Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence. Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

A tournament is held among $n$ teams, following such rules: a) every team plays all others once at home and once away.(i.e. double round-robin schedule) b) each team may participate in several away games in a week(from Sunday to Saturday). c) there is no away game arrangement for a team, if it has a home game in the same week. If the tournament finishes in 4 weeks, determine the maximum value of $n$.

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two. [color=#BF0000]Rewording of the last line for clarification:[/color] Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?