One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?
Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).
Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).
What is the largest number of elements in \(M\)?
Someone arranged a 10-volume collection of works in an arbitrary order. We call a “disturbance” a situation where there are two volumes for which a volume with a large number is located to the left. For this volume arrangement, we call the number \(S\) the number of all of the disturbances. What values can \(S\) take?
Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?
A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.
In how many ways can you rearrange the numbers from 1 to 100 so that the neighbouring numbers differ by no more than 1?
There are 18 sweets in one piles, and 23 in another. Two play a game: in one go one can eat one pile of sweets, and the other can be divided into two piles. The loser is one who cannot make a move, i.e. before this player’s turn there are two piles of sweets with one sweet in each. Who wins with a regular game?
Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?
In a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).