Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that $$2021B = A^m+B^2.$$ Prove that $|\text{det} A| \leq 1$.

Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$ let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$ Prove that the following statements are equivalent: $\textbf{i) }$ $|\det A | = 1$ $\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$ [i]Romeo Raicu[/i]

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

There are $ 16$ pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.

Consider the multiplicative group $ \left\{ \left.A_k:=\left(\begin{matrix} 2^k& 2^k\\2^k& 2^k\end{matrix}\right)\right| k\in\mathbb{Z} \right\} . $ [b]a)[/b] Prove that $A_xA_y=A_{x+y+1} , $ for all integers $ x,y. $ [b]b)[/b] Show that, for all integers $ t, $ the multiplicative group $ \left\{ A_{jt-1}|j\in\mathbb{Z} \right\} $ is a subgroup of $ G. $ [b]c)[/b] Determine the linear integer polynomials $ P $ for which it exists an isomorphism $ \left( G,\cdot \right)\stackrel{\eta}{\cong}\left( \mathbb{Z} ,+ \right) $ such that $ \eta\left( A_k \right) =P(k). $

Let $n>k$ and let $A_1,\ldots,A_k$ be real $n\times n$ matrices of rank $n-1$. Prove that $$A_1\cdots A_k\ne0.$$

Given the vector spaces $V,W$ with coefficients over a field $K$ and function $ \phi :V\to W$ satisfying the relation : $$\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)$$ for all $x,y \in V, \lambda \in K$. Such a function is called linear. Let $L\varphi=\{x\in V/\varphi(x)=0\}$ , and$M=\varphi(V)$ , prove that : (i) $L\varphi$ is subspace of $V$ and $M$ is subspace of $W$ (ii) $L\varphi={O}$ iff $\varphi$ is $1-1$ (iii) Dimension of $V$ equals to dimension of $L\varphi$ plus dimension of $M$ (iv) If $\theta : \mathbb{R}^3\to\mathbb{R}^3$ with $\theta(x,y,z)=(2x-z,x-y,x-3y+z)$, prove that $\theta$ is linear function . Find $L\theta=\{x\in {R}^3/\theta(x)=0\}$ and dimension of $M=\theta({R}^3)$.

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} $

Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)

Which of the following are true? $\textbf{(A)}~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R)$ $\textbf{(B)}~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible}$ $\textbf{(C)}~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC)$ $\textbf{(D)}~\text{None of the above}$

In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which: [b](i)[/b] $a_1 < a_2 < \cdots < a_n,$ [b](ii) [/b] $b_1 < b_2 <\cdots < b_n,$ [b](iii) [/b]$a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$

Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.

Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud. For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.

Let $k$ and $n$ be positive integers. A sequence $\left( A_1, \dots , A_k \right)$ of $n\times n$ real matrices is [i]preferred[/i] by Ivan the Confessor if $A_i^2\neq 0$ for $1\le i\le k$, but $A_iA_j=0$ for $1\le i$, $j\le k$ with $i\neq j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$. (Proposed by Fedor Petrov, St. Petersburg State University)

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 $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative: $ \begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix} $

[asy] real t=pi/12;real u=8*t; real cu=cos(u);real su=sin(u); draw(unitcircle); draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t))); draw((cu,su)--(cu,-su)); label("A",(cos(13*t),sin(13*t)),W); label("B",(cos(-t),sin(-t)),E); label("C",(cu,su),N); label("D",(cu,-su),S); label("E",(cu,sin(-t)),NE); label("2",((cu-1)/2,sin(-t)),N); label("6",((cu+1)/2,sin(-t)),N); label("3",(cu,(sin(-t)-su)/2),E); //Credit to Zimbalono for the diagram[/asy] Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is $\textbf{(A) }4\sqrt{5}\qquad\textbf{(B) }\sqrt{65}\qquad\textbf{(C) }2\sqrt{17}\qquad\textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$

The three row sums and the three column sums of the array \[\begin{bmatrix} 4 & 9 & 2 \\ 8 & 1 & 6 \\ 3 & 5 & 7 \end{bmatrix} \]are the same. What is the least number of entries that must be altered to make all six sums different from one another? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set \[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\] By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.

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 a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

Let $A=(a_{ij})$ be an $m\times n$ real matrix with at least one non-zero element. For each $i\in\{1,\ldots,m\}$, let $R_i=\sum_{j=1}^na_{ij}$ be the sum of the $i$-th row of the matrix $A$, and for each $j\in\{1,\ldots,n\}$, let $C_j =\sum_{i=1}^ma_{ij}$ be the sum of the $j$-th column of the matrix $A$. Prove that there exist indices $k\in\{1,\ldots,m\}$ and $l\in\{1,\ldots,n\}$ such that $$a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,$$or $$a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.$$

Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.

Let $ A \equal{} (a_{ik})$ be an $ n\times n$ matrix with nonnegative elements such that $ \sum_{k \equal{} 1}^n a_{ik} \equal{} 1$ for $ i \equal{} 1,...,n$. Show that, for every eigenvalue $ \lambda$ of $ A$, either $ |\lambda| < 1$ or there exists a positive integer $ k$ such that $ \lambda^k \equal{} 1$