A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and gets the following answers: $A_1$: On this island there is at least one scoundrel, $A_2$: On this island there are at least two scoundrels, $...$ $A_{n-1}$: On this island there are at least $n-1$ scoundrels, $A_n$: On this island everybody is a scoundrel. Can the journalist decide whether there are more scoundrels or more knights?

Alice and Bob play a game. Initially, they write the pair $(1012,1012)$ on the board. They alternate their turns with Alice going first. In each turn the player can turn the pair $(a,b)$ to either $(a-2, b+1), (a+1, b-2)$ or $(a-1, b)$ as long as the resulting pair has only nonnegative values. The game terminates, when there is no legal move possible. Alice wins if the game terminates at $(0,0)$ and Bob wins if the game terminates at $(0,1)$. Determine who has the winning strategy? [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

Mom brought Andriy and Olesya $4$ balls with the numbers $1, 2, 3$ and $4$ written on them (one on each ball). She held $2$ balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number. Does the mother know with certainty, which child has the ball with the higher number? [i]Proposed by Bogdan Rublov[/i]

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.

A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.

For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit $1$. Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\geqslant k$. Let $X$ denote the set of integers $\{ 0,1,\dotsc ,2^n-1\}$, let $Y$ denote the set of integers $\{ 0,1,\dotsc ,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\leqslant i\leqslant k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$. Player $B$ wins the game, if for every $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant k$, $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy? [i]Proposed by Levente Bodnár, Cambridge[/i]

Let $n\ge 3$ be an integer. Lucas and Matías play a game in a regular $n$-sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice. a) Determine all the $n\ge 3$ such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías. b) Determine all the $n\ge 3$ such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas. Note. The two players know the value of $n$ and see the polygon.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.