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?
There are 40 identical cords. If you set any cord on fire on one side, it burns, and if you set it alight on the other side, it will not burn. Ahmed arranges the cords in the form of a square (see the figure below, each cord makes up a side of a cell). Then, Helen arranges 12 fuses. Will Ahmed be able to lay out the cords in such a way that Helen will not be able to burn all of them?
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?
A box contains 111 red, blue, green, and white marbles. It is known that if we remove 100 marbles from the box, without looking, we will always have removed at least one marble of each colour. What is the minimum number of marbles we need to remove to guarantee that we have removed marbles of 3 different colours?
A box contains 100 red, blue, and white marbles. It is known that if we remove 26 marbles from the box, without looking, we will always have removed at least 10 marbles of one colour. What is the minimum number of marbles we need to remove to guarantee that we have removed 30 marbles of the same colour?
Author: A.V. Shapovalov
We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.
What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?
Harry thought of two positive numbers \(x\) and \(y\). He wrote down the numbers \(x + y\), \(x - y\), \(xy\) and \(x/y\) on a board and showed them to Sam, but did not say which number corresponded to which operation.
Prove that Sam can uniquely figure out \(x\) and \(y\).
Author: Shapovalov A.V.
Let \(A\) and \(B\) be two rectangles. From rectangles equal to \(A\), a rectangle similar to \(B\) was created.
Prove that from rectangles equal to \(B\), you can create a rectangle similar to \(A\).
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?