Given a positive integer $k$ and other two integers $b > w > 1.$ There are two strings of pearls, a string of $b$ black pearls and a string of $w$ white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step: [b](i)[/b] The strings are ordered by their lengths in a non-increasing order. If there are some strings of equal lengths, then the white ones precede the black ones. Then $k$ first ones (if they consist of more than one pearl) are chosen; if there are less than $k$ strings longer than 1, then one chooses all of them. [b](ii)[/b] Next, one cuts each chosen string into two parts differing in length by at most one. (For instance, if there are strings of $5, 4, 4, 2$ black pearls, strings of $8, 4, 3$ white pearls and $k = 4,$ then the strings of 8 white, 5 black, 4 white and 4 black pearls are cut into the parts $(4,4), (3,2), (2,2)$ and $(2,2)$ respectively.) The process stops immediately after the step when a first isolated white pearl appears. Prove that at this stage, there will still exist a string of at least two black pearls. [i]Proposed by Bill Sands, Thao Do, Canada[/i]

The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$, $b$, $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point? [i]Proposed by Evan Chen[/i]

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values. [i]Proposed by A. Golovanov[/i]

We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.

Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.

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] $\textbf{(A)}\ 115^\circ \qquad \textbf{(B)}\ 120^\circ \qquad \textbf{(C)}\ 130^\circ \qquad \textbf{(D)}\ 135^\circ \qquad \textbf{(E)}\ \text{not uniquely determined}$

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

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.

For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by \[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. $ a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?

Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.

It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers: $a)$ $0$, $2$, $4$, $6$ and $8$ $b)$ $3$, $4$, $5$, $6$ and $7$

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

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? $(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$ $(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$ $(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$ . A [i]move[/i] consists in choosing three consecutive vertices and changing the signs from the vertices , from $+$ to $-$ and from $-$ to $+$. [b]a)[/b] Prove that if $n=2015$ then for any initial configuration of signs , there exists a sequence of [i]moves[/i] such that we'll arrive at a configuration with only $+$ signs. [b]b)[/b] Prove that if $n=2016$ , then there exists an initial configuration of signs such that no matter how we make the [i]moves[/i] we'll never arrive at a configuration with only $+$ signs.

The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.

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.

Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

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 $i = 1, 2, \ldots, n$.

On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?

A little boy wrote the numbers $1,2,\cdots,2011$ on a blackboard. He picks any two numbers $x,y$, erases them with a sponge and writes the number $|x-y|$. This process continues until only one number is left. Prove that the number left is even.