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]

$10$ people are sitting at a round table. There are some nuts in front of each of them, $100$ nuts altogether. After a certain signal each person passes some of his nuts to the person sitting to his right . If he has an even number of nuts, he passes half of them; otherwise he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone will have exactly $10$ nuts. (A Shapovalov)

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time. Determine all $k$ such that $A$ can always win the game.

In a game two players alternately choose larger natural numbers. At each turn the difference between the new and the old number must be greater than zero but smaller than the old number. The original number is 2. The winner is considered to be the player who chooses the number $1987$. In a perfect game, which player wins?

Let $G$ be a complete directed graph with $2024$ vertices and let $k \leq 10^5$ be a positive integer. Angel and Boris play the following game: Angel colors $k$ of the edges in red and puts a pawn in one of the vertices. After that in each move, first Angel moves the pawn to a neighboring vertex and then Boris has to flip one of the non-colored edges. Boris wins if at some point Angel can't make a move. Find, depending on $G$ and $k$, whether or not Boris has a winning strategy.

Mari and Yuri play the next play. At first, there are two piles on the table, with $m$ and $n$ candies, respectively. At each turn, players eats one pile of candy from the table and distribute another pile of candy into two non-empty parts ,. Everything is done in turn and wins the player who can no longer share the pile (when there is only one candy left). Which player will win if both use the optimal strategy and Mari makes the first move?

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.

(a) Two players take turns taking $1, 2$ or $3$ stones at random from a given set of $3$ piles, in which initially on $11, 22$ and $33$ stones. If after the move of one of the players in any two groups the same number of stones will remain, this player has won. Who will win with the right game of both players? (b) Two players take turns taking $1$ or $2$ stones from one pile, randomly selected from a given set of $3$ ordered piles, in which at first $100, 200$ and $300$ stones, in order from left to right. Additionally it is forbidden to make a course at which, for some pair of the next handfuls, quantity of stones in the left will be more than the number of stones in the right. If after the move of one of the players of the stones in handfuls will not remain, then this player won. Who will win with the right game of both players? [hide=original wording] 1. Два гравця по черзi беруть 1, 2 чи 3 камiнця довiльним чином з заданого набору з 3 купок, в яких спочатку по 11, 22 i 33 камiнцiв. Якщо пiсля хода одного з гравцiв в якихось двух купках залишиться однакова кiлькiсть камiнцiв, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв? 2. Два гравця по черзi беруть 1 чи 2 камiнця з одної купки, довiльної вибраної з заданого набору з 3 впорядкованих купок, в яких спочатку по 100, 200 i 300 камiнцiв, в порядку злiва направо. Додатково забороняется робити ход при якому, для деякої пари сусiднiх купок, кiлькiсть камiнцiв в лiвiй стане бiльше нiж кiлькiсть камiнцiв в правiй. Якщо пiсля ходу одного з гравцiв камiнцiв в купках не залишиться, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв?[/hide]

From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?

Suppose that zeros and ones are written in the cells of an $n\times n$ board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed $\frac n2\left(1+\sqrt{4n-3}\right)$.

There are $n$ coins aligned in a row. In each step, it is allowed to choose a coin with the tail up (but not one of the outermost markers), remove it and reverse the closest coin to the left and the closest coin to the right of it. Initially, all the coins have tails up. Prove that one can achieve the state with only two coins remaining if and only if $n-1$ is not divisible by $3$.

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]

Let the [i]subbishop[/i] (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$, no two of which are not attacking. [i]V. Chukanov[/i]

Two positive integers $m$ and $n$ are written on the board. We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer). Depending on the numbers $m$ and $n$, specify all the pairs that can appear on the board in pairs. (Radovan Švarc)

A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$. Otherwise, the other player wins. Decide which of the players has a winning strategy. (Pavel Šalom)