Found problems: 1385
Isaak and Jeremy play the following game.
Isaak says to Jeremy that he thinks a few $2^n$ integers $k_1,..,k_{2^n}$.
Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth.
After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not.
Prove that Jeremy has bo way to be ''sure'' for his final decision.
(UK)
In a standard game of Rock–Paper–Scissors, two players repeatedly choose between rock, paper, and scissors, until they choose different options. Rock beats scissors, scissors beats paper, and paper beats rock. Nathan knows that on each turn, Richard randomly chooses paper with probability $33\%$, scissors with probability $44\%$, and rock with probability $23\%$. If Nathan plays optimally against Richard, the probability that Nathan wins is expressible as $a/b$ where $a$ and $b$ are coprime positive integers. Find $a + b$.
Let $n \geq 5$ be a positive integer. There are $n$ stars with values $1$ to $n$, respectively. Anya and Becky play a game. Before the game starts, Anya places the $n$ stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all $n$ such that Becky has a winning strategy.
[i]
Proposed by Ho-Chien Chen[/i]
In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).
The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.
$(a)$ Determine which player has a winning strategy for $N=2005$.
$(b)$ Determine which player has a winning strategy for $N=2004$.
$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
There are $n$ coins in a row. Two players take turns picking a coin and flipping it. The location of the heads and tails should not repeat. Loses the one who can not make a move. Which of player can always win, no matter how his opponent plays?
On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.
Who has a winning strategy: the starting player or his opponent?
$200$ soldiers occupy in a rectangle (military call it a square and educated military a carree): $20$ men (per row) times $10$ men (per column). In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the $10$ men considered we select the shortest (if some are of equal height, choose any of them). Call him $A$. Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these $20$, the tallest is chosen. Call him $B$. Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller: $A$ or $B$. Which colonel wins the bet?
In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors, as shown in the below diagram. As before, if two players choose the same move, then there is a draw. If three people each play a game of rock-paper-scissors-lizard-Spock at the same time by choosing one of the five moves at random, what is the probability that one player beats the other two?
[img]https://cdn.artofproblemsolving.com/attachments/6/0/3129da5998a2e872673e34351f786ffd47e1a1.png[/img]
The integers $1, 2, 3, 4, 5$ and $6$ are written on a board. You can perform the following kind of move: select two of the numbers, say $a$ and $b$, such that $4a - 2b$ is nonnegative; erase $a$ and $b$, then write down $4a - 2b$ on the board (hence replacing two of the numbers by just one). Continue performing such moves until only one number remains on the board. What is the smallest possible positive value of this last remaining number?
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?
$
\textbf{(A) }\frac{1}{2}\qquad
\textbf{(B) }\frac{13}{24} \qquad
\textbf{(C) }\frac{7}{12} \qquad
\textbf{(D) }\frac{5}{8} \qquad
\textbf{(E) }\frac{2}{3}\qquad
$
Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David?
[i]Proposed by Vlad Spătaru[/i]
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
Two players take turns placing an unused number from
{1, 2, 3, 4, 5, 6, 7, 8} into one of the empty squares in the array to the
right. The game ends once all the squares are filled. The first player
wins if the product of the numbers in the top row is greater. The second
player wins if the product of the numbers in the bottom row is greater. If both players play
with perfect strategy, who wins this game?
[asy]
unitsize(32);
int[][] a = {
{1, 2, 3, 4},
{5, 6, 7, 8}};
for (int i = 0; i < 4; ++i) {
for (int j = 0; j < 2; ++j) {
draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] > 0) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(16pt));
}
}
[/asy]
We have $105$ coins, among which we know that there are three fake ones. Authentic coins have all the same weight, which is greater than that of the false ones, which also have the same weight. Determine from can $26$ authentic coins be selected by weighing only two in one two pan balance.
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy.
[b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.
John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy?
[i](6 points)[/i]
Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?
Given an integer \( n \geq 1 \), Jo-Ané alternately writes crosses (\( \mathcal{X} \)) and circles (\( \mathcal{O}\)) in the cells of a square grid with \( 2n + 1 \) rows and \( 2n + 1 \) columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum \( \mathcal{X}+ \mathcal{O} \), where \( \mathcal{X} \) is the number of rows containing more crosses than circles and \( \mathcal{O} \) is the number of columns containing more circles than crosses.
Determine, in terms of \( n \), the highest possible score that Jo-Ané can obtain..
Pedro and Juan are playing the following game:
$-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$).
$-$ Each player can draw:
-- $1$ rock from one of the piles, or
-- $2$ rocks from one of the piles, or
-- $1$ rock from each pile, or
-- $2$ rock from one pile and $1$ from the other pile.
Each player must perform one of these four operations in their turns.
The looser is the one who takes the last rock.
Pedro plays first and has a winning strategy.
What are the three maximum possible values of ($X+Y$)?