the shape \[ g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), \] where $a_{ij}$ are all positive integers, $i, j \in \{ 2, 3 \}$. [i](Radu Toma)[/i]

Let be a real $ 4\times 4 $ real matrix with $ \text{det} \left( A^2-I\right) <0. $ Prove that there is a number $ \alpha\in (-1,1) $ so that $ A+\alpha I $ is singular. [i]Mihai Haivas[/i]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?

a)Let $A,B\in \mathcal{M}_3(\mathbb{R})$ such that $\text{rank}\ A>\text{rank}\ B$. Prove that $\text{rank}\ A^2\ge \text{rank}\ B^2$. b)Find the non-constant polynomials $f\in \mathbb{R}[X]$ such that $(\forall)A,B\in \mathcal{M}_4(\mathbb{R})$ with $\text{rank}\ A>\text{rank}\ B$, we have $\text{rank}\ f(A)>\text{rank}\ f(B)$.

Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$, $$a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if}\, i\ne j,\\ 2 & \text{if}\, i= j. \end{cases}$$

Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.

Show that for any natural numbers $ m,n\ge 3, $ the equation $ \Delta_n (x)=0 $ has exactly two distinct solutions, where $$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$

Let be eight real numbers $ 1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4. $ Prove that $$ \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\ a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\ a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\ a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\ \end{vmatrix} >0. $$ [i]Marian Andronache, Ion Savu[/i]

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

Let $A=[a_{ij}]_{n \times n}$ be a $n \times n$ matrix whose elements are all numbers which belong to set $\{1,2,\cdots ,n\}$. Prove that by swapping the columns of $A$ with each other we can produce matrix $B=[b_{ij}]_{n \times n}$ such that $K(B) \le n$ where $K(B)$ is the number of elements of set $\{(i,j) ; b_{ij} =j\}$.

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}$

Let $A\in M_n(\mathbb{C}) $ and $a\in \mathbb{C} $ such that $A-A^*=2aI_n $, where $A^*=(\overline{A})^T $ and $I_n$ is identity matrix. (i) Show that $|\det A|\ge |a|^n $. (ii) Show that if $|\det A|=|a|^n $ then $A=aI_n$.