All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
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).
Prove that there is no polyhedron that has exactly seven edges.
Does a continuous function that takes every real value exactly 3 times exist?
To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.
Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?
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.