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a) The vertices (corners) in a regular polygon with 10 sides are coloured black and white in an alternating fashion (i.e. one vertice is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same colour. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?

**A two-player game with matches.** There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

At a round table, 2015 people are sitting down, each of them is either a knight or a liar. Knights always tell the truth, liars always lie. They were given one card each, and on each card a number is written; all the numbers on the cards are different. Looking at the cards of their neighbours, each of those sitting at the table said: “My number is greater than that of each of my two neighbors.” After that, k of the sitting people said: “My number is less than that of each of my two neighbors.” At what maximum k could this occur?

Prove that the following inequalities hold for the Brockard angle $ \ varphi $:

a) $ \ varphi ^ {3} _ {} $$ \ le $ ($ \ alpha $ – $ \ varphi $) ($ \ beta $ – $ \ varphi $) ($ \ gamma $ – $ \ varphi $) ;

b) 8 $ \ varphi ^ {3} _ {} $$ \ le $$ \ alpha $$ \ beta $$ \ gamma $ (the Jiff inequality).

A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?

Two people take turns drawing noughts and crosses on a $9 \times 9$ grid. The first player uses crosses and the second player uses noughts. After they finish, the number of rows and columns where there are more crosses than noughts are counted, and these are the points which the first player receives. The number of rows and columns where there are more noughts than crosses are the second player’s points. The player who has the most points is the winner. Who wins, if the right strategy is used?

Consider a rectangular parallelepiped with size a) $4 \times 4 \times 4$; b) $4 \times 4 \times 3$; c) $4 \times 3 \times 3$, made up of unit cubes. Consider the game with two players where: in one turn a player is allowed to pierce through any row with a long wire, as long as there is at least one cube in the row with no wire. The loser is the player who cannot make a move. Who would win, if the right strategy is used?

There are twenty dots distributed along the circumference of circle. Consider the game with two players where: in one move a player is allowed to connect any two of the dots with a chord (aline going through the inside of the circle), as long as the chord does not intersect those previously drawn. The loser is the one who cannot make a move. Which player wins?

There are two piles of rocks: one with 30 rocks and the other with 20 rocks. In one turn a player is allowed to take any number of rocks but only from one of the piles. The loser is the player who has no rocks left to take. Who would win in a two player game, if the right strategy is used?

In each square of an 11×11 board there is a checker. Consider the game with two players where: in one move a player is allowed to take any amount of adjacent checkers from the board, as long as they checkers are in the same vertical column or in the same horizontal row. The winner is the player who removes the last checker. Which player wins the game?

There is a board of $10 \times 10$ squares. Consider the game with two players where: in one turn a player is allowed to cover any two adjacent squares with a domino (a $1 \times 2$ rectangle) so that the domino doesnt cover another domino. The loser is the one who cannot make a move. Which player would win, if the right strategy was used?

Two people take turns placing kings on squares of a $9 \times 9$ chessboard such that the kings cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

Two people take turns placing knights on a chessboard such that the knights cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

There are two piles of rocks, each with 7 rocks. Consider the game with two players where: in one turn you can take any amount of rocks, but only from one pile. The loser is the one who has no rocks left to take.

Two people take turns placing bishops on a chessboard such that the bishops cannot attack each other. Here, the colour of the bishops does not matter. (Note: bishops move and attack diagonally.) Which player wins the game, if the right strategy is used?

There are two stacks of coins on a table: in one of them there are 30 coins, and in the other – 20. You can take any number of coins from one stack per move. The player who cannot make a move is the one that loses. Which player wins with the correct strategy?

On the board the number 1 is written. Two players in turn add any number from 1 to 5 to the number on the board and write down the total instead. The player who first makes the number thirty on the board wins. Specify a winning strategy for the second player.

Two boys play the following game: they take turns placing rooks on a chessboard. The one who wins is the one whose last move leaves all the board cells filled. Who wins if both try to play with the best possible strategy?

On a board of size $8 \times 8$, two in turn colour the cells so that there are no corners of three coloured squares. The player who can’t make a move loses. Who wins with the right strategy?