Problems

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?

Find the area of the figures shown below.

Ben and Joe play chess. In addition to a chessboard, they have one rook, which they put in the lower right corner, and they move it in turns. It can only be moved upwards or to the left (for any number of cells). The player who can not make a move, loses. Joe goes first. Who will win with the correct method?

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

a) Prove that in any football team there are two players who were born on the same day of the week.

b) Prove that in the population of London, which is almost 9 million, there will be ten thousand people who celebrate their birthday on the same day.

Kai has a piece of ice in the shape of a “corner” (see the figure). The Snow Queen demanded that Kai cut it into four equal parts. How can he do this?

The king made a test for the future groom of his daughter. He put the princess in one of three rooms, a tiger in the other, and left the last room empty. It is known that the sign on the door where the princess is sitting is true, where the tiger is – it is false, and nothing is known about the sign on the third room. The tablets are as follows:

1 – room 3 is empty

2 – the tiger is in room 1

3 – this room is empty

Can the prince correctly guess the room with the princess?

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.

a) In the construction in the figure, move two matches so that there are five identical squares created. b) From the new figure, remove 3 matches so that only 3 squares remain.

How many squares are shown in the picture?