A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).
There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?
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 message is encrypted by replacing the letters of the source text with pairs of digits according to some table (known only to the sender and receiver) in which different letters of the alphabet correspond to different pairs of digits. The cryptographer was given the task to restore the encrypted text. In which case will it be easier for him to perform the task: if it is known that the first word of the second line is a “thermometer” or that the first word of the third line is “smother”? Justify your answer. (It is assumed that the cryptographic table is not known).
A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?
In one urn there are two white balls, in another two black ones, in the third – one white and one black. On each urn there was a sign indicating its contents: WW, BB, WB. Someone rehung the signs so that now each sign indicating the contents of the urn is incorrect. It is possible to remove a ball from any urn without looking into it. What is the minimum number of removals required to determine the composition of all three urns?
Around a table sit boys and girls. Prove that the number of pairs of neighbours of different sexes is even.
Could the difference of two integers multiplied by their product be equal to the number 1999?
a) There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?
b) The same question, if there are 20 coins, but you are allowed to turn over 19.