Found problems: 1385
There is a row with $2024$ cells. Ana and Beto take turns playing, with Ana going first. On each turn, the player selects an empty cell and places a digit in that space. Once all $2024$ cells are filled, the number obtained from reading left to right is considered, ignoring any leading zeros. Beto wins if the resulting number is a multiple of $99$, otherwise Ana wins. Determine which of the two players has a winning strategy and describe it.
Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?
A game consists of pieces of the shape of a regular tetrahedron of side $1$. Each face of each piece is painted in one of $n$ colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.
You own a calculator that computes exactly. It has all the standard buttons, including a button that replaces the number currently displayed with its arctangent, and a button that replaces whatever is currently displayed with its cosine. You turn on the calculator and it reads 0. You create a sequence by alternately clicking on the arctangent button and the cosine button. (The calculator is in radian mode.) Let $a_n$ be the value displayed after you've pressed the cosine button for the $n$th time. What is $\prod_{k=1}^{11} a_k$?
John and Mary play the following game. First they choose integers $n > m > 0$ and put $n$ sweets on an empty table. Then they start to make moves alternately. A move consists of choosing a nonnegative integer $k\le m$ and taking $k$ sweets away from the table (if $k = 0$ , nothing happens in fact). In doing so no value for $k$ can be chosen more than once (by none of the players) or can be greater than the number of sweets at the table at the moment of choice. The game is over when one of the players can make no more moves.
John and Mary decided that at the beginning Mary chooses the numbers $m$ and $n$ and then John determines whether the performer of the last move wins or looses. Can Mary choose $m$ and $n$ in such way that independently of John’s decision she will be able to win?
There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?
A board consists of $2021 \times 2021$ squares all of which are white, except for one corner square which is black. Alma and Bertha play the following game. At the beginning, there is a piece on the black square. In each turn, the player must move the piece to a new square in the same row or column as the one in which the piece is currently. All squares that the piece moves across, including the ending square but excluding the starting square, must be white, and all squares that the piece moves across, including the ending square, become black by this move. Alma begins, and the first player unable to move loses. Which player may prepare a strategy which secures her the victory?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/270d82f37b729bfe661f8a92cea8be67e5625c.png[/img]
[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccc}
& a & b & c & a \\
+ & & d & d & e \\
& & & d & e \\
\hline
d & f & f & d & d \\
\end{tabular}$
[b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$:
$$n^2 + m^2 = 111...111$$
[b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$.
[b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
Mugdho and Dipto play a game on a numbered row of $n \geq 5$ squares. At the beginning, a pebble is put on the first square and then the players make consecutive moves; Mugdho starts. During a move a player is allowed to choose one of the following:
[list]
[*] move the pebble one square rightward
[*] move the pebble four squares rightward
[*] move the pebble two squares leftward
[/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 $n$-th) wins. Determine for which values of $n$ each of the players has a winning strategy.
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
Two are playing the game "cats and rats" on the chess-board $8\times 8$. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions.
a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat?
b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.
Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A [i]move[/i] consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.
[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccc}
& a & b & c & a \\
+ & & d & d & e \\
& & & d & e \\
\hline
d & f & f & d & d \\
\end{tabular}$
[b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply?
[b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert?
[b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Three numbered tiles are arranged in a tray as shown:
[img]https://cdn.artofproblemsolving.com/attachments/d/0/d449364f92b7fae971fd348a82bafd25aa8ea1.jpg[/img]
Show that we cannot interchange the $1$ and the $3$ by a sequence of moves where we slide a tile to the adjacent vacant space.
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
Ana and Beto play on a grid of $2022 \times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?
For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$.
[list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$.
[*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list]
Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps.
[i]Proposed by Gerhard Woeginger, Austria[/i]
In a ping-pong game, it was planned to have a competition between any two players. But three players quit the game after having 2 competitions. In the end, the number of competitions played is 50. So the number of competitions between the three players is
$\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3$
Let $d$ be a positive real number. The scorpion tries to catch the flea on a $10\times 10$ chessboard. The length of the side of each small square of the chessboard is $1$. In this game, the flea and the scorpion move alternately. The flea is always on one of the $121$ vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than $d$.
At the beginning, the flea is at the center of the chessboard and the scorpion is at a point that he chooses on the boundary line. The flea is the first one to play. The flea is said to [i]escape[/i] if it reaches a point of the boundary line, which the scorpion can't reach in the next turn. Obviously, for big values of $d$, the scorpion has a strategy to prevent the flea's escape. For what values of $d$ can the flea escape? Justify your answer.
Players $ A,B$ alternately move a knight on a $ 1994\times 1994$ chessboard. Player $ A$ makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while $ B$ makes only vertical moves. Initally player $ A$ places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player $ A$ has a winning strategy.
Bowling Pins is a game played between two players in the following way:
We start with $ 14 $ bowling pins in a line:
$ X\quad X \quad X \quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X\quad X $
Players alternate turns. On each turn, the player can knock down one, two or three consecutive pins
at a time. For example:
Jing Jing bowls:
$ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad X \quad X \quad X \quad X \quad X \quad X \quad X $
Soumya bowls:
$ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad \quad\quad X \quad X \quad X \quad X \quad X \quad X $
Jing Jing bowls again:
$ X \quad X \quad\quad \quad\quad \:\:\:\: X \quad X \quad X \quad \quad\quad X \quad X \quad \quad \quad\quad\quad \quad\:\: X $
The player who knocks down the last pin wins.
In the above game, it is Soumya’s turn. If he plays perfectly from here, he has a winning strategy (In
fact, he has four different winning moves.)
Imagine it’s Jing Jing’s turn to play and the game looks as follows
$ X \quad \quad \quad\quad\quad \quad\:\: X\dots $
with 1 X on the left and a string of $ k $ consecutive X’s on the right.
For what values of $ k $ from $ 1 $ to $ 10 $ does she have a winning strategy?
The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.
[hide=Original Wording]The expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.[/hide]
Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers.
One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins).
In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice.
[i]Nikolai Beluhov[/i]