Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

Prove that for integer $n$ we have: \[n! \le \left( \frac{n+1}{2} \right)^n\] [size=75][i](please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)[/i][/size]

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.

Denote by $S_n$ the group of permutations of the sequence $(1,2,\dots,n).$ Suppose that $G$ is a subgroup of $S_n,$ such that for every $\pi\in G\setminus\{e\}$ there exists a unique $k\in \{1,2,\dots,n\}$ for which $\pi(k)=k.$ (Here $e$ is the unit element of the group $S_n.$) Show that this $k$ is the same for all $\pi \in G\setminus \{e\}.$

Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality: \[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]

There are a buch of 2000 stones. Two players play alternatively, following the next rules: ($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch. ($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play. The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.

Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that: \[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]

Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.

$2^{n-1}$ subsets are choosen from a set with $n$ elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.

Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that \[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\] Prove that Santa Claus can give each child a gift that this child likes.

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(m+f(n))=f(f(m))+f(n).\]

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that: (a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$; (b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$; (c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$. [i]Dan Schwarz[/i]

In the decimal expression of a $ 2009$-digit natural number there are only the digits $ 5$ and $ 8$. Prove that we can get a $ 2008$-digit number divisible by $ 11$ if we remove just one digit from the number.

Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?

A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1\equal{}1\minus{}a_0$,$ a_{n\plus{}1}\equal{}1\minus{}a_n(1\minus{}a_n)$, $ n\equal{}1,2,...$. Prove that for any positive integer $ n$, we have $ a_{0}a_{1}...a_{n}(\frac{1}{a_0}\plus{}\frac{1}{a_1}\plus{}...\plus{}\frac{1}{a_n})\equal{}1$