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

Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

In an $m\times n$ rectangular grid, where m and n are odd integers, $1\times 2$ dominoes are initially placed so as to exactly cover all but one of the $1\times 1$ squares at one corner of the grid. It is permitted to slide a domino towards the empty square, thus exposing another square. Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle. [i]A. Shapovalov[/i]

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$. Prove that if Alice picks $S$ to be of the form \[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\] for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.) [i]Proposed by Mark Sellke[/i]

Let be three complex numbers $ \alpha ,\beta ,\gamma $ such that $$ \begin{vmatrix} \left( \alpha -\beta \right)^2 & \left( \alpha -\beta \right)\left( \beta -\gamma \right) & \left( \beta -\gamma \right)^2 \\ \left( \beta -\gamma \right)^2 & \left( \beta -\gamma \right)\left( \gamma -\alpha \right) & \left( \gamma -\alpha \right)^2 \\ \left( \gamma -\alpha \right)^2 & \left( \gamma -\alpha \right)\left( \alpha -\beta \right) & \left( \alpha -\beta \right)^2\end{vmatrix} =0. $$ Prove that $ \alpha ,\beta ,\gamma $ are all equal, or their affixes represent a non-degenerate equilateral triangle. [i]Gheorghe Necșuleu[/i] and [i]Ion Necșuleu[/i]

Find all positive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$. [i]Proposed by Serbia[/i]

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.

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$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)$, $ \left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)$, $ \ldots$, $ \left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)$ of $ k$-tuples of finite sets satisfying the following two properties: (i) for every $ i$ and every $ j \neq j^{\prime}$, $ A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset$, and (ii) for every $ i\neq i^{\prime}$ there exist $ j\neq j^{\prime}$ for which $ A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset$. Prove that \[ \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1. \]

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.)

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]