Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
In a basket, there are 30 mushrooms. Among any 12 of them there is at least one brown one, and among any 20 mushrooms, there is at least one chanterelle. How many brown mushrooms and how many chanterelles are there in the basket?
So, the mother exclaimed - “It’s a miracle!", and immediately the mum, dad and the children went to the pet store. “But there are more than fifty bullfinches here, how will we decide?,” the younger brother nearly cried when he saw bullfinches. “Don’t worry,” said the eldest, “there are less than fifty of them”. “The main thing,” said the mother, “is that there is at least one!". “Yes, it’s funny,” Dad summed up, “of your three phrases, only one corresponds to reality.” Can you say how many bullfinches there was in the store, knowing that they bought the child a bullfinch?
Elephants, rhinoceroses, giraffes. In all zoos where there are elephants and rhinoceroses, there are no giraffes. In all zoos where there are rhinoceroses and there are no giraffes, there are elephants. Finally, in all zoos where there are elephants and giraffes, there are also rhinoceroses. Could there be a zoo in which there are elephants, but there are no giraffes and no rhinoceroses?
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?
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 some parts of the world, people write the date as follows: the number of the month, then the number of the day and finally the year. In other parts of the world, the number of the day comes first, then the month and finally the year. In one year, how many dates can be understood without knowing which of the two systems is being used?”