Found problems: 304
Axel and Berta play the following games: On a board are a number of positive integers. One move consists of a player exchanging a number $x$ on the board for two positive integers y and $z$ (not necessarily different), such that $y + z = x$. The game ends when the numbers on the board are relatively coprime in pairs. The player who made the last move has then lost the game. At the beginning of the game, only the number $2015$ is on the board. The two players make do their moves in turn and Berta begins. One of the players has a winning strategy. Who, and why?
Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?
Among a group of programmers, every two either know each other or do not know each other. Eleven of them are geniuses. Two companies hire them one at a time, alternately, and may not hire someone already hired by the other company. There are no conditions on which programmer a company may hire in the fi rst round. Thereafter, a company may only hire a programmer who knows another programmer already hired by that company. Is it possible for the company which hires second to hire ten of the geniuses, no matter what the hiring strategy of the other company may be?
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?
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?
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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Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.
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?
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)
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?
There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it.
P.S. Proposed by yours truly :-D
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game :
each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?
A game for two.
One gives a digit and the second substitutes it instead of a star in the following difference:
$$**** - **** = $$
Then the first gives the next digit, and so on $8$ times.
The first wants to obtain the greatest possible difference, the second -- the least. Prove that:
1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour.
2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.
A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?
The box contains a complete set of dominoes. Two players take turns choosing one dice from the box and placing them on the table, applying them to the already laid out chain on either of the two sides according to the rules of domino. The one who cannot make his next move loses. Who will win if they both played correctly?
(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win?
(b) What is the answer to this question if the second player is permitted to colour two squares at once?
(DG Azov)
PS. (a) for Juniors, (a),(b) for Seniors
There are $100$ diamonds on the pile, $50$ of which are genuine and $50$ false. We invited a peculiar expert who alone can recognize which are which. Every time we show him some three diamonds, he would pick two and tell (truthfully) how many of them are genuine . Decide whether we can surely detect all genuine diamonds regardless how the expert chooses the pairs to be considered.
Bob and Cob are playing a game on an infinite grid of hexagons. On Bob's turn, he chooses one hexagon that has not yet been chosen, and draws a segment from the center of the hexagon to the midpoints of three of its sides. On Cob's turn, he erases one of Bob's edges made on the previous turn. Bob wins if his edges form a closed loop. Can Bob guarantee to win in a finite amount of time? (Note that Bob may win before Cob can play his next turn.)
Proposed by [i]Jonathan He[/i]
On an infi nite grid, a square with four vertices lie at $(m, n)$, $(m-1, n)$, $(m,n-1)$, $(m-1, n-1)$ is denoted as cell $(m,n)$ $(m, n \in Z)$. Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways:
i) Remove one marble from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m - 1, n- 2)$ and cell $(m -2, n - 1)$.
ii) Remove two marbles from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m +1, n - 2)$ and cell $(m - 2, n +1)$.
Assume that initially, there are $n$ marbles at the cell $(1,n), (2,n - 1),..., (n, 1)$ (each cell contains one marble). Can we conduct an finite amount of moves such that both cells $(n + 1, n)$ and $(n, n + 1)$ have marbles?
Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$.
Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.
A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?