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]

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)

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

Given a (simple) graph $G$ with $n \geq 2$ vertices $v_1, v_2, \dots, v_n$ and $m \geq 1$ edges, Joël and Robert play the following game with $m$ coins: [list=i] [*]Joël first assigns to each vertex $v_i$ a non-negative integer $w_i$ such that $w_1+\cdots+w_n=m$. [*]Robert then chooses a (possibly empty) subset of edges, and for each edge chosen he places a coin on exactly one of its two endpoints, and then removes that edge from the graph. When he is done, the amount of coins on each vertex $v_i$ should not be greater than $w_i$. [*]Joël then does the same for all the remaining edges. [*]Joël wins if the number of coins on each vertex $v_i$ is equal to $w_i$. [/list] Determine all graphs $G$ for which Joël has a winning strategy.

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 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.

Tao plays the following game:given a constant $v>1$;for any positive integer $m$,the time between the $m^{th}$ round and the $(m+1)^{th}$ round of the game is $2^{-m}$ seconds;Tao chooses a circular safe area whose radius is $2^{-m+1}$ (with the border,and the choosing time won't be calculated) on the plane in the $m^{th}$ round;the chosen circular safe area in each round will keep its center fixed,and its radius will decrease at the speed $v$ in the rest of the time(if the radius decreases to $0$,erase the circular safe area);if it's possible to choose a circular safe area inside the union of the rest safe areas sometime before the $100^{th}$ round(including the $100^{th}$ round),then Tao wins the game.If Tao has a winning strategy,find the minimum value of $\biggl\lfloor\frac{1}{v-1}\biggr\rfloor$.

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Jose and Maria play the following game: Maria writes $2019$ positive integers different on the blackboard. Jose deletes some of them (possibly none, but not all) and write to the left of each of the remaining numbers a sign $+$or a sign $-$. Then the sum written on the board is calculated. If the result is a multiple of $2019$, Jose wins the game, if not, Maria wins. Determine which of the two has a winning strategy.

Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is: i) $3 \times 3$ ii) $3 \times 4$

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 Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if : $a)$ $k=9$? $b)$ $k=8$?

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game. Which player has a winning strategy? [i]Submitted by András Imolay, Budapest[/i]

Let $n\ge 3$ be an integer. Lucas and Matías play a game in a regular $n$-sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice. a) Determine all the $n\ge 3$ such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías. b) Determine all the $n\ge 3$ such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas. Note. The two players know the value of $n$ and see the polygon.

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$. At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$. The two players take turns, with Ariane starting. Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently. Depending on $n$, determine which of the two has a winning strategy.

Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game: (i) Alberto chooses a positive integer, (ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself), (iii) Alberto increases or decreases the Barbara’s number by $1$. Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?

A game is played with two heaps of $p$ and $q$ stones. Two players alternate playing, with $A$ starting. A player in turn takes away one heap and divides the other heap into two smaller ones. A player who cannot perform a legal move loses the game. For which values of $p$ and $q$ can $A$ force a victory?