Found problems: 1385
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
For a given natural number $n$, two players randomly (uniformly distributed) select a common number $0 \le j \le n$, and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that
\[ \sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), \quad (n \to \infty),\]
where $\gamma$ is the Euler constant.
Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.
In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?
Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.
(a) Two players take turns taking $1, 2$ or $3$ stones at random from a given set of $3$ piles, in which initially on $11, 22$ and $33$ stones. If after the move of one of the players in any two groups the same number of stones will remain, this player has won. Who will win with the right game of both players?
(b) Two players take turns taking $1$ or $2$ stones from one pile, randomly selected from a given set of $3$ ordered piles, in which at first $100, 200$ and $300$ stones, in order from left to right. Additionally it is forbidden to make a course at which, for some pair of the next handfuls, quantity of stones in the left will be more than the number of stones in the right. If after the move of one of the players of the stones in handfuls will not remain, then this player won. Who will win with the right game of both players?
[hide=original wording]
1. Два гравця по черзi беруть 1, 2 чи 3 камiнця довiльним чином з заданого набору з 3 купок, в
яких спочатку по 11, 22 i 33 камiнцiв. Якщо пiсля хода одного з гравцiв в якихось двух купках
залишиться однакова кiлькiсть камiнцiв, то цей гравець виграв. Хто виграє при правильнiй грi обох
гравцiв?
2. Два гравця по черзi беруть 1 чи 2 камiнця з одної купки, довiльної вибраної з заданого набору
з 3 впорядкованих купок, в яких спочатку по 100, 200 i 300 камiнцiв, в порядку злiва направо.
Додатково забороняется робити ход при якому, для деякої пари сусiднiх купок, кiлькiсть камiнцiв в
лiвiй стане бiльше нiж кiлькiсть камiнцiв в правiй. Якщо пiсля ходу одного з гравцiв камiнцiв в
купках не залишиться, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв?[/hide]
Alex thinks of a two-digit integer (any integer between $10$ and $99$). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was $65$, then by naming any of $64, 65, 66, 55$ or $75$ Greg will be answered “hot”, otherwise he will be answered “cold”.)
[list][b](a)[/b] Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts.
[b](b)[/b] Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than $24$ attempts.
[b](c)[/b] Is there a $22$ attempt winning strategy for Greg?[/list]
From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is
(a) greater than 34;
(b) greater than 37;
loses the game. Which player has a winning strategy?
A circle is divided by $2018$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?
Suppose that zeros and ones are written in the cells of an $n\times n$ board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed $\frac n2\left(1+\sqrt{4n-3}\right)$.
There are $n$ coins aligned in a row. In each step, it is allowed to choose a coin with the tail up (but not one of the outermost markers), remove it and reverse the closest coin to the left and the closest coin to the right of it. Initially, all the coins have tails up. Prove that one can achieve the state with only two coins remaining if and only if $n-1$ is not divisible by $3$.
Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy?
[i]Note: an empty graph is a graph with no edges.[/i]
On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?
A pile of $2000$ coins is given on a table. In each step, we choose a pile with at least three coins, remove one coin from it, and divide the rest of this pile into two piles (not necessarily of the same size). Is it possible that after several steps each pile on the table has exactly three coins?
The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?
There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move:
$\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes.
$\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. Then $B$ destroys all the cups having no pebbles.
$\bullet$ $B$ switches the places of two cups without telling $A$.
After finitely many moves, $A$ can guarantee that $n$ cups are destroyed. Find the maximum possible value of $n$.
(Note that $A$ doesn't see the cups while playing.)
[i]Proposed by Emre Osman[/i]
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]
[b]p1.[/b] Three positive integers sum to $16$. What is the least possible value of the sum of their squares?
[b]p2.[/b] Ben is thinking of an odd positive integer less than $1000$. Ben subtracts $ 1$ from his number and divides by $2$, resulting in another number. If his number is still odd, Ben repeats this procedure until he gets an even number. Given that the number he ends on is $2$, how many possible values are there for Ben’s original number?
[b]p3.[/b] Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\omega$. Tangents to $\omega$ at $ A$ and $ B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$?
[b]p4.[/b] How many non-decreasing sequences of five natural numbers have first term $ 1$, last term $ 11$, and have no three terms equal?
[b]p5.[/b] Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps?
[b]p6.[/b] Eric has two coins, which land heads $40\%$ and $60\%$ of the time respectively. He chooses a coin randomly and flips it four times. Given that the first three flips contained two heads and one tail, what is the probability that the last flip was heads?
[b]p7.[/b] In a five person rock-paper-scissors tournament, each player plays against every other player exactly once, with each game continuing until one player wins. After each game, the winner gets $ 1$ point, while the loser gets no points. Given that each player has a $50\%$ chance of defeating any other player, what is the probability that no two players end up with the same amount of points?
