Prove that for all natural $ m$, $ n$ polynomial $ \sum_{i \equal{} 0}^{m}\binom{n\plus{}i}{n}\cdot x^i$ has at most one real root.

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.

Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold: $\begin{array}{rl} i. & f(x^2) = f(x)^2 -2xf(x) \\ ii. & f(-x) = f(x-1)\\ iii. & 1<x<y \Longrightarrow f(x) < f(y). \end{array}$

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt. Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate. Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost. [i](4 points)[/i]

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]

The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$, \[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\] Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$.

Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders: (a) Each day, at least one task must be issued to a soldier. (b) No soldier may have more than two task or receive more than one tasks in a single day. (c) The lists of soldiers receiving tasks for two different days must not be the same. (d) The first sergeant violating any of these orders will be jailed. Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed? [i]M. Kulikov[/i]

Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

A bounded sequence $ (x_n)_{n\ge 1}$ of real numbers satisfies $ x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2}$ for all $ n \ge 1$. Prove that this sequence has a finite limit.

Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$ a) Prove that the sequence consists only of natural numbers. b) Check if there are terms of the sequence divisible by $2011$.

A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles. [i]A. Shapovalov[/i]

Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$