Found problems: 1385
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?
Pasha and Vova play the game crossing out the cells of the $3\times 101$ board by turns. At the start, the central cell is crossed out. By one move the player chooses the diagonal (there can be $1, 2$ or $3$ cells in the diagonal) and crosses out cells of this diagonal which are still uncrossed. At least one new cell must be crossed out by any player's move. Pasha begins, the one who can not make any move loses. Who has a winning strategy?
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.
Let $(G, *)$ a group of $n > 1$ elements, and let $g \in G$ be an element distinct from the identity.
Ana and Bob play with the group $G$ on the following way:
Starting with Ana and playing alternately, each player selects an element of $G$ that has not been selected before, until each element of $G$ have been selected or a player have selected the elements $a$ and $a *
g$ for some $a \in G$.
In that case it is said that the player loses and his opponent wins.
$a)$ If $n$ is odd, show that, independent of element $g$, one of the two players has
a winning strategy and determines which player
possesses such a strategy.
$b)$ If $n$ is even, show that there exists an element $g \in G$ for which none of the players
has a winning strategy.
Note: A group $(G, *)$ es a set $G$ together with a binary operation $* : G\times G \to G$ that satisfy the following properties
$(i)$ $*$ is asociative: $\forall a, b, c \in G (a * b) * c = a * (b * c)$;
$(ii)$ there exists an identity element $e \in G$ such that $\forall a \in G, a *e = e * a = a;$
$(iii)$ there exists inverse elements: $\forall a \in G \exists a^{-1} \in G$ such that $a*a^{-1} = a^{-1}
*a = e.$
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?
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".
[b](Condition)[/b] For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$.
It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.
In the game of Ric-Rac-Roe, two players take turns coloring squares of a $3 \times 3$ grid in their color; a player wins if they complete a row or column of their color on their turn. If Alice and Bob play this game, picking an uncolored square uniformly at random on their turn, what is the probability that they tie?
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 $m,n$ 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
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?
Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?
Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
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]
On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules:
$-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$;
$-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$;
$-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner.
Find which player has a winning strategy in each of the following cases:
a) $n_0=120$;
b) $n_0=\dsp \frac {3^{2002}-1}2$;
c) $n_0=\dsp \frac{3^{2002}+1}2$.
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]