A blackboard contains $2018$ instances of the digit $1$ separated by spaces. Georg and his mother play a game where they take turns filling in one of the spaces between the digits with either a $+$ or a $\times$. Georg begins, and the game ends when all spaces have been filled. Georg wins if the value of the expression is even, and his mother wins if it is odd. Which player may prepare a strategy which secures him/her victory?

Let $m$ be a positive integer. Two players, Axel and Elina play the game HAUKKU ($m$) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$. PS. As member Loppukilpailija noted, it should be written $m>1$, as the statement does not hold for $m = 1$.

Peter plays a solitaire game with a deck of cards, some of which are face-up while the others are face-down. Peter loses if all the cards are face-down. As long as at least one card is face up, Peter must choose a stack of consecutive cards from the deck, so that the top and the bottom cards of the stack are face-up. They may be the same card. Then Peter turns the whole stack over and puts it back into the deck in exactly the same place as before. Prove that Peter always loses. (A Shapovalov)

On the table there are $2016$ coins. Two players play the following game making alternating moves. In one move it is allowed to take $1, 2$ or $3$ coins. The player who takes the last coin wins. Which player has a winning strategy?

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

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.

A circular game board is divided into $n \ge 3$ sectors. Each sector is either empty or occupied by a marker. In each step one chooses an occupied sector, removes its marker and then switches each of the two adjacent sectors from occupied to empty or vice-versa. Starting with a single occupied sector, for which $n$ is it possible to end up with all empty sectors after finitely many steps?

(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]

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

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?

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)

Anna and Bob, with Anna starting first, alternately color the integers of the set $S = \{1, 2, ..., 2022 \}$ red or blue. At their turn each one can color any uncolored number of $S$ they wish with any color they wish. The game ends when all numbers of $S$ get colored. Let $N$ be the number of pairs $(a, b)$, where $a$ and $b$ are elements of $S$, such that $a$, $b$ have the same color, and $b - a = 3$. Anna wishes to maximize $N$. What is the maximum value of $N$ that she can achieve regardless of how Bob plays?

Let $n>2$ be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked. [asy]unitsize(.25cm); real s3=1.73205081; pair[] points={(-4,4*s3),(-2,4*s3),(0,4*s3),(2,4*s3),(4,4*s3),(-5,3*s3), (-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (5,3*s3), (-6,2*s3),(-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3),(6,2*s3),(-7,s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3),(7,s3),(-8,0), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (8,0),(-7,-s3),(-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (7,-s3), (-6,-2*s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (6,-2*s3), (-5,-3*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3), (5,-3*s3), (-4,-4*s3), (-2,-4*s3), (0,-4*s3), (2,-4*s3), (4,-4*s3)}; void draw_hexagon(pair p) { draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30))); } {for (int i=0;i<61;++i){draw_hexagon(points[i]);}} label((0,0), "\Large $*$"); [/asy] The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda and Magni take turns moving the stone: Anna begins, then Edda, then Magni, then Anna, and so on. The rules for each player's turn are: [list] [*] Anna has to move the stone to an adjacent tile, in any direction. [*] Edda has to move the stone straight by two tiles in any of the $6$ possible directions. [*] Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the $6$ possible directions. [/list] Find all $n$ for which Edda and Magni have a winning strategy.

There are $c$ red, $p$ blue, and $b$ white balls on a table. Two players $A$ and $B$ play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player $A$ begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of $c,p,b$ does player $A$ have a winning strategy?

Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were: $\textbf{(a)}~2,2,2$; $\textbf{(b)}~3,3,3$?

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]

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

A box contains $100$ tickets. Each ticket has a real number written on it. There are no restrictions on the type of number except that they are all different (they can be integers, rational, positive, negative, irrational, large or small). Of course there is one ticket that has the highest number and that is the winner. The game consists of drawing a ticket at random, looking at it and deciding whether to keep it or not. If we choose to keep him, it is verified if he was the oldest, in which case we win a million pesos (if we don't win, the game is over). If we don't think it's the biggest, we can discard it and draw another one, repeating the process until we like one or we run out of tickets. Going back to choose a previously discarded ticket is prohibited. Find a game strategy that gives at least a $25\%$ chance of winning.

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

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?

Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy. Proposed by [i]Demetres Christofides, Cyprus[/i]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]