Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have \[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \] Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.

An $n \times n$ complex matrix $A$ is called \emph{t-normal} if $AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Proposed by Shachar Carmeli, Weizmann Institute of Science

Find the determinant of this matrix: $\begin{bmatrix} 2 & 2 & 2 & 2 & 2 & 2 \\ 4 & 2 & 2 & 2 & 2 & 2 \\ 4 & 4 & 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 & 2 & 2 \\ 4 & 4 & 4& 4 & 4 & 2 \end{bmatrix} $

Let $A=(\alpha_{ij})$ be a $m\,x\,n$ matric and $B=(\beta_{kl})$ be a $n\,x\, m$ matric with $m>n$ . Prove that $D(A\cdot B)=0$.

In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player. Prove that at most $1024$ contestants participated in the tournament. [i]Submitted by Dávid Matolcsi, Budapest[/i]

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that: [b]a)[/b] $ \text{tr} (A) =\det (A) =0 $ [b]b)[/b] $ AB^nA=0, $ for any natural number $ n $

Let $ A$ be a $ n\times n$ matrix with complex elements. Prove that $ A^{\minus{}1} \equal{} \overline{A}$ if and only if there exists an invertible matrix $ B$ with complex elements such that $ A\equal{} B^{\minus{}1} \cdot \overline{B}$.

Let n, k be positive integers. Let $f_a(x) := ||x - a||^{2n}$ , where the vectors $x = (x_1, ..., x_k) , a\in R^k$ , and ||·|| is the Euclidean norm. Let the vector space $Q_{n, k}$ be generated by the functions $f_a$ ($a\in R^k$). What is the largest integer N for which $Q_{n, k}$ contains all polynomials of $x_1, ..., x_k$ whose total degree is at most N?

Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.

Let A be a symmetric matrix such that the sum of elements of any row is zero. Show that all elements in the main diagonal of cofator matrix of A are equal.

Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$. a) Prove that the matrix $A$ is not invertible. b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$.

Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$. (Proposed by Gerhard Woeginger, Austria)

Let be a natural number $ n\ge 2 $ and three $ n\times n $ complex matrices that have the properties that they commute pairwise, their sum is thrice the identity matrix, and their squares are the identity matrix. Prove that these three matrices are equal. [i]Marius Cavachi[/i]

Let $ A\in M_2\left( \mathbb{C}\right) $ such that $ \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. $ Prove that $ A^2\left( A^2+I_2\right) =2I_2. $

A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that a) $ 0$ is the only nilpotent element of $ A$; b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.