Two players play the following game starting with one pile of at least two stones. A player in turn chooses one of the piles and divides it into two or three nonempty piles. The player who cannot make a legal move loses the game. Which player has a winning strategy?

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy? Proposed by [i]Borche Joshevski, Macedonia[/i]

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

Hamza and Majid play a game on a horizontal $3 \times 2015$ white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal $1 \times 3$ rectangle. A legal move for Majid consists of painting three unit squares forming a vertical $3\times 1$ rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win? Lê Anh Vinh

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question: Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy? Indicate if Arnulfo and Berenice are right and answer the question asked by them.

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.

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]

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 $k$.

Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game. For each positive integer $n$, find a winning strategy for one of the players.

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]

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.

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

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]

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after the second player duplicated the move of the first player, they exchange hats. The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

Initially the number $2000$ is written down. The following operation is repeatedly performed: the sum of the $10$-th powers of the last number's digits is written down. Prove that in the infinite sequence thus obtained, some two numbers will be equal.

An integer is given $N> 1$. Arne and Britt play the following game: (1) Arne says a positive integer $A$. (2) Britt says an integer $B> 1$ that is either a divisor of $A$ or a multiple of $A$. ($A$ itself is a possibility.) (3) Arne says a new number $A$ that is either $B - 1, B$ or $B + 1$. The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number $N$ before the $50$th has been said. Otherwise, Arne wins. a) Show that Arne has a winning strategy if $N = 10$. b) Show that Britt has a winning strategy if $N = 24$. c) For which $N$ does Britt have a winning 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]

There are $8$ cards on which the numbers $1$, $2$, $\dots$, $8$ are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let $S$ be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer $N$ for which Bob can guarantee that $S$ is at least $N$.

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim? (A. Andjans, Riga)

Anna and Orjan play the following game: they start with a positive integer $n>1$, Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?