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.
There are two sets of numbers made up of 1s and \(-1\)s, and in each there are 2022 numbers. Prove that in some number of steps it is possible to turn the first set into the second one if for each step you are allowed to simultaneously change the sign of any 11 numbers of the starting set. (Two sets are considered the same if they have the same numbers in the same places.)
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.
The White Rook pursues a black horse on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play in order to take the horse? White makes the first move.
In a set there are 100 weights, each two of which differ in mass by no more than 20 g. Prove that these weights can be put on two cups of weighing scales, 50 pieces on each one, so that one cup of weights is lighter than the other by no more than 20 g.
Peter bought an automatic machine at the store, which for 5 pence multiplies any number entered into it by 3, and for 2 pence adds 4 to any number. Peter wants, starting with a unit that can be entered free of charge to get the number 1981 on the machine number whilst spending the smallest amount of money. How much will the calculations cost him? What happens if he wants to get the number 1982?
In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number (different letters correspond to different numbers). It is known that the products of the numbers of these two words are equal. Can both numbers be odd?
A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.
There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?