Found problems: 83
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
[i](Anant Mudgal)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.
There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ($A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?
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]
A magician has $300$ cards with numbers from $1$ to $300$ written on them, each number on exactly one card. The magician then lays these cards on a $3 \times 100$ rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over $k$ cards of his choice. What is the smallest value of $k$ for which it can happen that the opened cards definitely determine the exact positions of all other cards?
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.
General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!).
To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns.
Both of them must give a command and execute it in their turn.
Only two commands are possible: The command "[i]Divide![/i]" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier.
On the other hand, the command "[i]Rule![/i]" removes exactly one tin soldier from each company.
The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command.
a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each?
b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?
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]
Let $n> 1$ be a positive integer. Danielle chooses a number $N$ of $n$ digits but does not tell her students and they must find the sum of the digits of $N$. To achieve this, each student chooses and says once a number of $n$ digits to Danielle and she tells how many digits are in the correct location compared with $N$. Find the minimum number of students that must be in the class to ensure that students have a strategy to correctly find the sum of the digits of $N$ in any case and show a strategy in that case.
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).
At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?
[i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]
Anselmo and Claudio are playing alternatively a game with fruits in a box. The box initially has $32$ fruits. Anselmo plays first and each turn consists of taking away $1$, $2$ or $3$ fruits from the box or taking away $\frac{2}{3}$ of the fruits from the box (this is only possible when the number of the fruits left in the box is a multiple of $3$). The player that takes away the last fruit from the box wins. Which of these two players has a winning strategy? How should that player play in order to win?
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
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?
Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.
At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.
We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:
$(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.
$(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right.
$\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves.
$\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many 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?
Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks
picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors.
A move consists of an operation of one of the following three forms:
[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take
place, over all possible initial configurations.
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).
At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?
[i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]
There are $2^{2018}$ positions on a circle numbered from $1$ to $2^{2018}$ in a clockwise manner. Initially, two white marbles are placed at positions $2018$ and $2019$. Before the game starts, Ping chooses to place either a black marble or a white marble at each remaining position. At the start of the game, Ping is given an integer $n$ ($0\leq n\leq 2018$) and two marbles, one black and one white. He will then move around the circle, starting at position $2n$ and moving clockwise by $2n$ positions at a time. At the starting position and each position he reaches, Ping must switch the marble at that position with a marble of the other color he carries. If he cannot do so at any position, he loses the game. Is there a way to place the $2^{2018}-2$ remaining marbles so that Ping will never lose the game regardless of the number $n$ and the number of rounds he moves around the circle?
In the Jordan Building (the Olympiad building of High School Mandegar Alborz), Ali and Khosro are playing a game. First, Ali selects 2025 points on the plane such that no three points are collinear and no four points are concyclic. Then, Khosro selects a point, followed by Ali selecting another point, and then Khosro selects one more point. The circumcircle of these three points is drawn, and the number of points inside the circle is denoted by \( t \). If Khosro's goal is to maximize \( t \) and Ali's goal is to minimize \( t \), and both play optimally, determine the value of \( t \).
Proposed by Reza Tahernejad Karizi
Two students $A$ and $B$ play a game on a $20 \text{ x } 20$ chessboard. It is known that two squares are said to be [i]adjacent[/i] if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that $A$ will be the first one to move the chess piece, $A$ and $B$ will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among $A$ and $B$ has a winning strategy? Justify your claim.
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.
Let $N$ a positive integer.
In a spaceship there are $2 \cdot N$ people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows:
1) The first alien chooses any person as she wishes.
2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn.
3) The alien that can't play in her turn loses.
Prove that second player has a winning strategy [i]if, and only if[/i], the $2 \cdot N$ people can be divided in $N$ pairs in such a way that two people in the same pair are friends.
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 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.