Found problems: 622
On the board are written the natural numbers from $1$ to $2001$ inclusive. You have to delete some numbers so that among those that remain undeleted it is impossible to choose two different numbers such that the result of their multiplication is equal to one of the numbers that remain undeleted. What is the minimum number of numbers that must be deleted? For that amount, present an example showing which numbers are erased. Justify why, if fewer numbers are deleted, the desired property is not obtained.
Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps.
Find those pair of integers $(k,n)$ for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.
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]
Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)?
[i]Proposed by I. Bogdanov, K. Knop[/i]
$60$ symbols, each of which is either $X$ or $O$, are written consecutively on a strip of paper. This strip must then be cut into pieces with each piece containing symbols symmetric about their centre, e.g. $O, XX, OXXXXX, XOX$, etc.
(a) Prove that there is a way of cutting the strip so that there are no more than $24$ such pieces.
(b) Give an example of such an arrangement of the signs for which the number of pieces cannot be less than $15$.
(c) Try to improve the result of (b).
Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.
On a table lie $289$ coins that form a square array $17 \times 17$. All coins are facing with the crown up. In one move, it is possible to reverse any five coins lying in a row: vertical, horizontal or diagonal. Is it possible that after a number of such moves, all the coins to be arranged with tails up?
Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.
There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.
A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim?
(A. Andjans, Riga)
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened.
Decide whether there exists a strategy for player $A$ to win in a finite number of moves.
Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below.
[img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img]
After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy?
[i]Proposed by Bogdan Rublov[/i]
Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.
If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer.
[b]Comment.[/b] Batman and Robin decide their common strategy at the beginning.
[img]https://i.imgur.com/PeLBQNt.png[/img]
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
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 $x$. )
Rodolfo and Gabriela have $9$ chips numbered from $1$ to $9$ and they have fun with the following game: They remove the chips one by one and alternately (until they have $3$ chips each), with the following rules:
$\bullet$ Rodolfo begins the game, choosing a chip and in the following moves he must remove, each time, a chip three units greater than the last chip drawn by Gabriela.
$\bullet$ Gabriela, on her turn, chooses a first chip and in the following times she must draw, each time, a chip two units smaller than the last chip that she herself drew.
$\bullet$ The game is won by whoever gets the highest number by adding up their three tokens.
$\bullet$ If the game cannot be completed, a tie is declared.
If they play without making mistakes, how should Rodolfo play to be sure he doesn't lose?
Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way:
[list]
[*] At each round, you can have the guinea pigs taste any number of juice boxes.
[*] Conversely, a juice box can be tasted by any number of guinea pigs.
[*] After the round ends, any guinea pigs who tasted the poisoned juice die.
[/list]
Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?
Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.)
Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin.
Show that Amber can win the game, no matter which moves Brian makes.
Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)?
[i]Proposed by I. Bogdanov, K. Knop[/i]
Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins.
Find if one of the players has a winning strategy, and if so which player has a winning strategy.
[i]Théo Lenoir, France[/i]
Humberto and Luciano use the break between classes to have fun with the following game: Humberto writes a list of distinct positive integers on a green sheet of paper and hands it to Luciano. Luciano then writes on a board all the possible sums, without repetitions, of one or more different numbers written on the green sheet. For example, if Humberto writes $1$, $3$ and $4$ on the green sheet, Luciano will write $1$, $3$, $4$, $5$, $7$ and $8$ on the board.
(a) Let $n$ be a positive integer. Determine all positive integers $k$ such that Humberto can write a list of $n$ numbers on the green sheet in order to guarantee that Luciano will write exactly $k$ numbers on the board.
(b) Luciano now decides to write a list of $m$ distinct positive integers on a yellow sheet of paper. Determine the smallest positive integer $m$ such that it is possible for Luciano to write this list so that, for any list that Humberto writes on the green sheet, with a maximum of $2023$ numbers, not all the numbers on the yellow sheet will be written on the board.
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win?
[i]Proposed by Mykhailo Shtandenko[/i]
Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i] triple.