The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way : [list] [*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked [*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$. [/list] Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.

There are infinitely many boxes - initially one of them contains $n$ balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?

We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \). a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).

$ 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.

Sydney the squirrel is at $(0, 0)$ and is trying to get to $(2021, 2022)$. She can move only by reflecting her position over any line that can be formed by connecting two lattice points, provided that the reflection puts her on another lattice point. Is it possible for Sydney to reach $(2021, 2022)$?

Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: [b](i)[/b] Each vertice is colored by exactly one color. [b](ii)[/b] Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. [i]Alternative statement:[/i] In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.

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 $n > 3$ 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.

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}$.

Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.

The numbers $1, 2, 3, \ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\geqslant 2b$, he erases them, and in their place writes the number $a-2b$. Find all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard.

In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

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.

1) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy? 2) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b+2$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?

A regular $ (5 \times 5)$-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

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

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.

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.

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.

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]

Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals: $ \textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \minus{}17$