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?
Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
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 \(y\).
The functions \(f\) and \(g\) are defined on the entire number line and are reciprocal. It is known that \(f\) is represented as a sum of a linear and a periodic function: \(f (x) = kx + h (x)\), where \(k\) is a number, and \(h\) is a periodic function. Prove that \(g\) is also represented in this form.