Found problems: 622
Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this?
[i]Andrey Arzhantsev[/i]
The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?
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?
Let $n$ be a positive integer. Lavi Dopes has two boards $n \times n$. On the first board, he writes an integer in each of his $n^2$ squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if $n = 3$ and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this.
Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board.
(a) If $n = 4$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
(b) If $n = 5$, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
There are two numbers on a board, $1/2009$ and $1/2008$. Alex and Ben play the following game. At each move, Alex names a number $x$ (of his choice), while Ben responds by increasing one of the numbers on the board (of his choice) by $x$. Alex wins if at some moment one of the numbers on the board becomes $1$. Can Alex win (no matter how Ben plays)?
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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Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?
Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions:
• \( f(0) = 0 \);
• \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \).
For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian.
For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type:
\[
\text{How many integers } a \text{ are there such that } f(a) = b?
\]
given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).
The maze is an $8 \times 8 $square, each cell contains $1 \times 1$ which has one of four arrows drawn (up, down, right, left). The upper side of the upper right cell is the exit from the maze.In the lower left cell there is a chip that, with each move, moves one square in the direction indicated by the arrow. After each move, the shooter in the cell in which there was just a chip rotates $90^o$ clockwise. If a chip must move, taking it outside the $8 \times 8$ square, it remains in place, and the arrow also rotates $90^o$ clockwise. Prove that sooner or later, the chip will come out of the maze.
Three legendary knights are fighting against a multiheaded dragon.
Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads?
Alexey Zaslavsky
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
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).
During a fight, each of the $2001$ roosters has torn out exactly one feather of another rooster, and each rooster has lost a feather. It turned out that among any three roosters there is one who hasn’t torn out a feather from any of the other two roosters. Find the smallest $k$ with the following property: It is always possible to kill $k$ roosters and place the rest into two henhouses in such a way that no two roosters, one of which has torn out a feather from the other one, stay in the same henhouse.
Give and Take divide $100$ coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has $9$ handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does?
(A Shapovalov)
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;
Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.
[i]Proposed by Hans Zantema, Netherlands[/i]
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 $m = 1$.
$1992$ vectors are given in the plane. Two players pick unpicked vectors alternately. The winner is the one whose vectors sum to a vector with larger magnitude (or they draw if the magnitudes are the same). Can the first player always avoid losing?
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]
Let $\alpha, \beta$ be two rational numbers strictly between 0 and 1. Alice and Bob play a game. At the start of the game, Alice chooses a positive integer $n$. Knowing that, Bob then chooses a positive integer $T$. They then do the following for $T$ rounds: at the $i$th round, Bob chooses a set $X_i$ of $n$ positive integers that form a complete residue system modulo $n$. Then Alice chooses a subset $Y_i$ of $X_i$ such that the sum of elements in $Y_i$ is at most $\alpha$ times the sum of elements in $X_i$. After the $T$ rounds, Alice wins if it is possible to pick an integer $s$ between 0 and $n-1$ such that there are at least $\beta T$ positive integers among the elements in $Y_1, Y_2, . . . , Y_T$ (counted with multiplicities) that are equal to $s \pmod n$, and Bob wins otherwise.
Find all pairs $(\alpha, \beta)$ of rational numbers strictly between 0 and 1 such that Alice has a winning strategy.
[i]Proposed by Hans[/i]