[b]p8.[/b] Let $\vartriangle ABC$ have $\angle A = \angle B = 75^o$. Points $D, E$, and $F$ are on sides $BC$, $CA$, and $AB$, respectively, so that $EF$ is parallel to $BC$, $EF \perp DE$, and $DE = EF$. Find the ratio of $\vartriangle DEF$’s area to $\vartriangle ABC$’s area.
[b]p9.[/b] Suppose $a, b, c$ are positive integers such that $a+b =\sqrt{c^2 + 336}$ and $a-b =\sqrt{c^2 - 336}$. Find $a+b+c$.
[b]p10.[/b] How many times on a $12$-hour analog clock are there, such that when the minute and hour hands are swapped, the result is still a valid time? (Note that the minute and hour hands move continuously, and don’t always necessarily point to exact minute/hour marks.)
[b]p11.[/b] Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$.
[b]p12.[/b] Find the value of $\sqrt{3 \sqrt{3^3 \sqrt{3^5 \sqrt{...}}}}$, where the exponents are the odd natural numbers, in increasing order.
[b]p13.[/b] Jamesu and Fhomas challenge each other to a game of Square Dance, played on a $9 \times 9$ square grid. On Jamesu’s turn, he colors in a $2\times 2$ square of uncolored cells pink. On Fhomas’s turn, he colors in a $1 \times 1$ square of uncolored cells purple. Once Jamesu can no longer make a move, Fhomas gets to color in the rest of the cells purple. If Jamesu goes first, what the maximum number of cells that Fhomas can color purple, assuming both players play optimally in trying to maximize the number of squares of their color?
[b]p14.[/b] Triangle $ABC$ is inscribed in circle $\omega$. The tangents to $\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\angle BAC = 60^o$ and the area of $\vartriangle BDE$ is twice the area of $\vartriangle CDE$, what is $\frac{AB}{AC}$ ?
[b]p15.[/b] Fhomas and Jamesu are now having a number duel. First, Fhomas chooses a natural number $n$. Then, starting with Jamesu, each of them take turns making the following moves: if $n$ is composite, the player can pick any prime divisor $p$ of $n$, and replace $n$ by $n - p$, if $n$ is prime, the player can replace n by $n - 1$. The player who is faced with $ 1$, and hence unable to make a move, loses. How many different numbers $2 \le n \le 2019$ can Fhomas choose such that he has a winning strategy, assuming Jamesu plays optimally?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let the [i]subbishop[/i] (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$, no two of which are not attacking.
[i]V. Chukanov[/i]
Two positive integers $m$ and $n$ are written on the board.
We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer).
Depending on the numbers $m$ and $n$, specify all the pairs that can appear on the board in pairs.
(Radovan Švarc)
A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$. Otherwise, the other player wins. Decide which of the players has a winning strategy.
(Pavel Šalom)
Let $n \ge 4$ and $k$ be positive integers. We consider $n$ lines in the plane between which there are not two parallel nor three concurrent. In each of the $\frac{n(n-1)}{2}$ points of intersection of these lines, $k$ coins are placed. Ana and Beto play the following game in turns: each player, in turn, chooses one of those points that does not share one of the $n$ lines with the point chosen immediately before by the other player, and removes a coin from that point. Ana starts and can choose any point. The player who cannot make his move loses. Determine based on $n$ and $k$ who has a winning strategy.
Dua has all the odd natural numbers less than 20. Asija has all the even numbers less than 21. They play the following game. In each round, they take a number from each other and after every round, they may fix two or more consecutive numbers so that their opponent cannot take these fixed numbers in the next round. The game is won by the player who attains 10 consecutive numbers first. Does either player have a winning strategy?
The game of Greed starts with an initial configuration of one or more piles of stones.
Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$. At each turn, a player has two choices:
• take one stone from any one of the piles (a simple move);
• take one stone from each of the remaining piles (a greedy move).
The player who takes the last stone wins.
Consider the following two initial configurations:
(a) There are $2018$ piles, with either $20$ or $18$ stones in each pile.
(b) There are four piles, with $17, 18, 19$, and $20$ stones, respectively.
In each case, find an appropriate strategy that guarantees victory to one of the players.
Al is bored of Rock Paper Scissors, and wants to invent a new game: $Z-Y-X-W-V.$ Two players, each choose to play either $Z, Y, X, W,$ or $V.$ If they play the same thing, the result is a tie. However, Al must come up with a ’pecking order’, that is, he must decide which plays beat which. For each of the $10$ pairs of distinct plays that the two players can make, Al randomly decides a winner. For example, he could decide that $W$ beats $Y$ and that $Z$ beats $X,$ etc. What is the probability that after Al makes all of these $10$ choices, the game is balanced, that is, playing each letter results in an equal probability of winning?