The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner. Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.

A table with three rows and 100 columns is given. Initially, in the left cell of each row there are $400\cdot 3^{100}$ chips. At one move, Petya marks some (but at least one) chips on the table, and then Vasya chooses one of the three rows. After that, all marked chips in the selected row are shifted to the right by a cell, and all marked chips in the other rows are removed from the table. Petya wins if one of the chips goes beyond the right edge of the table; Vasya wins if all the chips are removed. Who has a winning strategy? [i]Proposed by P. Svyatokum, A. Khuzieva and D. Shabanov[/i]

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?

Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays. (A Kanel)

Nils is playing a game with a bag originally containing $n$ red and one black marble. He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if it is black, it decreases by $x$. The game is over after $n$ moves when there is only a single marble left. In each move Nils chooses $x$ so that he ensures a final fortune greater or equal to $Y$ . What is the largest possible value of $Y$?

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?

Bob proposes the following game to Johanna. The board in the figure is an equilateral triangle subdivided in turn into $256$ small equilateral triangles, one of which is painted in black. Bob chooses any point inside the board and places a small token. Johanna can make three types of plays. Each of them consists of choosing any of the $3$ vertices of the board and move the token to the midpoint between the current position of the tile and the chosen vertex. In the second figure we see an example of a move in which Johana chose vertex $A$. Johanna wins if she manages to place her piece inside the triangle black. Prove that Johanna can always win in at most $4$ moves. [asy] unitsize(8 cm); pair A, B, C; int i; A = dir(60); C = (0,0); B = (1,0); fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle); for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); } label("$A$", A, N); label("$B$", B, SE); label("$C$", C, SW); [/asy] [asy] unitsize(8 cm); pair A, B, C, X, Y, Z; int i; A = dir(60); C = (0,0); B = (1,0); X = 9.2/16*(1,0) + 3.3/16*dir(60); Y = (A + X)/2; Z = rotate(60,X)*(Y); fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle); for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); } draw(A--X, dotted); draw(arc(Z,abs(X - Y),-12,40), Arrow(6)); label("$A$", A, N); label("$B$", B, SE); label("$C$", C, SW); dot(A); dot(X); dot(Y); [/asy]

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

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.

Antonio and Beltran have impeccable logical reasoning, they put on a hat with a integer between $0$ and $19$ (including both) so that each of them sees the number that has the other (but cannot see his own number), and they must try to guess the number that have on their hat. They have a timer that a bell rings every minute and the moment it rings. This is when they must say if they know the number on their hat. A third person tells them: ''the sum of the numbers is $6$ or $11$ or $19$''. At that moment it begins to run time. After a minute the bell rings and neither of them says anything. The second minute passes , the doorbell rings and neither of us says anything. Time continues to pass and when the bell rings for the tenth time Antonio says that he already knows what is his number. Just determine the number each has in his hat.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges. Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game: • First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas. • Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants. • After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron. Determine if there is a strategy for which Lucas can always properly discover the four special sweets.

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block. Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.

Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way: [list] [*] At each round, you can have the guinea pigs taste any number of juice boxes. [*] Conversely, a juice box can be tasted by any number of guinea pigs. [*] After the round ends, any guinea pigs who tasted the poisoned juice die. [/list] Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?

For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle. Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game. Determine all pairs $(m,n)$ for which Renate has a winning strategy.

Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]

Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is: i) $3 \times 3$ ii) $3 \times 4$

The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$. To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$, be determined? (b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$, be determined? (Theresia Eisenkölbl, Clemens Heuberger)

Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy. [b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.

Let $G$ be a complete directed graph with $2024$ vertices and let $k \leq 10^5$ be a positive integer. Angel and Boris play the following game: Angel colors $k$ of the edges in red and puts a pawn in one of the vertices. After that in each move, first Angel moves the pawn to a neighboring vertex and then Boris has to flip one of the non-colored edges. Boris wins if at some point Angel can't make a move. Find, depending on $G$ and $k$, whether or not Boris has a winning strategy.

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.

Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins? (Oleksii Masalitin)

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.