Found problems: 622
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?
Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following:
$$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$
Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if
$a) p = 10^9 + 7$;
$b) p = 10^9 + 9$.
[i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime.
[i]Proposed by Ivan Novak[/i]
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]
There are totally $16$ teams participating in a football tournament, each team playing with every other exactly $1$ time. In each match, the winner gains $3$ points, the loser gains $0$ point and each teams gain $1$ point for the tie match. Suppose that at the end of the tournament, each team gains the same number of points. Prove that there are at least $4$ teams that have the same number of winning matches, the same number of losing matches and the same number of tie matches.
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.
Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ?
Ivan Mitrofanov
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?
Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:
- Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
- Then Viktor without seeing the table, chooses a positive integer $n$;
- After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them;
- Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before.
If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes.
[i]Proposed by Viktor Simjanoski[/i]
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time.
Determine all $k$ such that $A$ can always win the game.
The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge).
[i]Proposed by Norman Do, Australia[/i]
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)
Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.
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
In the game "cat and mouse" the cat chases the mouse in either labyrinth $A, B$ or $C$ .
[img]https://cdn.artofproblemsolving.com/attachments/4/5/429d106736946011f4607cf95956dcb0937c84.png[/img]
The cat makes the first move starting at the point marked "$K$" , moving along a marked line to an adjacent point . The mouse then moves , under the same rules, starting from the point marked "$M$" . Then the cat moves again, and so on . If, at a point of time , the cat and mouse are at the same point the cat eats the mouse.
Is there available to the cat a strategy which would enable it to catch the mouse , in cases $A, B$ and $C$?
(A. Sosinskiy, Moscow)
Initially the number $2000$ is written down. The following operation is repeatedly performed: the sum of the $10$-th powers of the last number's digits is written down. Prove that in the infinite sequence thus obtained, some two numbers will be equal.
Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game.
Prove that Johan can always get a score that is at least as high as Julian’s.
Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?
[i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]
Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards $1, 3, 5,7,..., 2n-1$
b) Bruno has cards $2, 4,6, 8,...,2n$
During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.
Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy?
$Agnijo \ Banerjee \ , United \ Kingdom$
Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician.
Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.
Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written.
They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important.
Thus if Ahmad starts by writing $3$ and $-2$, the first five moves (or steps) are as shown:
(a) Step 1 (Ahmad) $3$ and $-2$
(b) Step 2 (Salem) $1$ and $-6$
(c) Step 3 (Ahmad) $-5$ and $-6$
(d) Step 4 (Salem) $-11$ and $30$
(e) Step 5 (Ahmad) $19$ and $-330$
(i) Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2.
(ii) What pair of integers should Ahmad write so that the game finishes at step 4?
(iii) Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps.
(iv) Ahmad and Salem decide to change the game. The first player writes three numbers on the board, $u, v$ and $w$. The second player then writes the three numbers $u + v + w,uv + vw + wu$ and $uvw$, and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.
Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy?
Proposed by [i]Borche Joshevski, Macedonia[/i]
Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$.
There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line.
Prove that every two players from the opposite teams will eventually play against each other.