Prove that there is no polyhedron that has exactly seven edges.
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.
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.
Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?