A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.
\(N\) points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the \(k\) colours. Prove that if \(N > \lfloor k!e\rfloor\), then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.
A square is divided into triangles (see the figure). How many ways are there to paint exactly one third of the square? Small triangles cannot be painted partially.
a) There are three identical large vessels. In one there are 3 litres of syrup, in the other – 20 litres of water, and the third is empty. You can pour all the liquid from one vessel into another or into a sink. You can choose two vessels and pour into one of them liquid from the third, until the liquid levels in the selected vessels are equal. How can you get 10 litres of diluted 30% syrup?
b) The same, but there is \(N\) l of water. At what integer values of \(N\) can you get 10 liters of diluted 30% syrup?
Monica is in a broken space buggy at a distance of 18 km from the Lunar base, in which Rachel sits. There is a stable radio communication system between them. The air reserve in the space buggy is enough for 3 hours, in addition, Monica has an air cylinder for the spacesuit, with an air reserve of 1 hour. Rachel has a lot of cylinders with an air supply of 2 hours each. Rachel can not carry more than two cylinders at the same time (one of them she uses herself). The speed of movement on the Moon in the suit is 6 km/h. Could Rachel save Monica and not die herself?
301 schoolchildren came to the school’s New Year’s party in the city of Moscow. Some of them always tell the truth, and the rest always lie. Each of some 200 students said: “If I leave the hall, then among the remaining students, the majority will be liars.” Each of the other schoolchildren said: “If I leave the room, then among the remaining students, there will be twice as many liars as those who speak the truth.” How many liars were at the party?
Two play the following game. There is a pile of stones. The first takes either 1 stone or 10 stones with each turn. The second takes either m or n stones with every turn. They take turns, beginning with the first player. He who can not make a move, loses. It is known that for any initial quantity of stones, the first one can always play in such a way as to win (for any strategy of the second player). What values can m and n take?
On the left bank of the river, there were 5 physicists and 5 chemists. All of them need to cross to the right bank. There is a two-seater boat. On the right bank at any time there can not be exactly three chemists or exactly three physicists. How do they all cross over by making 9 trips to the right side?
A group of children from two classes came to an after school club: Jack, Ben, Fred, Louis, Claudia, Janine and Charlie. To the question: “How many of your classmates are here?” everyone honestly answered with either “Two” or “Three”. But the boys thought that they were only being asked about the boy classmates, and the girls correctly understood that they were asking about everyone. Is Charlie a boy or a girl?