Found problems: 83
2018 All-Russian Olympiad, 5
On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?
2016 Iran MO (3rd Round), 2
A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps).
In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$
First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them,
Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw.
Assume that both players play at their best. What will be the result of the game ?
[i]Proposed by Mahyar Sefidgaran[/i]
2025 JBMO TST - Turkey, 2
Let $n$ be a positive integer. Aslı and Zehra are playing a game on an $n\times n$ grid. Initially, $10n^2$ stones are placed on some of the unit squares of this grid.
On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.
For which values of $n$, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
2020 LIMIT Category 1, 8
Kunal and Arnab play a game as follows. Initially there are $2$ piles of coins with $x$ and $y$ coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of $(x,y)$ which ensure Kunal's victory against Arnab given both os them play optimally.
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[i]You are required to find an exhaustive set of solutions[/i]
2003 Baltic Way, 8
There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?
2002 BAMO, 3
A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?
2018 IFYM, Sozopol, 7
$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.
2022 Switzerland Team Selection Test, 8
Johann and Nicole are playing a game on the coordinate plane. First, Johann draws any polygon $\mathcal{S}$ and then Nicole can shift $\mathcal{S}$ to wherever she wants. Johann wins if there exists a point with coordinates $(x, y)$ in the interior of $\mathcal{S}$, where $x$ and $y$ are coprime integers. Otherwise, Nicole wins. Determine who has a winning strategy.
2022 JHMT HS, 3
Andy, Bella, and Chris are playing in a trivia contest. Andy has $21,200$ points, Bella has $23,600$ points, and Chris has $11,200$ points. They have reached the final round, which works as follows:
[list]
[*] they are given a hint as to what the only question of the round will be about;
[*] then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on);
[*] then, they will be shown the question, where they will have $30$ seconds to individually submit a response (a player does not know any of the other players' answers);
[*] finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided).
[/list]
Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.
2021 Malaysia IMONST 2, 2
Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.
2020 OMpD, 2
A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?
2021 JHMT HS, 7
A number line with the integers $1$ through $20,$ from left to right, is drawn. Ten coins are placed along this number line, with one coin at each odd number on the line. A legal move consists of moving one coin from its current position to a position of strictly greater value on the number line that is not already occupied by another coin. How many ways can we perform two legal moves in sequence, starting from the initial position of the coins (different two-move sequences that result in the same position are considered distinct)?
2021 Serbia JBMO TSTs, 3
Two players play the following game: alternatively they write numbers $1$ or $0$ in the vertices of an $n$-gon.
First player starts the game and wins if after any of his moves there exists a triangle, whose vertices are three consecutive vertices of the $n$-gon, such that the sum of numbers in it's vertices is divisible by $3$.
Second player wins if he prevents this.
Determine which player has a winning strategy if:
a) $n=2019$
b) $n=2020$
c) $n=2021$
2017 CentroAmerican, 1
The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules:
i) The endpoints must coincide with vertices of the marked equilateral triangles.
ii) The segment must be made up of one or more of the sides of the triangles.
iii) The segment cannot contain any point (endpoints included) of a previously colored segment.
Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.
2019 Baltic Way, 6
Alice and Bob play the following game. They write the expressions $x + y$, $x - y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables $x$ and $y$, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable.
Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result, wins. Which player, if any, has a winning strategy?
2023 Tuymaada Olympiad, 4
Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?
2019 IFYM, Sozopol, 2
Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.
2016 Tournament Of Towns, 6
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
[i](Anant Mudgal)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
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.
2016 IFYM, Sozopol, 6
Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?
2022 Iran MO (3rd Round), 5
Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it away. This action repeats until when there is no remaining card for Ali.
Amin wants to pick cards in a way that the sum of the number of his cards is maximized and Ali wants to choose cards in a way that the sum of the number of Amin's cards is minimized. Find the most value of $k$ such that Amin can play in a way that is sure the sum of the number of his cards will be at least equal to $k$.
2016 Germany Team Selection Test, 3
In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on.
If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won.
Prove that Kain can force a win in a finite number of moves.
2016 India Regional Mathematical Olympiad, 6
A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.
2024 Brazil EGMO TST, 2
Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy.
[b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.
2021 Taiwan TST Round 1, C
Let $n$ and $k$ be positive integers satisfying $k\leq2n^2$. Lee and Sunny play a game with a $2n\times2n$ grid paper. First, Lee writes a non-negative real number no greater than $1$ in each of the cells, so that the sum of all numbers on the paper is $k$. Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most $1$. There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.)
Let $M$ be the number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize $M$. Find the value of $M$ when Lee and Sunny both play optimally.