There are two bowls on a table, one white and one black. In the white bowl there $2019$ balls. Players $A$ and $B$ play a game where they make every other move ($A$ begins). One move consists is $\bullet$ to move one or your balls from one bowl to the other, or $\bullet$ to remove a ball from the white bowl, with the condition that the resulting position (that is, the number of bullets in the two bowls) have not occurred before. The player who has no valid move to make loses. Can any of the players be sure to win? If so, which one?

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 $n <2m$.

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?

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 $n <2m$.

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).

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Three players $A,B$ and $C$ take turns removing stones from a pile of $N$ stones. They move in the order $A,B,C,A,B,C,…A$. The game begins, and the one who takes out the last stone loses the game. The players $A$ and $C$ team up against $B$ , they agree on a joint strategy. $B$ can take in each play $1,2,3,4$ or $5$ stones, while $A$ and $C$, they can each get $1,2$ or $3$ stones each turn. Determine for what values ​​of $N$ have winning strategy $A$ and $C$, and for what values ​​the winning strategy is from $B$. .

Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the [i]starting position [/i]of the game. Henning and Paul play with the following rules: $\bullet$ They take turns alternatively where Henning begins. $\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles. $\bullet$ The player how take the last coin wins. Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$ $(a)$ Prove that $A$ has infinity many elements. $(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)

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

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

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.

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

Ana and Natalia alternately play on a $ n \times n$ board (Ana rolls first and $n> 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), without leaving the board. The winner is whoever manages to place their token on the opponent's token. Determine if either of them can secure victory after a finite number of turns.

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.

Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time. Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders. Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.

Let $n$ be an even integer greater than $2$. On the vertices of a regular polygon with n sides we can place red or blue chips. Two players, $A$ and $B$, play alternating turns of the next mode: each player, on their turn, chooses two vertices that have no tiles and places on one of them a red chip and in the other a blue chip. The goal of $A$ is to get three vertices consecutive with tiles of the same color. $B$'s goal is to prevent this from happening. To the beginning of the game there are no tiles in any of the vertices. Show that regardless of who starts to play, Player $B$ can always achieve his goal.

There is a row with $2024$ cells. Ana and Beto take turns playing, with Ana going first. On each turn, the player selects an empty cell and places a digit in that space. Once all $2024$ cells are filled, the number obtained from reading left to right is considered, ignoring any leading zeros. Beto wins if the resulting number is a multiple of $99$, otherwise Ana wins. Determine which of the two players has a winning strategy and describe it.

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker. If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer. [b]Comment.[/b] Batman and Robin decide their common strategy at the beginning. [img]https://i.imgur.com/PeLBQNt.png[/img]

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.

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).

The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.

All the cells in a $8* 8$ board are colored white. Omar and Asaad play the following game: in the beginning Omar colors $n$ cells red, then Asaad chooses $4$ rows and $4$ columns and colors them black. Omar wins if there is at least one red cell. Find the least possible value for n such that Omar can always win regardless of Asaad's move.