Found problems: 1385
Consider the starting position in a game of bughouse. Exhibit a sequence of moves
on both boards, indicating the chronology, such that at the end:
(a) The positions on both boards are the same as the original positions.
(b) It is White to play on one board, but Black to play on the other.
(c) All four players still have the right to castle subsequently (equivalently, the kings and rooks
haven’t moved).
for each of the following cases :
(a) without moving any pawns.
(b) without moving any queen.
In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the fi rst player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?
An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of
\[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\]
How many guesses are needed for Banana to figure out the quadruple Ana chose?
Alice and Bob are playing a game on a graph with $n\ge3$ vertices. At each moment, Alice needs to choose two vertices so that the graph is connected even if one of them (along with the edges incident to it) is removed. Each turn, Bob removes one edge in the graph, and upon the removal, Alice needs to re-select the two vertices if necessary. However, Bob has to guarantee that after each removal, any two vertices in the graph are still connected via at most $k$ intermediate vertices. Here $0\le k\le n-2$ is some given integer. Suppose that Bob always knows which two vertices Alice chooses, and that initially, the graph is a complete graph. Alice's objective is to change her choice of the two vertices as few times as possible, and Bob's objective is to make Alive re-select as many times as possible. If both Alice and Bob are sufficiently smart, how many times will Alice change her choice of the two vertices?
(usjl)
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?
[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy?
[b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$.
[b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers.
[b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$.
[b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area?
[img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n$ be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
$(i)\:$ Choose a number of the form $2^j$, with $j$ a non-negative integer, and place it in an empty square.
$(ii)\:$ Choose two (not necessarily consecutive) squares containing the same number, say $2^j$. Replace the number in one of the squares with $2^{j+1}$ and erase the number in the other square.
At the end of the game, one square contains the number $2^n$, while the other squares are empty. Determine, as a function of $n$, the maximum number of turns Alex can make.
Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so?
[img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]
Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth.
[b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps?
[b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?
Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win?
(Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.)
[i]Proposed by Lewis Chen[/i]
Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins.
(a) Determine which player has a winning strategy if $N = 2011$.
(b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy.
(This is based on ISL 2004, Problem C5.)
There are two piles of stones with $1012$ stones each. Ann and Ben play a game. In every move, a player removes two stones from one of the piles and adds one to the other pile. Ann goes first. The first player to remove the last stone in one of the piles wins the game. Which player has a winning strategy and why?
[b]p1.[/b] Solve the equation: $\sqrt{x} +\sqrt{x + 1} - \sqrt{x + 2} = 0$.
[b]p2.[/b] Solve the inequality: $\ln (x^2 + 3x + 2) \le 0$.
[b]p3.[/b] In the trapezoid $ABCD$ ($AD \parallel BC$) $|AD|+|AB| = |BC|+|CD|$. Find the ratio of the length of the sides $AB$ and $CD$ ($|AB|/|CD|$).
[b]p4.[/b] Gollum gave Bilbo a new riddle. He put $64$ stones that are either white or black on an $8 \times 8$ chess board (one piece per each of $64$ squares). At every move Bilbo can replace all stones of any horizontal or vertical row by stones of the opposite color (white by black and black by white). Bilbo can make as many moves as he needs. Bilbo needs to get a position when in every horizontal and in every vertical row the number of white stones is greater than or equal to the number of black stones. Can Bilbo solve the riddle and what should be his solution?
[b]p5.[/b] Two trolls Tom and Bert caught Bilbo and offered him a game. Each player got a bag with white, yellow, and black stones. The game started with Tom putting some number of stones from his bag on the table, then Bert added some number of stones from his bag, and then Bilbo added some stones from his bag. After that three players started making moves. At each move a player chooses two stones of different colors, takes them away from the table, and puts on the table a stone of the color different from the colors of chosen stones. Game ends when stones of one color only remain on the table. If the remaining stones are white Tom wins and eats Bilbo, if they are yellow, Bert wins and eats Bilbo, if they are black, Bilbo wins and is set free. Can you help Bilbo to save his life by offering him a winning strategy?
[b]p6.[/b] There are four roads in Mirkwood that are straight lines. Bilbo, Gandalf, Legolas, and Thorin were travelling along these roads, each along a different road, at a different constant speed. During their trips Bilbo met Gandalf, and both Bilbo and Gandalf met Legolas and Thorin, but neither three of them met at the same time. When meeting they did not stop and did not change the road, the speed, and the direction. Did Legolas meet Thorin? Justify your answer.
PS. You should use hide for answers.
A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?
The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is
known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides.
[i]D. Tereshin[/i]
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?
5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types:
Type $1$: Piece like Type $2$ but with one square at the right of the bottom square.
Type $2$: Piece of $2$ consecutive squares, one over another.
Type $3$: Piece of $1$ square.
At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy.
Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.
A rook stands at a corner of an $m \times n$ squared board. Two players move the rook in turn (vertically or horizontally through any numbers of squares). As the rook moves, it paints the squares that it visits (stopping or passing through). The rook is not allowed to pass through or stop at the painted squares. The player who cannot move, loses. Who has a guaranteed win: the first player (who starts the game) or the other, and how should he/she play?
(B Begun)
Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules :
[b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that
\[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2.
\]
[b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that
\[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}}
\]
is a prime raised to a positive integer power.
Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does :
[b]a.)[/b] $ {\mathcal A}$ have a winning strategy?
[b]b.)[/b] $ {\mathcal B}$ have a winning strategy?
[b]c.)[/b] Neither player have a winning strategy?
Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position.
(Sergey Dorichenko)
On a $1\times n$ board there are $n-1$ separating edges between neighbouring cells. Initially, none of the edges contain matches. During a move of size $0 < k < n$ a player chooses a $1\times k$ sub-board which contains no matches inside, and places a matchstick on all of the separating edges bordering the sub-board that don’t already have one.
A move is considered legal if at least one matchstick can be placed and if either $k = 1$ or $k{}$ is divisible by 4. Two players take turns making moves, the player in turn must choose one of the available legal moves of the largest size $0 < k < n$ and play it. If someone does not have a legal move, the game ends and that player loses.
[i]Beat the organisers twice in a row in this game! First the organisers determine the value of $n{}$, then you get to choose whether you want to play as the first or the second player.[/i]
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
Arne and Berit are playing a game. They have chosen positive integers $m$ and $n$ with $n\geq 4$ and $m \leq 2n + 1$. Arne begins by choosing a number from the set $\{1, 2, \dots , n \}$, and writes it on a blackboard. Then Berit picks another number from the same set, and writes it on the board. They continue alternating turns, always choosing numbers that are not already on the blackboard. When the sum of all the numbers on the board exceeds or equals $m$, the game is over, and whoever wrote the last number has won. For which combinations of $m$ and $n$ does Arne have a winning strategy?
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]