Let $n$ be a positive integer. Lavi Dopes has two boards $n \times n$. On the first board, he writes an integer in each of his $n^2$ squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if $n = 3$ and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this. Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board. (a) If $n = 4$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board? (b) If $n = 5$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of \[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\] How many guesses are needed for Banana to figure out the quadruple Ana chose?

There is a queue of $n{}$ girls on one side of a tennis table, and a queue of $n{}$ boys on the other side. Both the girls and the boys are numbered from $1{}$ to $n{}$ in the order they stand. The first game is played by the girl and the boy with the number $1{}$ and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if $n{}$ is odd, then a girl and a boy with odd numbers played in the last game. [i]Proposed by A. Gribalko[/i]

Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

Two players take turns playing on a $3\times1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allows them to win, no matter how well his opponent plays.

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is $0$? (Maxim Didin)

Given $n \ge 2$ points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position $P$ to one of the $n-1$ other points $Q$ and draws the segment $PQ$. This move is not allowed if the segment $PQ$ is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each $n$, which of the two players has a winning strategy.

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]

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.

Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.

$A$ 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]

Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$ and $n$ where $m, n \ge 3$. Proposed by [i]Toghrul Abbasov, Azerbaijan[/i]

There is a black token in the lower-left corner of a board $m \times n$ ($m, n \ge 3$), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first. In his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens. Vasyl wins if at some point of the game white tokens are in the same cell. For which values of $m, n$ can Petryk prevent him from winning? [i](Proposed by Arsenii Nikolaiev)[/i]

Given a positive integer $k$ and other two integers $b > w > 1.$ There are two strings of pearls, a string of $b$ black pearls and a string of $w$ white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step: [b](i)[/b] The strings are ordered by their lengths in a non-increasing order. If there are some strings of equal lengths, then the white ones precede the black ones. Then $k$ first ones (if they consist of more than one pearl) are chosen; if there are less than $k$ strings longer than 1, then one chooses all of them. [b](ii)[/b] Next, one cuts each chosen string into two parts differing in length by at most one. (For instance, if there are strings of $5, 4, 4, 2$ black pearls, strings of $8, 4, 3$ white pearls and $k = 4,$ then the strings of 8 white, 5 black, 4 white and 4 black pearls are cut into the parts $(4,4), (3,2), (2,2)$ and $(2,2)$ respectively.) The process stops immediately after the step when a first isolated white pearl appears. Prove that at this stage, there will still exist a string of at least two black pearls. [i]Proposed by Bill Sands, Thao Do, Canada[/i]

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below: [img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img] Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses. (a) Show that, even if Chris and Michael play cooperatively, the game will come to an end. (b) Which player has a winning strategy?

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

The box contains a complete set of dominoes. Two players take turns choosing one dice from the box and placing them on the table, applying them to the already laid out chain on either of the two sides according to the rules of domino. The one who cannot make his next move loses. Who will win if they both played correctly?

Let $n \ge 4$ and $k$ be positive integers. We consider $n$ lines in the plane between which there are not two parallel nor three concurrent. In each of the $\frac{n(n-1)}{2}$ points of intersection of these lines, $k$ coins are placed. Ana and Beto play the following game in turns: each player, in turn, chooses one of those points that does not share one of the $n$ lines with the point chosen immediately before by the other player, and removes a coin from that point. Ana starts and can choose any point. The player who cannot make his move loses. Determine based on $n$ and $k$ who has a winning strategy.

Alice and Bob play the following game on a square grid with $2024 \times 2024$ unit squares. They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. \\ On the $k$-th turn, let $n_k$ be the least integer such that $n_k\geqslant\tfrac{k}{2024}$. If there is at least one square without a sticker, then the player taking the turn: [list = i] [*] selects at most $n_k$ unit squares on the grid such that at least one of the chosen unit squares does not have a sticker. [*] covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first. [/list] At the end of their turn, a player wins if there exist $123$ unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. \\ Does Alice have a winning strategy? [i]Proposed by Erik Paemurru, Estonia[/i]

Given integer $n \geq 3$. There are $n$ dots marked $1$ to $n$ clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up $n$ pairs from $\{ (i,j)|1 \leq i < j \leq n \}$ and for every pairs $(i,j)$. B starts from the point marked $i$ and choose to walk clockwise or counterclockwise to the point marked $j$. And B invert the status of all passing lights (bright $\leftrightarrow$ dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end? [i]Proposed by BlessingOfHeaven[/i] [img]https://pbs.twimg.com/profile_images/1014932415201120256/u9KAaMZ4_400x400.jpg[/img]