Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$. Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$ otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.

Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?

On an infinite square grid, Gru and his $2022$ minions play a game, making moves in a cyclic order with Gru first. On any move, the current player selects $2$ adjacent cells of their choice, and paints their shared border. A border cannot be painted over more than once. Gru wins if after any move there is a $2 \times 1$ or $1 \times 2$ subgrid with its border (comprising of $6$ segments) completely colored, but the $1$ segment inside it uncolored. Can he guarantee a win? [i]Evan Chang[/i] [size=50](oops)[/size]

There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move: $\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes. $\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. Then $B$ destroys all the cups having no pebbles. $\bullet$ $B$ switches the places of two cups without telling $A$. After finitely many moves, $A$ can guarantee that $n$ cups are destroyed. Find the maximum possible value of $n$. (Note that $A$ doesn't see the cups while playing.) [i]Proposed by Emre Osman[/i]

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

$n$ points are marked on the board points that are vertices of the regular $n$ -gon. One of the points is a chip. Two players take turns moving it to the other marked point and at the same time draw a segment that connects them. If two points already connected by a segment, such a move is prohibited. A player who can't make a move, lose. Which of the players can guarantee victory?

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 $k$.