Found problems: 1385
Every square of a $3\times3$ board is assigned a sign $+$ or $-$. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are $-$ after finitely many moves?
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order:
[list=i]
[*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$
[*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$
[*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$
[/list]
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$
[i]Proposed by Gerhard Woeginger, Austria[/i]
Yatta and Yogi play a game in which they begin with a pile of $n$ stones. The players take turns removing $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones from the pile. That is, when it is a player's turn to remove stones, that player may remove from $1$ to $8$ stones, but [i]cannot[/i] remove exactly $4$ stones. The player who removes the last stone [i]loses[/i]. Yogi goes first and finds that he has a winning position, meaning that so long as he plays perfectly, Yatta cannot defeat him. For how many positive integers $n$ from $100$ to $2008$ inclusive is this the case?
There are $n$ matches on the table ($n > 1$). Two players take turns shooting them from the table. On the first move, the player removes any number of matches from the table from $1$ to $n - 1$, and then each time you can take no more matches from the table, than the partner took with the previous move. The one who took the last match wins.. Find all $n$ for which the first player can provide win for yourself.
Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers.
Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.
Alice and Bob play a game on a numbered row of $n \ge 5$ squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following:
[list]
[*] move the pebble one square forward;
[*] move the pebble four squares forward;
[*] move the pebble two squares backwards.
[/list]
All of the possible moves are only allowed if the pebble stays within the borders of the square row.
The player who moves the pebble to the last square (a.k.a $n\text{-th}$) wins. Determine for which values of $n$ each of the players has a winning strategy.
Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:
$(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.
$(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right.
$\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves.
$\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.
Let $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than $25$, not necessarily different, such that their sum is at least $200$. The second player wins if he can select some of these numbers so that their sum $S$ satisfies the condition $200-N\le S\le 200+N$. What is the smallest value of $N$ for which the second player has a winning strategy?
There are 3 heaps with $100,101,102$ stones. Ilya and Kostya play next game. Every step they take one stone from some heap, but not from same, that was on previous step. They make his steps in turn, Ilya make first step. Player loses if can not make step. Who has winning strategy?
Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are
[list=a]
[*]at least a hundred copies of number 5?
[*]at least a hundred copies of number 10?
[/list]
[i]Andrey Arzhantsev[/i]
Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$. Then player $II$ chooses another square and marks it with $X$. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.
A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges.
Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game:
• First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas.
• Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants.
• After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron.
Determine if there is a strategy for which Lucas can always properly discover the four special sweets.
Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.
Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$. At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$. The two players take turns, with Ariane starting.
Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently.
Depending on $n$, determine which of the two has a winning strategy.
Given a $n\times n$ table with non-negative real entries such that the sums of entries in each column and row are equal, a player plays the following game: The step of the game consists of choosing $n$ cells, no two of which share a column or a row, and subtracting the same number from each of the entries of the $n$ cells, provided that the resulting table has all non-negative entries. Prove that the player can change all entries to zeros.
The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win?
[i]Alexandr Gribalko[/i]
Alberto wants to organize a poker game with his friends this evening. Bruno and Barbara together go to gym once in three evenings, whereas Carla, Corrado, Dario and Davide are busy once in two evenings (not necessarily the same day). Moreover, Dario is not willing to play with Davide, since they have a quarrel over a girl. A poker game requires at least four persons (including Alberto). What is the probability that the game will be played?
Let $n\geq 2$ be an integer. Alice and Bob play a game concerning a country made of $n$ islands. Exactly two of those $n$ islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands $I_1$ and $I_2$ such that:
$\bullet$ $I_1$ and $I_2$ are not already connected by a bridge.
$\bullet$ at least one of the two islands $I_1$ and $I_2$ is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.)
As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each $n\geq 2$) who has a winning strategy.
([i]Note:[/i] It is allowed to construct a bridge passing above another bridge.)
$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$.
Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?
Three boxes with at least one marble in each are given. In each step we double the number of marbles in one of the boxes, taking the required number of boxes from one of the other two boxes. Is it always possible to have one of the boxes empty after several steps?
[b]p1[/b]. Consider the following four statements referring to themselves:
1. At least one of these statements is true.
2. At least two of these statements are false.
3. At least three of these statements are true.
4. All four of these statements are false.
Determine which statements are true and which are false. Justify your answer.
[b]p2.[/b] Let $f(x) = a_{2017}x^{2017} + a_{2016}x^{2016} + ... + a_1x + a_0$ where the coefficients $a_0, a_1, ... , a_{2017}$ have not yet been determined. Alice and Bob play the following game:
$\bullet$ Alice and Bob alternate choosing nonzero integer values for the coefficients, with Alice going first. (For example, Alice’s first move could be to set $a_{18}$ to $-3$.)
$\bullet$ After all of the coefficients have been chosen:
- If f(x) has an integer root then Alice wins.
- If f(x) does not have an integer root then Bob wins.
Determine which player has a winning strategy and what the strategy is. Make sure to justify your answer.
[b]p3.[/b] Suppose that a circle can be inscribed in a polygon $P$ with $2017$ equal sides. Prove that $P$ is a regular polygon; that is, all angles of $P$ are also equal.
[b]p4.[/b] A $3 \times 3 \times 3$ cube of cheese is sliced into twenty-seven $ 1 \times 1 \times 1$ blocks. A mouse starts anywhere on the outside and eats one of the $1\times 1\times 1$ cubes. He then moves to an adjacent cube (in any direction), eats that cube, and continues until he has eaten all $27$ cubes. (Two cubes are considered adjacent if they share a face.) Prove that no matter what strategy the mouse uses, he cannot eat the middle cube last.
[Note: One should neglect gravity – intermediate configurations don’t collapse.]
p5. Suppose that a constant $c > 0$ and an infinite sequence of real numbers $x_0, x_1, x_2, ...$ satisfy
$x_{k+1} =\frac{x_k + 1}{1 - cx_k}$ for all $k \ge 0$. Prove that the sequence $x_0, x_1, x_2, ....$ contains infinitely many positive terms and also contains infinitely many negative terms.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first).
The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies?
Proposed by Denis Afrizonov
The expression
\[ \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \]
is written on the blackboard. Two players, $ A $ and $ B $, play a game, taking turns. Player $ A $ takes the first turn. In each turn, the player on turn replaces a symbol $ \Box $ by a positive integer. After all the symbols $\Box$ are replace, player $A$ replaces each of the signs $\pm$ by either + or -, independently of each other. Player $ A $ wins if the value of the expression on the blackboard is not divisible by any of the numbers $ 11, 12, \cdots, 18 $. Otherwise, player $ B$ wins. Determine which player has a winning strategy.