Let $n \geq 3$ be an integer. Consider the set $A=\{1,2,3,\ldots,n\}$, in each move, we replace the numbers $i, j$ by the numbers $i+j$ and $|i-j|$. After doing such moves all of the numbers are equal to $k$. Find all possible values for $k$.

Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with lamp $k+1$,..., lamp $n-1$ with lamp $n$, lamp $n$ with lamp $1$. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.

Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define \[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \] We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?

Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\] [i]Fajar Yuliawan, Bandung[/i]

Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions: $(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater. $(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal. Find all possible values of $n$.

Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?

In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds: [list] [*]Some student from $A$ knows all students from $B$. [*]Some student from $B$ knows all students from $C$. [*] Some student from $C$ knows all students from $A$.[/list]

Show that for each $n \ge 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a, b\in S$.

Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

Prove that each finite set of integers can be arranged without intersection.

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.

Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by \[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\] Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.

Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that (i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$; (ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$; (iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$. Prove that $a_1,a_2$ and $a_3$ cannot all be prime.

Find the set of all $ a \in \mathbb{R}$ for which there is no infinite sequene $ (x_n)_{n \geq 0} \subset \mathbb{R}$ satisfying $ x_0 \equal{} a,$ and for $ n \equal{} 0,1, \ldots$ we have \[ x_{n\plus{}1} \equal{} \frac{x_n \plus{} \alpha}{\beta x_n \plus{} 1}\] where $ \alpha \beta > 0.$

Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively. Let $ f: Z \rightarrow R$ be a function satisfying: (i) $ f(n) \ge 0$ for all $ n \in Z$ (ii) $ f(mn)\equal{}f(m)f(n)$ for all $ m,n \in Z$ (iii) $ f(m\plus{}n) \le max(f(m),f(n))$ for all $ m,n \in Z$ (a) Prove that $ f(n) \le 1$ for all $ n \in Z$ (b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) \equal{} 1$

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , \[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

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 $X=\mathbb{N}$.

A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$