Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions : (a) $ x_{0} \equal{} x_{n} \equal{} 0$, and (b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$. Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.

Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote \[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \] Initially, some pieces are put on the each point in $Q$, with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ; (2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ; (3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$. Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$, no matter how the pieces are distributed initially.

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

[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.

Finitely many cards are placed in two stacks, with more cards in the left stack than the right. Each card has one or more distinct names written on it, although different cards may share some names. For each name, we define a “shuffle” by moving every card that has this name written on it to the opposite stack. Prove that it is always possible to end up with more cards in the right stack by picking several distinct names, and doing in turn the shuffle corresponding to each name.

$ 1994$ girls are seated at a round table. Initially one girl holds $ n$ tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if $ n < 1994$, the game must terminate. b.) Show that if $ n \equal{} 1994$ it cannot terminate.

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.

Consider a $2^k$-tuple of numbers $(a_1,a_2,\dots,a_{2^k})$ all equal to $1$ or $-1$. In one step, we transform it to $(a_1a_2,a_2a_3,\dots,a_{2^k}a_1)$. Prove that eventually, we will obtain a $2^k$-tuple consisting only of $1$'s.

The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles? [asy] unitsize(45); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0)); dot((0,0),ds); label("$A$", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("$B$", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("$C$", (1.96,1.09),NE*lsf); dot((4,0),ds); label("$D$", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy] $ \textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi $

Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

We are given an $n \times n$ board, where $n$ is an odd number. In each cell of the board either $+1$ or $-1$ is written. Let $a_k$ and $b_k$ denote them products of numbers in the $k^{th}$ row and in the $k^{th}$ column respectively. Prove that the sum $a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n$ cannot be equal to zero.

A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?

There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later). After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? [i]I. Izmest’ev[/i]

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]

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

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then \[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?