Alice and Bob are playing game on an $n \times n$ grid. Alice goes first, and they take turns drawing a black point from the coordinate set \[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\] There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers $n$ such that Alice has a winning strategy. [i]Proposed by chengbilly[/i]

Alex the Kat and Kelvin the Frog play a game on a complete graph with $n$ vertices. Kelvin goes first, and the players take turns selecting either a single edge to remove from the graph, or a single vertex to remove from the graph. Removing a vertex also removes all edges incident to that vertex. The player who removes the final vertex wins the game. Assuming both players play perfectly, for which positive integers $n$ does Kelvin have a winning strategy?

On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. [i]Note: The order of the numbers of the list is not important.[/i]

Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$, in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$, there exists a box containing $n$ ballots. For each $n$, find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?

There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?

Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.

Numbers $1,2,3...,100$ are written on a board. $A$ and $B$ plays the following game: They take turns choosing a number from the board and deleting them. $A$ starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?

There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens? [i](5 points)[/i]

Let $S$ the set of natural numbers from $1$ up to $1001$ , $S=\{1,2,...,1001\}$. Lisandro thinks of a number $N$ of $S$ , and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of $S$, Lisandro reads it and tells Carla how many subsets of her list contain $N$ . If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than $3$ are allowed. What is the smallest total number of subsets that allow Carla to find $N$ for sure?

Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii)The first person who cannot spell an incantation loses the contest. Answer the following questions: a) If A says '$STAGEPREIMO$' first, then who will win? b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.

A chocolate bar has five lengthwise dents and eight crosswise ones, which can be used to break up the bar into sections (one can get a total of $ 9 \times 6 = 54$ cells). Two players play the following game with such a bar. At each move (the two players move alternatively) one player breaks off a section of width one from the bar along a single dent and eats it, the other player does the same with what’s left of the bar, and so on. When one of the players breaks up a section of width two into two strips of width one, he eats one of the strips and the other player eats the other strip. Prove that the player who has the first move can play so as to eat at least $6$ cells more than his opponent (no matter how his opponent plays). (R Fedorov)

There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given: \\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$) \\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$). \\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?

Ana and Beto play on a blackboard where all integers from 1 to 2024 (inclusive) are written. On each turn, Ana chooses three numbers $a,b,c$ written on the board and then Beto erases them and writes one of the following numbers: $$a+b-c, a-b+c, ~\text{or}~ -a+b+c.$$ The game ends when only two numbers are left on the board and Ana cannot play. If the sum of the final numbers is a multiple of 3, Beto wins. Otherwise, Ana wins. ¿Who has a winning strategy?