Two players take turns alternatively and remove a number from $1,2,\dots,1000$. Players can not remove a number that differ with a number already removed by $1$ also they can not remove a number such that it sums up with another removed number to $1001$. The player who can not move loses. Determine the winner.

Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.

Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

We have 11 boxes. On a move, we can choose 10 of them and put one ball in each of the boxes chosen. Two players move alternately. The one who gets a box of 21 balls wins. Which of the two players has winning strategy?

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

Let $S=\{13, 133, \cdots\}$ be the set of the positive integers of the form $133 \cdots 3$. Consider a horizontal row of $2022$ cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a $2022$-digit number $N$. Borja wins if $N$ is divisible by a number in $S$, otherwise Ana wins. Find which player has a winning strategy and describe it.

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: [center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center] [center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center] Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy. [i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?

In a one-player game, you have three cards. At the beginning, a nonnegative integer is written on each of the cards, and the sum of these three integers is $2006$. At each step, you can select two of the three chards, subtract $1$ from the integer written on each of these two cards - as long as the resulting integers are still nonnegative -, and add $1$ to the integer written on the third card. You play this game until you can’t perform a step anymore because two of the cards have $0$’s written on them. Assume that, at this moment, the third card has a $1$ written on it. Prove that I can tell you which card contains the $1$ without knowing how exactly you proceeded in your game, but only knowing the starting configuration (i. e., the numbers written on the cards at the beginning of the game) and the fact that at the end, you were left with two $0$’s and a $1$.

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. 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)

An integer $n > 1$ is given. Two players in turns mark points on a circle. First Player uses red color while Second Player uses blue color. The game is over when each player marks $n$ points. Then each player nds the arc of maximal length with ends of his color, which does not contain any other marked points. A player wins if his arc is longer (if the lengths are equal, or both players have no such arcs, the game ends in a draw). Which player has a winning strategy?

There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?