Problems

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.

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 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

Each of the three axes has one rotating pin and a fixed arrow. The gears are connected in series. On the first gear there are 33 teeth, on the second – 10, on the third – 7. On each tooth of the first gear one symbol or letter of the following string of letters and symbols is written in the clockwise direction in the following order:

A B V C D E F G H I J K L M N O P Q R S T U W X Y Z ! ? \(>\) \(<\) $ £ €

On the teeth of the second and third gears in increasing order the numbers 0 to 9 and 0 to 6 are written respectively in a clockwise direction. When the arrow of the first axis points to a letter, the arrows of the other two axes point to numbers.

The letters and symbols of the message are encrypted in sequence. Encryption is performed by rotating the first gear anti-clockwise until the first possible letter or symbol that can be encrypted is landed on by the arrow. At this point, the numbers indicated by the second and third arrows are consistently written out. At the beginning of the encryption, the 1st wheel points to the letter A, and the arrows of the 2nd and 3rd wheels to the number 0.

Encrypt the Slavic name OLIMPIADA.

A message is encrypted using numbers where each number corresponds to a different letter of the alphabet. Decipher the following encoded text:

1317247191772413816720713813920257178

A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.

Each side in the triangle \(ABC\) is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division (the points \(A\), \(B\), \(C\) cannot be the vertices of triangles) in which neither side is parallel to either side of the triangle \(ABC\)?

How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?

a) One person had a basement illuminated by three electric bulbs. Switches of these bulbs are located outside the basement, so that having switched on any of the switches, the owner has to go down to the basement to see which lamp switches on. One day he came up with a way to determine for each switch which bulb it switched on, descending into the basement exactly once. What is the method?

b) If he goes down to the basement exactly twice, how many bulbs can he identify the switches for?

Peter thought of a number between 1 to 200. What is the fewest number of questions for which you can guess the number if Peter answers

a) “yes ” or “no”;

b) “yes”, “no” or “I do not know”

for every question?