Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?

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?

Let $n>1$ be a given positive integer. 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 $n$ 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 $n$ loses. Who wins if every player wants to win? Find answer for each $n>1$. [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.

Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$. Then player $II$ chooses another square and marks it with $X$. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.

Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following: $$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$ Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if $a) p = 10^9 + 7$; $b) p = 10^9 + 9$. [i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime. [i]Proposed by Ivan Novak[/i]

Arne and Bertil play a game on an $11 \times 11$ grid. Arne starts. He has a game piece that is placed on the center od the grid at the beginning of the game. At each move he moves the piece one step horizontally or vertically. Bertil places a wall along each move any of an optional four squares. Arne is not allowed to move his piece through a wall. Arne wins if he manages to move the pice out of the board, while Bertil wins if he manages to prevent Arne from doing that. Who wins if from the beginning there are no walls on the game board and both players play optimally?

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block. Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.

Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?

On the circumference of a circle there are red and blue points. One may add a red point and change the colour of both its neighbours (to the other colour) or remove a red point and change the colour of both its previous neighbours. Initially there are two red points. Prove that there is no sequence of allowed operations which leads to the configuration consisting of two blue points. (K Kazarnovskiy, Moscow)

Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable

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.

All the cells in a $8* 8$ board are colored white. Omar and Asaad play the following game: in the beginning Omar colors $n$ cells red, then Asaad chooses $4$ rows and $4$ columns and colors them black. Omar wins if there is at least one red cell. Find the least possible value for n such that Omar can always win regardless of Asaad's move.

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]

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]

Ana and Beto are playing a game. Ana writes a whole number on the board. Beto then has the right to erase the number and add $2$ to it, or erase the number and subtract $3$, as many times as he wants. Beto wins if he can get $2021$ after a finite number of stages; otherwise, Ana wins. Which player has a winning strategy?

(Game) At the beginning of the game the organisers place $4$ piles of paper disks onto the table. The player who is in turn takes away a pile, then divides one of the remaining piles into two nonempty piles. Whoever is unable to move, loses. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

Two players play the following game. The first player starts by writing either $0$ or $1$ and then, on his every move, chooses either $0$ or $1$ and writes it to the right of the existing digits until there are $1999$ digits. Each time the first player puts down a digit (except the first one) , the second player chooses two digits among those already written and swaps them. Can the second player guarantee that after his last move the line of digits will be symmetrical about the middle digit? (I Izmestiev)

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]

There are two bowls on the table, in one there are $p$, in the other $q$ stones ($p, q \in N*$ ). Two players $A$ and $B$ take turns playing, starting with $A$. Who's turn: $\bullet$ takes a stone from one of the bowls $\bullet$or removes one stone from each bowl $\bullet$ or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can $A$ and under what conditions can $B$ force the win? The answer must be justified.