Found problems: 304
There is a pile with 2022 rocks. Ana y Beto play by turns to the following game, starting with Ana: in each turn, if there are $n$ rocks in the pile, the player can remove $S(n)$ rocks or $n-S(n)$ rocks, where $S(n)$ is the sum of the the digits of $n$. The person who removes the last rock wins. Determine which of the two players has a winning strategy and describe it.
Suppose you are playing a game against Daniel. There are $2017$ chips on a table. During your turn, if you can write the number of chips on the table as a sum of two cubes of not necessarily distinct, nonnegative integers, then you win. Otherwise, you can take some number of chips between $1$ and $6$ inclusive off the table. (You may not leave fewer than $0$ chips on the table.) Daniel can also do the same on his turn. You make the first move, and you and Daniel always make the optimal move during turns. Who should win the game? Explain.
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?
There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ($A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?
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?
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?
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)
The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right
A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.
Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
Vaggelis has a box that contains $2015$ white and $2015$ black balls. In every step, he follows the procedure below:
He choses randomly two balls from the box. If they are both blacks, he paints one white and he keeps it in the box, and throw the other one out of the box. If they are both white, he keeps one in the box and throws the other out. If they are one white and one black, he throws the white out, and keeps the black in the box.
He continues this procedure, until three balls remain in the box. He then looks inside and he sees that there are balls of both colors. How many white balls does he see then, and how many black?
A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction?
(B, Frenkin)
Anna and Basilis play a game writing numbers on a board as follows:
The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$.
Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.
Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods:
[i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts.
[i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut.
Finally, stage 3 varies among the three methods as follows:
[i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps.
[i]Method 2:[/i] Nickolas takes the two middle size heaps.
[i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice.
Determine the most and the least profitable method for Nickolas.
Bob proposes the following game to Johanna. The board in the figure is an equilateral triangle subdivided in turn into $256$ small equilateral triangles, one of which is painted in black. Bob chooses any point inside the board and places a small token. Johanna can make three types of plays. Each of them consists of choosing any of the $3$ vertices of the board and move the token to the midpoint between the current position of the tile and the chosen vertex. In the second figure we see an example of a move in which Johana chose vertex $A$. Johanna wins if she manages to place her piece inside the triangle black. Prove that Johanna can always win in at most $4$ moves.
[asy]
unitsize(8 cm);
pair A, B, C;
int i;
A = dir(60);
C = (0,0);
B = (1,0);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7));
draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) {
draw(interp(A,B,i/16)--interp(A,C,i/16));
draw(interp(B,C,i/16)--interp(B,A,i/16));
draw(interp(C,A,i/16)--interp(C,B,i/16));
}
label("$A$", A, N);
label("$B$", B, SE);
label("$C$", C, SW);
[/asy]
[asy]
unitsize(8 cm);
pair A, B, C, X, Y, Z;
int i;
A = dir(60);
C = (0,0);
B = (1,0);
X = 9.2/16*(1,0) + 3.3/16*dir(60);
Y = (A + X)/2;
Z = rotate(60,X)*(Y);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7));
draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) {
draw(interp(A,B,i/16)--interp(A,C,i/16));
draw(interp(B,C,i/16)--interp(B,A,i/16));
draw(interp(C,A,i/16)--interp(C,B,i/16));
}
draw(A--X, dotted);
draw(arc(Z,abs(X - Y),-12,40), Arrow(6));
label("$A$", A, N);
label("$B$", B, SE);
label("$C$", C, SW);
dot(A);
dot(X);
dot(Y);
[/asy]
The figure shows a game board with $16$ squares. At the start of the game, two cars are placed in different squares. Two players $A$ and $B$ alternately take turns, and A starts. In each turn, the player chooses one of the cars and moves it one or more squares to the right. The left-most car may never overtake or land on the same square as the right-most car. The first player which is unable to move loses.
[img]https://cdn.artofproblemsolving.com/attachments/1/b/8d6f40fac4983d6aa9bd076392c91a6d200f6a.png[/img]
(a) Prove that A can win regardless of how $B$ plays, if the two cars start as shown in the figure.
(b) Determine all starting positions in which $B$ can win regardless of how $A$ plays.
There are $9999$ rods with lengths $1, 2, ..., 9998, 9999$. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd.
Who has a winning strategy?
( "Sisyphian Labour" )
There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step.
Can Aid stop him?
( S . Yeliseyev)
Alina and Bogdan play a game on a $2\times n$ rectangular grid ($n\ge 2$) whose sides of length $2$ are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?
Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?
We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game.
Which player has a winning strategy, and what is it? (The answer may depend on $n$)
There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)
Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?
Alina and Bogdan play the following game. They have a heap and $330$ stones in it. They take turns. In one turn it is allowed to take from the heap exactly $1$, exactly $n$ or exactly $m$ stones. The player who takes the last stone wins. Before the beginning Alina says the number $n$, ($1 < n < 10$). After that Bogdan says the number $m$, ($m \ne n, 1 < m < 10$). Alina goes first. Which of the two players has a winning strategy? What if initially there are 2018 stones in the heap?
adapted from a Belarus Olympiad problem