In a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).
There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?
How many ways can I schedule the first round of the Russian Football Championship, in which 16 teams are playing? (It is important to note who is the host team).
A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.
The following text is obtained by encoding the original message using Caesar Cipher.
WKHVLAWKROBPSLDGRIFUBSWRJUDSKBGHGLFDWHGWKHWRILIWLHWKBHDURIWKHEULWLVKVHFUHWVHUYLFH.
The following text is also obtained from the same original text:
KYVJZOKYFCPDGZRUFWTIPGKFXIRGYPUVUZTRKVUKYVKFWZWKZVKYPVRIFWKYVSIZKZJYJVTIVKJVIMZTV.
Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.
What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.
The key of the cipher, called the “swivelling grid”, is a stencil made from a square sheet of chequered paper of size \(n \times n\) (where \(n\) is even). Some of the cells are cut out. One side of the stencil is marked. When this stencil is placed onto a blank sheet of paper in four possible ways (marked side up, right, down or left), its cut-outs completely cover the entire area of the square, where each cell is found under the cut-out exactly once. The letters of the message, that have length \(n^2\), are successively written into the cut-outs of the stencil, where the sheet of paper is placed on a blank sheet of paper with the marked side up. After filling in all of the cut-outs of the stencil with the letters of the message, the stencil is placed in the next position, etc. After removing the stencil from the sheet of paper, there is an encrypted message.
Find the number of different keys for an arbitrary even number \(n\).
A cryptogram is given:
Restore the numerical values of the letters under which all of the equalities are valid, if different letters correspond to different digits. Arrange the letters in order of increasing numerical value and to find the required string of letters.
The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?