Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A [i]step[/i] is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it. [b]a)[/b] Show that, for any $ m,n\ge 5, $ there exists a succession of [i]steps[/i] that make the rectagle to be single-colored. [b]b)[/b] What about $ m=n+1=5? $

A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps). In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$ First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them, Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw. Assume that both players play at their best. What will be the result of the game ? [i]Proposed by Mahyar Sefidgaran[/i]

Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule: First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest. This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest. How many gram of gold can Smilla make sure to win?

Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen.

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table. Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.

On a board there are written the integers from $1$ to $119$. Two players, $A$ and $B$, make a move by turn. A $move$ consists in erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player $A$ makes the first move. Find the highest integer $k$, such that the player $A$ can be sure that his score is not smaller than $k$.

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares [i]forbidden[/i], meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game. What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)

All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy? [I]Proposed by A. Khrabrov[/i]

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.) [i]Proposed by Steven Liu[/i]

Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle. Determine who wins each of the following games: (a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$. (b) The starting rectangles are $100 \times 100$ and $100 \times 500$.

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo: [b]Rule 1:[/b] Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise. [b]Rule 2:[/b] Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed. Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

Tanya and Serezha have a heap of $2016$ candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?

Let $N$ a positive integer. In a spaceship there are $2 \cdot N$ people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows: 1) The first alien chooses any person as she wishes. 2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn. 3) The alien that can't play in her turn loses. Prove that second player has a winning strategy [i]if, and only if[/i], the $2 \cdot N$ people can be divided in $N$ pairs in such a way that two people in the same pair are friends.

On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement. Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?

The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.