There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

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.

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.

The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player wins, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

$2019$ 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)

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?

Amy and Bec play the following game. Initially, there are three piles, each containing $2020$ stones. The players take turns to make a move, with Amy going first. Each move consists of choosing one of the piles available, removing the unchosen pile(s) from the game, and then dividing the chosen pile into $2$ or $3$ non-empty piles. A player loses the game if he/she is unable to make a move. Prove that Bec can always win the game, no matter how Amy plays.

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

Twenty cubes of the same size and appearance are made of either aluminum or of heavier duralumin. How can one find the number of duralumin cubes using not more than $11$ weighings on a balance without weights? (We assume that all cubes can be made of aluminum, but not all of duralumin.)

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie.

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)

Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.

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 $N=29$

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]

Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first. The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins. Which of the two has a winning strategy? [i](Birgit Vera Schmidt)[/i]

Sixty-four dice with the numbers ”one” to ”six” are placed on one table and formed into a square with eight horizontal and eight vertical rows of cubes pushed together. By rotating the dice, while maintaining their place, we want to finally have all sixty-four dice the "one" points upwards. Each dice however, may not be turned individually, but only every eight dice in a horizontal or vertical row together by $90^o$ to the longitudinal axis of this row may turn. Prove that it is always possible to solve the dice by repeatedly applying the permitted type of rotation to the required end position.

Alma and Bertha play the following game. There are $100$ round, $200$ triangular and $200$ square pieces on a table. In each move a player must remove two pieces, but it cannot be a triangle and a square. Alma starts, and one loses if one is unable to move or if there are no pieces left when it is one’s turn. Which player has a winning strategy?

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?

Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i] triple.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

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).