$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties: (1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s. (2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.

Eight students solved $8$ problems. a) It turned out that each problem was solved by $5$ students. Prove that there are two students such that each problem is solved by at least one of them. b) If it turned out that each problem was solved by $4$ students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.) Proposed by S. Tokarev

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 $\angle DFE$.

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.

Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.

Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

$A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}$. Find $\text{rank}(A)$.

Let be a natural number $ n $ and a matrix $ A\in\mathcal{M}_n(\mathbb{R}) $ having the property that sum of the squares of all its elements is strictly less than $ 1. $ Prove that the matrices $ I\pm A $ are invertible.

The point $ (\minus{}3, 2)$ is rotated $ 90^\circ$ clockwise around the origin to point $ B$. Point $ B$ is then reflected over the line $ y \equal{} x$ to point $ C$. What are the coordinates of $ C$? $ \textbf{(A)}\ ( \minus{} 3, \minus{} 2)\qquad \textbf{(B)}\ ( \minus{} 2, \minus{} 3)\qquad \textbf{(C)}\ (2, \minus{} 3)\qquad \textbf{(D)}\ (2,3)\qquad \textbf{(E)}\ (3,2)$

Let be a $ 2\times 2 $ real matrix such that $ \det \left( A^6+64I \right) =0. $ Show that $ \det A=4. $ [i]Viorel Botea[/i]

Let $n\in \mathbb{N},\ n\ge 2$. For any matrix $A\in \mathcal{M}_n(\mathbb{C})$, let $m(A)$ be the number of non-zero minors of $A$. Prove that: a)$m(I_n)=2^n-1$; b)If $A\in \mathcal{M}_n(\mathbb{C})$ is non-singular, then $m(A)\ge 2^n-1$. [i]Marius Ghergu[/i]

Let $A=a_{ij}$ is simetrical real matrix. Prove that : $\sum_i e^{a_{ii}} \leq tr (e^A)$

Investigate the convergence of the integral \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]

Mientka Publishing Company prices its bestseller [i]Where's Walter?[/i] as follows: \[C(n) \equal{} \begin{cases} 12n, &\text{if } 1 \le n \le 24\\ 11n, &\text{if } 25 \le n \le 48\\ 10n, &\text{if } 49 \le n \end{cases}\] where $ n$ is the number of books ordered, and $ C(n)$ is the cost in dollars of $ n$ books. Notice that $ 25$ books cost less than $ 24$ books. For how many values of $ n$ is it cheaper to buy more than $ n$ books than to buy exactly $ n$ books? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$. [i]Proposed by Ray Li[/i]

Let $A=[a_{ij}]_{n\times n}$ be a matrix with nonnegative entries such that $$\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.$$ (a) Prove that $|\det A|\le1$. (b) If $|\det A|=1$ and $\lambda\in\mathbb C$ is an arbitrary eigenvalue of $A$, show that $|\lambda|=1$.

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

Let be an odd integer $ n\ge 3 $ and an $ n\times n $ real matrix $ A $ whose determinant is positive and such that $ A+\text{adj} A=2A^{-1} . $ Prove that $ A^{2010} +\text{adj}^{2010} A =2A^{-2010} . $ [i]Lucian Petrescu[/i]

$A,B \in M_{2\times 2}(C)$ Prove that: $Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

Consider the $ 4\times 4 $ integer matrices that have the property that each one of them multiplied by its transpose is $ 4I. $ [b]a)[/b] Show that the product of the elements of such a matrix is either $ 0, $ either $ 1. $ [b]b)[/b] How many such matrices have the property that the product of its elements is $ 0? $

Let $(a_{i,j})^n_{i,j=1}$ be a real matrix such that $a_{i,i}=0$ for $i=1,2,\ldots,n$. Prove that there exists a set $\mathcal J\subset\{1,2,\ldots,n\}$ of indices such that $$\sum_{\begin{smallmatrix}i\in\mathcal J\\j\notin\mathcal J\end{smallmatrix}}a_{i,j}+\sum_{\begin{smallmatrix}i\notin\mathcal J\\j\in\mathcal J\end{smallmatrix}}a_{i,j}\ge\frac12\sum_{i,j=1}^na_{i,j}.$$