Problems

A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?

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.

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?

Let \(x\) be a natural number. Among the statements:

\(2x\) is more than 70;

\(x\) is less than 100;

\(3x\) is greater than 25;

\(x\) is not less than 10;

\(x\) is greater than 5;

three are true and two are false. What is \(x\)?

It is known that any person has at most 400,000 hairs on their head. Given that the population of London is not less than 8 million, prove that there are 20 Londoners with the same number of hairs on their heads.

The key of the cipher, called the “lattice”, is a rectangular stencil of size 6 by 10 cells. In the stencil, 15 cells are cut out so that when applied to a rectangular sheet of paper of size 6 by 10, its cut-outs completely cover the entire area of the sheet in four possible ways. The letters of the string (without spaces) are successively entered into the cut-outs of the stencil (in rows, in each line from left to right) at each of its four possible positions. Find the original string of letters if, after encryption, the following text appeared in the sheet of paper