Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements. a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1; b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1. [i]After an idea of Harazi[/i]

Consider a matrix of size $8 \times 8$, containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules: -Multiply all entries in some row by 2. -Subtract 1 from all entries in some column. Prove that one can transform the given matrix into the zero matrix.

Show that: $$ 2015\in\left\{ x_1+2x_2+3x_3\cdots +2015x_{2015}\big| x_1,x_2,\ldots ,x_{2015}\in \{ -2,3\}\right\}\not\ni 2016. $$

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 $12$.

Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$. [i]J. Szucs[/i]

If the expression $ \begin{pmatrix}a & c \\ d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\ x & x \end{pmatrix} = 3$: $\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\ \textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\ \textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\ \textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\ \textbf{(E)}\ \text{None of these.}$

Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed. [i]B. Gyires[/i]

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.

Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that: [b]a)[/b] $ \det (A+N)=\det (A) $ [b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $

Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.

A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$. [i]Proposed by Ethan Tan[/i]

Let $A,B$ be two $n\times n$ complex matrices of the same rank, and let $k$ be a positive integer. Prove that $A^{k+1}B^k = A$ if and only if $B^{k+1}A^k = B$.

Let be two distinct $ 2\times 2 $ real matrices having the property that there exists a natural number such that these matrices raised to this number are equal, and these matrices raised to the successor of this number are also equal. Prove that these matrices raised to any power greater than $ 2 $ are equal.

An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called [i]Naissus[/i] if all products of $n$ numbers written in $n$ [i]scattered[/i] cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for $(a) n = 8;$ $(b) n = 10?$ ($n$ cells are [i]scattered[/i] if no two are in the same row or column.) [i]Proposed by Marko Djikic[/i]

Let $n$ be a positive integer. Let $A,B,$ and $C$ be three $n$-dimensional vector subspaces of $\mathbb{R}^{2n}$ with the property that $A \cap B = B \cap C = C \cap A = \{0\}$. Prove that there exist bases $\{a_1,a_2, \dots, a_n\}$ of $A$, $\{b_1,b_2, \dots, b_n\}$ of $B$, and $\{c_1,c_2, \dots, c_n\}$ of $C$ with the property that for each $i \in \{1,2, \dots, n\}$, the vectors $a_i,b_i,$ and $c_i$ are linearly dependent.

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let \[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). \] Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.

Given is a prime number $p$ and natural $n$ such that $p \geq n \geq 3$. Set $A$ is made of sequences of lenght $n$ with elements from the set $\{0,1,2,...,p-1\}$ and have the following property: For arbitrary two sequence $(x_1,...,x_n)$ and $(y_1,...,y_n)$ from the set $A$ there exist three different numbers $k,l,m$ such that: $x_k \not = y_k$, $x_l \not = y_l$, $x_m \not = y_m$. Find the largest possible cardinality of $A$.

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 $\textsl{Good}$.

Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$. [i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]

Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that \[ AB = \left(% \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ \end{array}% \right). \] Find $BA$.

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying \[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\] [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]