Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle. Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of the lamp; if $i$ is turned on, a [i]signal[/i] of range $i$ is sent by this lamp, and it will be received only by the next $i$ lamps which follow $i$, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed. The example in attachment, for $n=4$, ilustrates a configuration where lamps $2$ and $4$ are initially turned on. Lamp $2$ sends signal only for the lamps $3$ e $4$, while lamp $4$ sends signal for lamps $1$, $2$, $3$ e $4$. Therefore, we verify that lamps $1$ e $2$ received only one signal, while lamps $3$ e $4$ received two signals. Therefore, in the next configuration, lamps $1$ e $4$ will be turned on, while lamps $2$ e $3$ will be turned off. Let $\Psi$ to be the set of all $2^n$ possible configurations, where $0 \le \ell \le n$ random lamps are turned on. We define a function $f: \Psi \rightarrow \Psi$ where, if $\xi$ is a configuration of lamps, then $f(\xi)$ is the configurations obtained after we perform the [i]cool procedure[/i] described above. Determine all values of $n$ for which $f$ is bijective.

In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$. A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary? [i]Proposed by Michael Kural[/i]

Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements. Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $$A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.$$ Prove that $A^6=I$.

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: [list][*]Every row consists of distinct numbers; [*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; [*]$x_{17}=x_{28}=x_{39}=0$; [*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. [*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list] Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.

Let real numbers $ a,b,c$. Prove that $ \sqrt{2(a^2\plus{}b^2)}\plus{}\sqrt{2(b^2\plus{}c^2)}\plus{}\sqrt{2(c^2\plus{}a^2)}\ge \sqrt{3(a\plus{}b)^2\plus{}3(b\plus{}c)^2\plus{}3(c\plus{}a)^2}$.

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

Given a natural number $ n\ge 2 $ and an $ n\times n $ matrix with integer entries, consider the multiplicative monoid $$ M=\{ M_k=I+kA| k\in \mathbb{Z} \} . $$ [b]a)[/b] Prove that $ M $ is a commutative group if the [url=https://en.wikipedia.org/wiki/Nilpotent_matrix]index[/url] of $ A $ is $ 2. $ [b]b)[/b] Prove that all elements of $ M $ are units if $ M_1,M_2,\ldots M_{2n} $ are all units.

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that $TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.

Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$

Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that \[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.

Consider the set \[M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in\mathcal{M}_2(\mathbb{C})\ |\ ab = cd \right\}.\] a) Give an example of matrix $A\in M$ such that $A^{2017}\in M$ and $A^{2019}\in M$, but $A^{2018}\notin M$. b) Show that if $A\in M$ and there exists the integer number $k\ge 1$ such that $A^k \in M$, $A^{k + 1}\in M$ si $A^{k + 2} \in M$, then $A^n\in M$, for any integer number $n\ge 1$.

In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$ Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left( \begin{array}{cc} 1-n & 1 \\ -n(n+1) & n+2 \end{array} \right)$ on the $xy$ plane. Answer the following questions: (1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines. (2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$. (3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$ [i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]

[color=darkred]A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be [i]permutable[/i] if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two [i]permutable[/i] rows, then its determinant is equal to $0$ .[/color]

Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left( \begin{array}{cc} 1 & 2 \\ 3 & 6 \end{array} \right)$