Found problems: 1385
Given a positive integer $n$, all of its positive integer divisors are written on a board. Two players $A$ and $B$ play the following game:
Each turn, each player colors one of these divisors either red or blue. They may choose whichever color they wish, but they may only color numbers that have not been colored before. The game ends once every number has been colored. $A$ wins if the product of all of the red numbers is a perfect square, or if no number has been colored red, $B$ wins otherwise. If $A$ goes first, determine who has a winning strategy for each $n$.
Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if:
(a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$;
(b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?
On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game.
Alice starts and they alternately play the following moves:
$\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$
$\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$.
Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps.
Prove that Alice has a winning strategy.
(Daniel Holmes)
There are $n$ cards. Max and Lewis play, alternately, the following game
Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.
Let \( n \geq 2 \) be an integer. Two players, Alice and Bob, play the following game on the complete graph \( K_n \): They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.
Prove that there exists a positive number \(\varepsilon\), Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding
\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]
A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
Given two natural numbers $a < b$, Xavier and Ze play the following game. First, Xavier writes $a$ consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has $b$ numbers, with the condition that the sum of the $b$ numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for $a = 4$ and $b = 7$, if Xavier wrote the numbers $3,4,5,6,3,3,4$, Ze could win, separating these numbers into groups $3,3 ,4,4$ and $3,5,6$. For what values of $a$ and $b$ can Xavier guarantee victory?
Let $n$ be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him.
After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when:
(a) $n=2013^{2012}$
(b) $n=2013^{2013}$
A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.
Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn.
During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses.
Determine which player can always win this game.
Given a board $3\times3$ and $9$ cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.
Given $2012$ stones divided into several groups, a [i]legal move[/i] is to merge two of the groups into one, as long as the size of the new group is less than or equal to $51$. Two players, $A$ and $B$, take turns making legal moves, starting with $A$. Initially, each stone is in a separate group. The player who cannot make a legal move on their turn loses.
Determine which of the two players has a winning strategy and provide that strategy.
Two players, \(A\) (first player) and \(B\), take alternate turns in playing a game using 2016 chips as follows: [i]the player whose turn it is, must remove \(s\) chips from the remaining pile of chips, where \(s \in \{ 2,4,5 \}\)[/i]. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).
Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk.
Starting from A, they take turns and do one of the following:
(1) Pick a bottle with at least $100$ml of milk, and drink half of it.
(2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle.
Whoever cannot do any operations loses the game. Who has a winning strategy?
[i]
Proposed by Chu-Lan Kao and usjl[/i]
Alice and Bob are playing the following game: They have an $8\times8$ chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black. Or he does nothing. After that, Alice may decide to continue the game or not. The goal of Alice is to maximize the number of connected components of black grids, on the other hand, Bob wants to minimize that number. If both of them are extremely smart, how many connected components will be in the end of the game?
Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules:
a) They move alternately in turns.
b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously.
c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step.
d) Whoever reaches the last step (listed with $2022$) wins.
Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.
[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$.
[b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer.
[b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy.
[b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers.
[b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$.
[b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$
where $m|496$ means $496$ is divisible by $m$.
[b]p7.[/b] What is the value of
$${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$
[b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Ana and Bob are playing the following game.
[list]
[*] First, Bob draws triangle $ABC$ and a point $P$ inside it.
[*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$.
[*] Finally, Ana draw a triangle $V_1V_2V_3$.
[/list]
For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$ to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.
$n$ players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of $k(\le n)$ players, if it is able to line the players up so that each player won every player at the back, we call the set [i]ranked[/i]. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.
Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were:
$\textbf{(a)}~2,2,2$;
$\textbf{(b)}~3,3,3$?
Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.
The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner.
Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.
A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$.
Anna and Bob play a game on this grid.
Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first.
1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line.
2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted.
As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins.
Does Anna have a winning strategy?
[i]Proposed by Demetres Christofides, Cyprus[/i]
On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$. Luis erases $100$ of these numbers, then Martin erases another $100$. Martin wins if the sum of the $200$ erased numbers equals the sum of those not deleted; otherwise, he wins Luis. Which of the two has a winning strategy? What if Luis deletes $101$ numbers and Martín deletes $99$?
In each case, explain how the player with the winning strategy can ensure victory.