Problems

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?

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?

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

a) ${\phi }^3\le (\alpha -\phi )(\beta -\phi )(\gamma -\phi )$ ;

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

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?

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) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex 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 color. 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.

Two players in turn put coins on a round table, in such a way that they do not overlap. The player who can not make a move loses.

Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence $1,2,\dots ,100,101$. After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.

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?

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?