Found problems: 29
2016 Kyiv Mathematical Festival, P3
Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What
maximal number of the cells can remain unpainted when the game stops?
2019 Bulgaria EGMO TST, 3
$A$ 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]
2014 BAMO, 5
A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.
2020 Peru Cono Sur TST., P8
Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it.
What number will be written if both play optimally?
2024 Iran MO (2nd Round), 2
Sahand and Gholam play on a $1403\times 1403$ table. Initially all the unit square cells are white. For each row and column there is a key for it (totally 2806 keys). Starting with Sahand players take turn alternatively pushing a button that has not been pushed yet, until all the buttons are pushed. When Sahand pushes a button all the cells of that row or column become black, regardless of the previous colors. When Gholam pushes a button all the cells of that row or column become red, regardless of the previous colors. Finally, Gholam's score equals the number of red squares minus the number of black squares and Sahand's score equals the number of black squares minus the number of red squares. Determine the minimum number of scores Gholam can guarantee without if both players play their best moves.
2020 Brazil Undergrad MO, Problem 5
Let $N$ a positive integer.
In a spaceship there are $2 \cdot N$ people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows:
1) The first alien chooses any person as she wishes.
2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn.
3) The alien that can't play in her turn loses.
Prove that second player has a winning strategy [i]if, and only if[/i], the $2 \cdot N$ people can be divided in $N$ pairs in such a way that two people in the same pair are friends.
2021 Israel TST, 1
An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of
\[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\]
How many guesses are needed for Banana to figure out the quadruple Ana chose?
2016 Brazil National Olympiad, 3
Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game:
Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\).
If someone can't play, he loses.
We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player.
Else, we call it [i]loser[/i].
Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)
1998 Brazil National Olympiad, 3
Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses.
For each $(n, d)$ which player has a winning strategy?
1961 All-Soviet Union Olympiad, 5
Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods:
[i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts.
[i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut.
Finally, stage 3 varies among the three methods as follows:
[i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps.
[i]Method 2:[/i] Nickolas takes the two middle size heaps.
[i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice.
Determine the most and the least profitable method for Nickolas.
2019 Peru Cono Sur TST, P5
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.
[b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal.
[b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.
2021 Olympic Revenge, 4
On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement.
Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?
2018 Brazil National Olympiad, 2
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.
[b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal.
[b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.
2016 CentroAmerican, 4
The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$, the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.
2005 Colombia Team Selection Test, 6
$A$ 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]
2024 Belarus Team Selection Test, 3.3
Olya and Tolya are playing a game on $[0,1]$ segment. In the beginning it is white. In the first round Tolya chooses a number $0 \leq l \leq 1$, and then Olya chooses a subsegment of $[0,1]$ of length $l$ and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts $2024$ rounds. Let $L$ be the sum of length of white segments after the end of the game. If $L > \frac{1}{2}$ Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win?
[i]A. Naradzetski[/i]
1989 Brazil National Olympiad, 4
A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second.
At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A.
B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins.
[list=a]
[*] Which player has a winning strategy?
[*] Suppose now both players has $k$ chips numbered 1 to $k$. Which player has a winning strategy?
[*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy?
[/list]
JOM 2013, 4.
Let $n$ be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him.
After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when:
(a) $n=2013^{2012}$
(b) $n=2013^{2013}$
2014 BAMO, 3
Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are $20$ cookies on a red plate and $14$ on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.
2004 IMO Shortlist, 5
$A$ 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]
2017 Benelux, 2
Let $n\geq 2$ be an integer. Alice and Bob play a game concerning a country made of $n$ islands. Exactly two of those $n$ islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands $I_1$ and $I_2$ such that:
$\bullet$ $I_1$ and $I_2$ are not already connected by a bridge.
$\bullet$ at least one of the two islands $I_1$ and $I_2$ is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.)
As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each $n\geq 2$) who has a winning strategy.
([i]Note:[/i] It is allowed to construct a bridge passing above another bridge.)
2004 Bosnia and Herzegovina Team Selection Test, 4
On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.
2000 Saint Petersburg Mathematical Olympiad, 9.5
The numbers $1,2,\dots,2000$ are written on the board. Two players are playing a game with alternating moves. A move consists of erasing two number $a,b$ and writing $a^b$. After some time only one number is left. The first player wins, if the numbers last digit is $2$, $7$ or $8$. If not, the second player wins. Who has a winning strategy?
[I]Proposed by V. Frank[/i]
2018 SIMO, Bonus
Simon plays a game on an $n\times n$ grid of cells. Initially, each cell is filled with an integer. Every minute, Simon picks a cell satisfying the following:
[list]
[*] The magnitude of the integer in the chosen cell is less than $n^{n^n}$
[*] The sum of all the integers in the neighboring cells (sharing one side with the chosen cell) is non-zero
[/list]
Simon then adds each integer in a neighboring cell to the chosen cell.
Show that Simon will eventually not be able to make any valid moves.
2016 Lusophon Mathematical Olympiad, 4
$8$ CPLP football teams competed in a championship in which each team played one and only time with each of the other teams. In football, each win is worth $3$ points, each draw is worth $1$ point and the defeated team does not score. In that championship four teams were in first place with $15$ points and the others four came in second with $N$ points each. Knowing that there were $12$ draws throughout the championship, determine $N$.