The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?
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 chequered strip of \(1 \times N\) is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.
Which of the players can always win (no matter how their opponent played)?
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?
Hannah and Emma have three coins. On different sides of one coin there are scissors and paper, on the sides of another coin – a rock and scissors, on the sides of the third – paper and a rock. Scissors defeat paper, paper defeats rock and rock wins against scissors. First, Hannah chooses a coin, then Emma, then they throw their coins and see who wins (if the same image appears on both, then it’s a draw). They do this many times. Is it possible for Emma to choose a coin so that the probability of her winning is higher than that of Hannah?
A rectangular chocolate bar size \(5 \times 10\) is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice \(1\times 1\) (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?
There is a system of equations \[\begin{aligned} * x + * y + * z &= 0,\\ * x + * y + * z &= 0,\\ * x + * y + * z &= 0. \end{aligned}\] Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.
Two players play on a square field of size \(99 \times 99\), which has been split onto cells of size \(1 \times 1\). The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.
A White Rook pursues a black bishop on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play to take the bishop? White makes the first move.
A monkey escaped from it’s cage in the zoo. Two guards are trying to catch it. The monkey and the guards run along the zoo lanes. There are six straight lanes in the zoo: three long ones form an equilateral triangle and three short ones connect the middles of the triangle sides. Every moment of the time the monkey and the guards can see each other. Will the guards be able to catch the monkey, if it runs three times faster than the guards? (In the beginning of the chase the guards are in one of the triangle vertices and the monkey is in another one.)