Problems

Several natives of an island met up (each either a liar or a knight), and everyone said to everyone else: “You are all liars.” How many knights were there among them?

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

In a 10-storey house, 1 person lives on the first floor, 2 on the second floor, 3 on the third, 4 on the fourth, ..., 10 on the tenth. On which floor does the elevator stop most often?

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Solve the equation \(\lfloor x^3\rfloor + \lfloor x^2\rfloor + \lfloor x\rfloor = \{x\} - 1\).

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice (without the Pythagorean theorem).

The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).