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\(^2\)?
Cut an equilateral triangle into 4 smaller equilateral triangles. Then can another equilateral triangle be cut into 7 smaller equilateral triangles (triangles do not necessarily have to be identical)?
Back in the days when a young mathematician was even younger he could only draw digits “4” and “7”. While looking through the old notes his mother found one piece of paper on which he wrote the numbers with digit sums equal to 18, 22 and 26. Which numbers could be written on this piece of paper?
Michael decided to buy new equipment for his daily exercises. There is a wide choice of barbells in the sports shop. All of them weigh an integer amount of kilograms. He recently got his job so he is a bit stingy and wants to buy as few barbells as possible. Michael has only one condition about the weights: he wants to be able to lift any integer amount of kilograms from 1 kg to 15 kg. What is the smallest amount of barbells he needs to buy and how many kilograms do they have to weigh?