Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal. [i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.

A sequence $a_n$ is defined by $\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0$. Find $\lim_{n\to\infty} \frac{a_n}{n^3}$.

Let $n$ be an odd natural number and $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $(A-B)^2=O_n.$ Prove that $\det(AB-BA)=0.$

Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$?

Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$. 1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$ 2) Prove that $f$ is linear.

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

On a chess table $n\times m$ we call a [i]move [/i] the following succesion of operations (i) choosing some unmarked squares, any two not lying on the same row or column; (ii) marking them with 1; (iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move). We call a [i]game [/i]a succession of moves that end in the moment that we cannot make any more moves. What is the maximum possible sum of the numbers on the table at the end of a game?

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

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

Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)

Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $

Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.

Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

For any $n\times n$ unitary matrices $A,B$, prove that $|\det (A+2B)|\leq 3^n$.

Given an integer $n\ge 2$, compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$, where all $n$-element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$.

Let $A,B\in \mathcal{M}_n(\mathbb{R})$ such that $B^2=I_n$ and $A^2=AB+I_n$. Prove that: \[\det A\le \left(\frac{1+\sqrt{5}}{2}\right)^n\]

$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$).

Let $\mathbb R$ be the real numbers. Set $S = \{1, -1\}$ and define a function $\operatorname{sign} : \mathbb R \to S$ by \[ \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} \] Fix an odd integer $n$. Determine whether one can find $n^2+n$ real numbers $a_{ij}, b_i \in S$ (here $1 \le i, j \le n$) with the following property: Suppose we take any choice of $x_1, x_2, \dots, x_n \in S$ and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right), \quad \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then $z=x_1 x_2 \dots x_n$.

Let $A$ be a $5\times10$ matrix with real entries, and let $A^{\text T}$ denote its transpose. Suppose every $5\times1$ matrix with real entries can be written in the form $A\mathbf u$ where $\mathbf u$ is a $10\times1$ matrix with real entries. Prove that every $5\times1$ matrix with real entries can be written in the form $AA^{\text T}\mathbf v$ where $\mathbf v$ is a $5\times1$ matrix with real entries.

Do either (1) or (2): (1) Prove that the determinant of the matrix $$\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix}$$ is given by $(1+a^2 +b^2 +c^2)^{3}$. (2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.