Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.

Given two natural numbers $a < b$, Xavier and Ze play the following game. First, Xavier writes $a$ consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has $b$ numbers, with the condition that the sum of the $b$ numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for $a = 4$ and $b = 7$, if Xavier wrote the numbers $3,4,5,6,3,3,4$, Ze could win, separating these numbers into groups $3,3 ,4,4$ and $3,5,6$. For what values of $a$ and $b$ can Xavier guarantee victory?

Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few $2^n$ integers $k_1,..,k_{2^n}$. Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth. After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision. (UK)

$10$ 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)

The main building of ETH Zurich is a rectangle divided into unit squares. Every side of a square is a wall, with certain walls having doors. The outer wall of the main building has no doors. A number of participants of the SMO have gathered in the main building lost. You can only move from one square to another through doors. We have indicates that there is a walkable path between every two squares of the main building. Cyril wants the participants to find each other again by having everyone on the same square leads. To do this, he can give them the following instructions via walkie-talkie: North, East, South or West. After each instruction, each participant simultaneously attempts a square in that direction to go. If there is no door in the corresponding wall, he remains standing. Show that Cyril can reach his goal after a finite number of directions, no matter which one square the participants at the beginning. [hide=original wording]Das Hauptgebäude der ETH Zürich ist ein in Einheitsquadrate unterteiltes Rechteck. Jede Seite eines Quadrates ist eine Wand, wobei gewisse Wände Türen haben. Die Aussenwand des Hauptgebäudes hat keine Türen. Eine Anzahl von Teilnehmern der SMO hat sich im Hauptgebäude verirrt. Sie können sich nur durch Türen von einem Quadrat zum anderen bewegen. Wir nehmen an, dass zwischen je zwei Quadraten des Hauptgebäudes ein begehbarer Weg existiert. Cyril möchte erreichen, dass sich die Teilnehmer wieder nden, indem er alle auf dasselbe Quadrat führt. Dazu kann er ihnen per Walkie-Talkie folgende Anweisungen geben: Nord, Ost, Süd oder West. Nach jeder Anweisung versucht jeder Teilnehmer gleichzeitig, ein Quadrat in diese Richtung zu gehen. Falls in der entsprechenden Wand keine Türe ist, bleibt er stehen. Zeige, dass Cyril sein Ziel nach endlich vielen Anweisungen erreichen kann, egal auf welchen Quadraten sich die Teilnehmer am Anfang benden. [/hide]

Let $n> 1$ be an odd integer. On a square surface have been placed $n^2 - 1$ white slabs and a black slab on the center. Two workers $A$ and $B$ take turns removing them, betting that whoever removes black will lose. First $A$ picks a slab; if it has row number $i \ge (n + 1) / 2$, then it will remove all tiles from rows with number greater than or equal to$ i$, while if $i <(n + 1) / 2$, then it will remove all tiles from the rows with lesser number or equal to $i$. Proceed in a similar way with columns. Then $B$ chooses one of the remaining tiles and repeats the process. Determine who has a winning strategy and describe it. Note: Row and column numbering is ascending from top to bottom and from left to right.

Consider a game played on the integers in the closed interval $[1, n]$. The game begins with some tokens placed in $[1, n]$. At each turn, tokens are added or removed from$ [1, n]$ using the following rule: For each integer $k \in [1, n]$, if exactly one of $k - 1$ and $k + 1$ has a token, place a token at $k$ for the next turn, otherwise leave k blank for the next turn. We call a position [i]static [/i] if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.

The checker is standing on the corner field of a $n\times n$ chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts. a) Prove that for even $n$ the first can always win, and if $n$ is odd, than the second can always win. b) Who wins if the checker stands initially on the neighbour to the corner field?

Ana & Bruno decide to play a game with the following rules.: a) Ana has cards $1, 3, 5,7,..., 2n-1$ b) Bruno has cards $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.

Let $n\geq 3$ be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular $n$-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with $0$ points. In their turn, a player colours a vertex $V$ which has not been coloured and gains $k$ points where $k$ is the number of already coloured neighbouring vertices of $V$. (Thus, $k$ is either $0$, $1$ or $2$.) The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all $n\geq 3$ for which Alice has a winning strategy and all $n\geq 3$ for which Bob has a winning strategy.

Three players $A,B$ and $C$ take turns removing stones from a pile of $N$ stones. They move in the order $A,B,C,A,B,C,…A$. The game begins, and the one who takes out the last stone loses the game. The players $A$ and $C$ team up against $B$ , they agree on a joint strategy. $B$ can take in each play $1,2,3,4$ or $5$ stones, while $A$ and $C$, they can each get $1,2$ or $3$ stones each turn. Determine for what values ​​of $N$ have winning strategy $A$ and $C$, and for what values ​​the winning strategy is from $B$. .

Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it. P.S. Proposed by yours truly :-D

Agustín and Lucas, by turns, each time mark a box that has not yet been marked on a $101\times 101$ grid board. Augustine starts the game. You cannot check a box that already has two checked boxes in its row or column. The one who can't make his move loses. Decide which of the two players has a winning strategy.

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right

Suppose you are playing a game against Daniel. There are $2017$ chips on a table. During your turn, if you can write the number of chips on the table as a sum of two cubes of not necessarily distinct, nonnegative integers, then you win. Otherwise, you can take some number of chips between $1$ and $6$ inclusive off the table. (You may not leave fewer than $0$ chips on the table.) Daniel can also do the same on his turn. You make the first move, and you and Daniel always make the optimal move during turns. Who should win the game? Explain.

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Also compare shortlist 2009, combinatorics problem C1.[/i]

Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations: 1. Choose two non-empty piles and take one stone from each pile. 2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones. The player who cannot perform either of operations 1 and 2 loses. Determine who has the winning strategy based on $x$ and $y$.

Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

There are $100$ diamonds on the pile, $50$ of which are genuine and $50$ false. We invited a peculiar expert who alone can recognize which are which. Every time we show him some three diamonds, he would pick two and tell (truthfully) how many of them are genuine . Decide whether we can surely detect all genuine diamonds regardless how the expert chooses the pairs to be considered.

There are a) $2022$, b) $2023$ plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.