The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?
Two pirates Bob and John were boasting that they could make the strongest coctail. Bob mixed together rum and gin, and John mixed vodka and port. It is known that rum is stronger than vodka, and gin is stronger than port. Can it be that John’s drink was stronger than Bob’s?
Is it possible that the sum and the product of some given natural numbers is equal to 99?
Selena wrote down some positive numbers. She added up those numbers, and the resulting sum was greater than 10. Then she decided to add up the squares of those numbers. Could it be possible that the sum of the squares of the numbers was less than 0.1?
There were two retired couples Robinsons and Morrises who lived next to each other in a quiet street. They loved animals, especially cats and dogs, but did not consider themselves fit enough to have the actual animals in the house. Instead, they were collecting stamps depicting cats and dogs. Mr Robinson had some stamps with cats and dogs, Mrs Robinson had her own stamps with cats and dogs, and so did Mr and Mrs Morris. It was known that Mrs Robinson had bigger proportion of stamps with cats (the number of stamps with cats to the number of all stamps she owned, i.e. stamps with cats and dogs) than Mrs Morris, and Mr Robinson had bigger proportion of stamps with cats than Mr Morris. Does it mean that the proportion of stamps with cats Mr & Mrs Robinson owned together was larger than proportion of stamps with cats owned by Mr & Mrs Morris?
Anna, Sasha, and India were running races on a sports day. Could it be that Anna was faster than Sasha in more than half of the races, Sasha was faster than India in more than half of the races, and India was faster than Anna in more than half of the races?
Manraj wrote down a fraction, then he added 1 to the nominator and 100 to the denominator of the fraction. Could it be that the new fraction is bigger than the original one?
There are 10 strongman and 10 acrobats performing in a circus. At the beginning of the performance each strongman carried an acrobat to the arena, and at the end of the performance each acrobat carried a strongman offstage. It is known that each strongman carried an acrobat who weighed less than himself. Could it be that
(a) each acrobat carried a strongman lighter than himself? (b) there were nine acrobats each carrying a strongman lighter than himself?
The board of directors of a company consists of 4 people – one chairman and three ordinary members. The board has a meeting each month, where they decide on the amount of compensation each of them receives for serving on the board. According to the procedure the chairman proposes a new compensation scheme for all the members of the board, and all the members except the chairman vote for the new scheme subsequently. It is known that a member of the board votes for the scheme only if his/her compensation increases more or the same than everybody else’s, otherwise he/she votes against the scheme. The decisions are made according to majority rule. Can the chairman increase his/her compensation by 10 times, and simultaneously decrease every other member’s compensation by 10 times after several board meetings?
Is it possible to place several non-overlapping squares inside one big square with side length 1m if
(a) the sum of perimeters of smaller squares is equal to 100 m? (b) the sum of areas of smaller squares is equal to 100 m\(^2\)?