Found problems: 304
All the cells in a $8* 8$ board are colored white. Omar and Asaad play the following game: in the beginning Omar colors $n$ cells red, then Asaad chooses $4$ rows and $4$ columns and colors them black. Omar wins if there is at least one red cell. Find the least possible value for n such that Omar can always win regardless of Asaad's move.
A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?
On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. )
( F . L . Nazarov)
We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.
Three persons $A,B,C$, are playing the following game:
A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$.
For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?
On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?
In the middle cell of the $1 \times 2005$ strip there is a chip. Two players each queues move it: first, the first player moves the piece one cell in any direction, then the second one moves it $2$ cells, the $1$st - by $4$ cells, the 2nd by $8$, etc. (the $k$-th shift occurs by $2^{k-1}$ cells). That, whoever cannot make another move loses. Who can win regardless of the opponent's play?
Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.
Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?
The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?
On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game.
Alice starts and they alternately play the following moves:
$\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$
$\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$.
Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps.
Prove that Alice has a winning strategy.
(Daniel Holmes)
An integer $n > 1$ is given. Two players in turns mark points on a circle. First Player uses red color while Second Player uses blue color. The game is over when each player marks $n$ points. Then each player nds the arc of maximal length with ends of his color, which does not contain any other marked points. A player wins if his arc is longer (if the lengths are equal, or both players have no such arcs, the game ends in a draw). Which player has a winning strategy?
Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?
Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn.
During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses.
Determine which player can always win this game.
On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$.
Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore.
[i]Note: The order of the numbers of the list is not important.[/i]
The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game :
each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?
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 $S$ the set of natural numbers from $1$ up to $1001$ , $S=\{1,2,...,1001\}$. Lisandro thinks of a number $N$ of $S$ , and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of $S$,
Lisandro reads it and tells Carla how many subsets of her list contain $N$ . If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than $3$ are allowed. What is the smallest total number of subsets that allow Carla to find $N$ for sure?
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).
At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?
[i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]
$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?
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.
LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?
There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam.
(S. Fomin, Leningrad)
Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors.
We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$.
I. Bogdanov