Found problems: 70
Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.
[b]a)[/b] Prove that Andrey can make moves in such a way that Sasha will never win.
[b]b)[/b] Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?
[i](N. Naradzetski)[/i]
A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal
a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough?
b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough?
c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?
Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.
Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game:
Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\).
If someone can't play, he loses.
We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player.
Else, we call it [i]loser[/i].
Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)
Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule:
First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.
This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.
How many gram of gold can Smilla make sure to win?
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy?
[I]Proposed by A. Khrabrov[/i]
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$. A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by $2014$, divide by $2014$. The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$, $a\ge 2015$ such that all numbers of the form $an+b$ will not be the starting integer, where $n$ is any positive integer.
Find the minimum number of starting integer where Navi wins.
Juty and Azgor plays the following game on a \((2n+1) \times (2n+1)\) board with Juty moving first. Initially all cells are colored white. On Juty's turn, she colors a white cell green and on Azgor's turn, he colors a white cell red. The game ends after they color all the cells of the board. Juty wins if all the green cells are connected, i.e. given any two green cells, there is at least one chain of neighbouring green cells connecting them (we call two cells [i]neighboring[/i] if they share at least one corner), otherwise Azgor wins. Determine which player has a winning strategy.
[i]Proposed by Atonu Roy Chowdhury[/i]
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?
Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
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])
Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:
[b]Rule 1:[/b] Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise.
[b]Rule 2:[/b] Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed.
Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.
A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?
Peter wrote $100$ distinct integers on a board. Basil needs to fill the cells of a table $100\times{100}$ with integers so that the sum in each rectangle $1\times{3}$ (either vertical, or horizontal) is equal to one of the numbers written on the board. Find the greatest $n$ such that, regardless of numbers written by Peter, Basil can fill the table so that it would contain each of numbers $(1,2,...,n)$ at least once (and possibly some other integers).
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.
Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle.
Determine who wins each of the following games:
(a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$.
(b) The starting rectangles are $100 \times 100$ and $100 \times 500$.
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?
Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A [i]step[/i] is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it.
[b]a)[/b] Show that, for any $ m,n\ge 5, $ there exists a succession of [i]steps[/i] that make the rectagle to be single-colored.
[b]b)[/b] What about $ m=n+1=5? $
A [i]Nim-style game[/i] is defined as follows. Two positive integers $k$ and $n$ are specified, along with a finite set $S$ of $k$-tuples of integers (not necessarily positive). At the start of the game, the $k$-tuple $(n, 0, 0, ..., 0)$ is written on the blackboard.
A legal move consists of erasing the tuple $(a_1,a_2,...,a_k)$ which is written on the blackboard and replacing it with $(a_1+b_1, a_2+b_2, ..., a_k+b_k)$, where $(b_1, b_2, ..., b_k)$ is an element of the set $S$. Two players take turns making legal moves, and the first to write a negative integer loses. In the event that neither player is ever forced to write a negative integer, the game is a draw.
Prove that there is a choice of $k$ and $S$ with the following property: the first player has a winning strategy if $n$ is a power of 2, and otherwise the second player has a winning strategy.
[i]Proposed by Linus Hamilton[/i]
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?
There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.
[i]Czech Republic[/i]