Found problems: 117
Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have 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) $3 \times 4$
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)
Axel and Berta play the following games: On a board are a number of positive integers. One move consists of a player exchanging a number $x$ on the board for two positive integers y and $z$ (not necessarily different), such that $y + z = x$. The game ends when the numbers on the board are relatively coprime in pairs. The player who made the last move has then lost the game. At the beginning of the game, only the number $2015$ is on the board. The two players make do their moves in turn and Berta begins. One of the players has a winning strategy. Who, and why?
Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?
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.
$2019$ coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from $ 1$ to $99$, the second player in one move can take an even number of coins from $2$ to $100$. The player who can not make a move is lost. Who has the winning strategy in this game?
Josie and Ross are playing a game on a $20 \times 20$ chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie’s turn, she selects any two different empty cells, and places one white stone in each of them. On Ross’ turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are $ 8$ consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?
Arthur and Renate play a game on a $7 \times 7$ board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?
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 $B$.
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Two people, $A$ and $B$, play the game of blowing up a balloon. The balloon will explode only when the volume of the balloon $V>2014$ mL. $A$ blows in $1$ mL first, and then they takes turns blowing. It is agreed that the gas blown by each person must not be less than the gas blown by the other party last time and should not be more than twice the amount of gas the other party blew last time. The agreement is that the person who blows up the balloon loses. Who has a winning strategy ? Briefly explain it. (Do not consider the change in volume caused by the change in tension when the balloon is inflated).
Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on.
The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?
The archipelago Barrantes - $n$ is a group of islands connected by bridges as follows: there are a main island (Humberto), in the first step I place an island below Humberto and one above from Humberto and I connect these 2 islands to Humberto. I put $2$ islands to the left of these $2$ new islands and I connect them with a bridge to the island that they have on their right. In the second step I take the last $2$ islands and I apply the same process that I applied to Humberto. In the third step I apply the same process to the $4$ new islands. We repeat this step n times we reflect the archipelago that we have on a vertical line to the right of Humberto. We connect Humberto with his reflection and so we have the archipelago Barrantes -$n$. However, the archipelago Barrantes -$n$ exists on a small planet cylindrical, so that the islands to the left of the archipelago are in fact the islands that are connected to the islands on the right. The figure shows the Barrantes archipelago -$2$, The islands at the edges are still numbered to show how the archipelago connects around the cylindrical world, the island numbered $1$ on the left is the same as the island numbered $1$ on the right.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/803d95ce742c2739729fdb4d74af59d4d0652f.png[/img]
One day two bands of pirates arrive at the archipelago Barrantes - $n$: The pirates Black Beard and the Straw Hat Pirates. Blackbeard proposes a game to Straw Hat: The first player conquers an island, the next player must conquer an island connected to the island that was conquered in the previous turn (clearly not conquered on a previous shift). The one who cannot conquer any island in his turn loses. Straw Hat decides to give the first turn to Blackbeard. Prove that Straw Hat has a winning strategy for every $n$.
Let $m$ be a positive integer.
Two players, Axel and Elina play the game HAUKKU ($m$) proceeds as follows:
Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$.
PS. As member Loppukilpailija noted, it should be written $m>1$, as the statement does not hold for $m = 1$.
Alicia and Bob take turns writing words on a blackboard.
The rules are as follows:
a) Any word that has been written cannot be rewritten.
b) A player can only write a permutation of the previous word, or can simply simply remove one letter (whatever you want) from the previous word.
c) The first person who cannot write another word loses.
If Alice starts by typing the word ''Olympics" and Bob's next turn, who, do you think, has a winning strategy and what is it?
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.
On a board there are written the integers from $1$ to $119$. Two players, $A$ and $B$, make a move by turn. A $move$ consists in erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player $A$ makes the first move. Find the highest integer $k$, such that the player $A$ can be sure that his score is not smaller than $k$.
There is a row with $2024$ cells. Ana and Beto take turns playing, with Ana going first. On each turn, the player selects an empty cell and places a digit in that space. Once all $2024$ cells are filled, the number obtained from reading left to right is considered, ignoring any leading zeros. Beto wins if the resulting number is a multiple of $99$, otherwise Ana wins. Determine which of the two players has a winning strategy and describe it.
John and Mary play the following game. First they choose integers $n > m > 0$ and put $n$ sweets on an empty table. Then they start to make moves alternately. A move consists of choosing a nonnegative integer $k\le m$ and taking $k$ sweets away from the table (if $k = 0$ , nothing happens in fact). In doing so no value for $k$ can be chosen more than once (by none of the players) or can be greater than the number of sweets at the table at the moment of choice. The game is over when one of the players can make no more moves.
John and Mary decided that at the beginning Mary chooses the numbers $m$ and $n$ and then John determines whether the performer of the last move wins or looses. Can Mary choose $m$ and $n$ in such way that independently of John’s decision she will be able to win?
John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy?
[i](6 points)[/i]
Ana and Bogdan play the following turn based game: Ana starts with a pile of $n$ ($n \ge 3$) stones. At his turn each player has to split one pile. The winner is the player who can make at his turn all the piles to have at most two stones. Depending on $n$, determine which player has a winning strategy.
A box contains $100$ tickets. Each ticket has a real number written on it. There are no restrictions on the type of number except that they are all different (they can be integers, rational, positive, negative, irrational, large or small). Of course there is one ticket that has the highest number and that is the winner.
The game consists of drawing a ticket at random, looking at it and deciding whether to keep it or not. If we choose to keep him, it is verified if he was the oldest, in which case we win a million pesos (if we don't win, the game is over). If we don't think it's the biggest, we can discard it and draw another one, repeating the process until we like one or we run out of tickets. Going back to choose a previously discarded ticket is prohibited.
Find a game strategy that gives at least a $25\%$ chance of winning.
Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below:
[img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img]
Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses.
(a) Show that, even if Chris and Michael play cooperatively, the game will come to an end.
(b) Which player has a winning strategy?
Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$.
Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.
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.