Ana and Bruno have an $8 \times 8$ checkered board. Ana paints each of the $64$ squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the $4$ squares where they intersect. Bruno's goal is for these $4$ squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

Let $n\ge 3$ be an integer. Lucas and Matías play a game in a regular $n$-sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice. a) Determine all the $n\ge 3$ such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías. b) Determine all the $n\ge 3$ such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas. Note. The two players know the value of $n$ and see the polygon.

Three players $A, B$ and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, $....$, $A$ starts the game and the one who takes the last stone loses. Players $A$ and $C$ They form a team against $B$, they agree on a strategy joint. $B$ can take $1, 2, 3, 4$ or $5$ stones on each move, while that $A$ and $C$ can each draw $1, 2$ or $3$ stones in each turn. Determine for which values of $N$ have winning strategies $A$ and $C$ , and for what values the winning strategy is $B$'s.

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces Anja and Bernd play in turns according to the following rule: [i]When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. [/i] Who has a winning strategy when Anja starts?

The numbers from $1$ to $500$ are written on the board. Two players $A$ and $B$ erase alternately one number at a time, and $A$ deletes the first number. If the sum of the last two number on the board is divisible by $3$, $B$ wins, otherwise $A$ wins. Which player can lay out a strategy that ensures this player's victory?

Josie and Ross are playing a game on a $20 \times 20$ chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie’s turn, she selects any two different empty cells, and places one white stone in each of them. On Ross’ turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are $ 8$ consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?

Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game: (i) Alberto chooses a positive integer, (ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself), (iii) Alberto increases or decreases the Barbara’s number by $1$. Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?

Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

There are $9999$ rods with lengths $1, 2, ..., 9998, 9999$. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd. Who has a winning strategy?

Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.

Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$. Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy? [i](6 points)[/i]

On the table there are $2016$ coins. Two players play the following game making alternating moves. In one move it is allowed to take $1, 2$ or $3$ coins. The player who takes the last coin wins. Which player has a winning strategy?

Two people, $A$ and $B$, play the game of blowing up a balloon. The balloon will explode only when the volume of the balloon $V>2014$ mL. $A$ blows in $1$ mL first, and then they takes turns blowing. It is agreed that the gas blown by each person must not be less than the gas blown by the other party last time and should not be more than twice the amount of gas the other party blew last time. The agreement is that the person who blows up the balloon loses. Who has a winning strategy ? Briefly explain it. (Do not consider the change in volume caused by the change in tension when the balloon is inflated).

Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon $\mathcal{S}$ and then Nicole can shift $\mathcal{S}$ to wherever she wants. Johann wins if there exists a point with coordinates $(x, y)$ in the interior of $\mathcal{S}$, where $x$ and $y$ are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.

John and Mary play the following game. First they choose integers $n > m > 0$ and put $n$ sweets on an empty table. Then they start to make moves alternately. A move consists of choosing a nonnegative integer $k\le m$ and taking $k$ sweets away from the table (if $k = 0$ , nothing happens in fact). In doing so no value for $k$ can be chosen more than once (by none of the players) or can be greater than the number of sweets at the table at the moment of choice. The game is over when one of the players can make no more moves. John and Mary decided that at the beginning Mary chooses the numbers $m$ and $n$ and then John determines whether the performer of the last move wins or looses. Can Mary choose $m$ and $n$ in such way that independently of John’s decision she will be able to win?

Let $m$ be a positive integer. Two players, Axel and Elina play the game HAUKKU ($m$) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$. PS. As member Loppukilpailija noted, it should be written $m>1$, as the statement does not hold for $m = 1$.

Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$. I. Bogdanov

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.