Found problems: 622
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]
The "Sea battle" game.
a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely?
b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.
Agustín and Lucas, by turns, each time mark a box that has not yet been marked on a $101\times 101$ grid board. Augustine starts the game. You cannot check a box that already has two checked boxes in its row or column. The one who can't make his move loses. Decide which of the two players has a winning strategy.
Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.
[b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal.
[b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.
Anna and Orjan play the following game: they start with a positive integer $n>1$, Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?
In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?
Dragos, the early ruler of Moldavia, and Maria the Oracle play the following game. Firstly, Maria chooses a set $S$ of prime numbers. Then Dragos gives an infinite sequence $x_1, x_2, ...$ of distinct positive integers. Then Maria picks a positive integer $M$ and a prime number $p$ from her set $S$. Finally, Dragos picks a positive integer $N$ and the
game ends. Dragos wins if and only if for all integers $n \ge N$ the number $x_n$ is divisible by $p^M$; otherwise, Maria wins. Who has a winning strategy if the set S must be: $\hspace{5px}$a) finite; $\hspace{5px}$b) infinite?
Proposed by [i]Boris Stanković, Bosnia and Herzegovina[/i]
Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it.
What number will be written if both play optimally?
At a quiz show there are three doors. Behind exactly one of the doors, a prize is hidden. You may ask the quizmaster whether the prize is behind the left-hand door. You may also ask whether the prize is behind the right-hand door. You may ask each of these two questions multiple times, in any order that you like. Each time, the quizmaster will answer ‘yes’ or ‘no’. The quizmaster is allowed to lie at most $10$ times. You have to announce in advance how many questions you will be asking (but which questions you will ask may depend on the answers of the quizmaster).
What is the smallest number you can announce, such that you can still determine with absolute certainty the door behind which the prize is hidden?
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:
$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.
The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
[i]Proposed by Amine Natik, Morocco[/i]
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?
You are playing a game called "Hovse."
Initially you have the number $0$ on a blackboard.
If at any moment the number $x$ is written on the board, you can either:
$\bullet$ replace $x$ with $3x + 1$
$\bullet$ replace $x$ with $9x + 1$
$\bullet$ replace $x$ with $27x + 3$
$\bullet$ or replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor $.
However, you are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"
Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations:
1. Choose two non-empty piles and take one stone from each pile.
2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones.
The player who cannot perform either of operations 1 and 2 loses.
Determine who has the winning strategy based on $x$ and $y$.
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
Alma and Bertha play the following game. There are $100$ round, $200$ triangular and $200$ square pieces on a table. In each move a player must remove two pieces, but it cannot be a triangle and a square. Alma starts, and one loses if one is unable to move or if there are no pieces left when it is one’s turn. Which player has a winning strategy?
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?
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
Georg writes a positive integer $a$ on a blackboard. As long as there is a number on the blackboard, he does the following each day:
$\bullet$ If the last digit in the number on the blackboard is less than or equal to $5$, he erases that last digit.
(If there is only this digit, the blackboard thus becomes empty.)
$\bullet$ Otherwise he erases the entire number and writes $9$ times the number.
Can Georg choose $a$ in such a way that the blackboard never becomes empty?
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]
Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers.
One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins).
In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice.
[i]Nikolai Beluhov[/i]
Kolya and Dima play a game on an $8\times 8$ board, making moves in turn. During his turn, Kolya must put one cross in any empty cell (i.e., in a cell in which a cross has not yet been drawn and which has not yet been covered with a domino). Dima must cover two adjacent cells with a domino (which are not yet covered with other dominoes), in which there are an even number of crosses in total (0 or 2). The one who can't make a move loses. Which of does the player have a winning strategy, if
[list=a]
[*]Dima makes the first move?
[*]Kolya makes the first move?
[/list]
[i]Proposed by M. Didin[/i]
The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$. To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end.
(a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$, be determined?
(b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$, be determined?
(Theresia Eisenkölbl, Clemens Heuberger)
Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses.
An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise.
Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.
The natural numbers $1$ through $2003$ are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is $k$, the order of the first $k$ terms is reversed. Prove that after several operations number $1$ will occur on the first place.