In a simple graph $G$, an operation is defined as taking two neighbor vertices $u,v$ which have a common neighbor, deleting the edge between $u,v$ and adding a new vertex $w$ whose neighbors are exactly the common neighbors of $u$ and $v$. Starting with the complete graph $G=K_n$ where $n\ge 3$ is a positive integer, find the maximum number of operations that can be applied. Proposed by[i] Deniz Can Karaçelebi[/i]

At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained? $(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$ $(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$ $(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$

The numbers $1,2,\dots,2000$ are written on the board. Two players are playing a game with alternating moves. A move consists of erasing two number $a,b$ and writing $a^b$. After some time only one number is left. The first player wins, if the numbers last digit is $2$, $7$ or $8$. If not, the second player wins. Who has a winning strategy? [I]Proposed by V. Frank[/i]

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that $ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done. [i]Yudi Satria, Jakarta[/i]

In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. [i]Maxim Didin[/i]

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.)

Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$. (a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$. (b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression \[\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)\] does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.

Five numbers 1,2,3,4,5 are written on a blackboard. A student may erase any two of the numbers a and b on the board and write the numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?

[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.

A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Now steps are performed according to the following scheme: (i) At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put into each of the neighbouring holes. (ii) No pebbles are ever removed from hole $0$. (iii) The game ends if there is no hole with a positive label that contains at least two pebbles. Show that the game always terminates, and that the number of pebbles in hole $0$ at the end of the game is independent of the specific sequence of steps. Determine this number.

Triangles $ABC$ and $ABD$ are isosceles with $AB =AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C + \angle D$ is [asy] size(130); defaultpen(linewidth(0.8) + fontsize(11pt)); pair A, B, C, D, E; real angle = 70; B = origin; A = dir(angle); D = dir(90-angle); C = rotate(2*(90-angle), A) * B; draw(A--B--C--cycle); draw(B--D--A); E = extension(B, D, C, A); draw(rightanglemark(B, E, A, 1.5)); label("$A$", A, dir(90)); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, dir(0)); label("$E$", E, 1.5*dir(340)); [/asy] $\textbf{(A)}\ 115^\circ \qquad \textbf{(B)}\ 120^\circ \qquad \textbf{(C)}\ 130^\circ \qquad \textbf{(D)}\ 135^\circ \qquad \textbf{(E)}\ \text{not uniquely determined}$

A group of $67$ students pass their examination consisting of $6$ questions, labeled with the numbers $1$ to $6$. A correct answer to question $n$ is quoted $n$ points and for an incorrect answer to the same question a student loses $n$ point. a) Find the least possible positive difference between any $2$ final scores b) Show that at least $4$ participants have the same final score c) Show that at least $2$ students gave identical answer to all six questions.

Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that $ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$ $ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$. Prove that $ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M. [i]A. Khrabrov[/i]

For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: (1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$, (2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and (3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.