7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.
\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
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.