Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.) [i]Proposed by Milan Haiman[/i]

The vertices of the regular $n$-gon are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center of the $n$-gon. The winner is a player if after his move it is possible to get any vertex from any other vertex moving along segments. For each integer $n\geqslant 3$ determine who has a winning strategy.

Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.

Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are $20$ cookies on a red plate and $14$ on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.

A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.

Ana and Beto play on a grid of $2022 \times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?

Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$. Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$s are pairwise different $(i=1,2,\cdots ,6)$. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy? [I]Proposed by A. Khrabrov[/i]

$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$. Wins the one who makes the bag empty. For each value of $n$, find who has the winning strategy.

Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon $\mathcal{S}$ and then Nicole can shift $\mathcal{S}$ to wherever she wants. Johann wins if there exists a point with coordinates $(x, y)$ in the interior of $\mathcal{S}$, where $x$ and $y$ are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.

A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.

There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.

There are $50$ coins in a row; each coin has a value. Two people are playing a game alternating moves. In one move a player can take either the leftmost or the rightmost coin. Who can always accumulate coins whose total value is at least the value of the coins of the opponent?

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$, in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it : 1 -either with a positive divisor k different from 1 and k 2- or with k+1 At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?

A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.

Alice and Bob play the following game. They write the expressions $x + y$, $x - y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables $x$ and $y$, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable. Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result, wins. Which player, if any, has a winning strategy?