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\(^2\)?
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?
(a) Well, Michael was just a beginner that time. Don’t judge him much. He has made a considerable progress over the last month. Now he is planning to do any integer amount of kilograms from 1 kg to 31 kg. What is the smallest number of barbells one needs to have in order to do such weights?
(b) Michael is doing just fine with weights up to 31 kg. Assume he is getting promotion soon, so he can afford a new set of weights. Can you already suggest which set will be the smallest if he decides to do all integer weights from 1 kg to 63 kg?
(c) From 1 kg to 64 kg?
(d) From 1 kg to 129 kg?
You have a two pan set of scales. You have a black box which weighs a random integer amount of kilograms.
(a) The weight of this box varies from 1 kg to 40 kg. Find a set of 4 integer weights which can be used to determine the weight of the box. You are allowed to put weights on both pans (even next to the black box).
(b) A red box can weight any integer amount of kilograms up to 100 kg. Is there a set of 5 integer weights adding up to 100 kg which allows us to determine the weight of the red box?
(a) A traveller decided to stay in the motel. He has no money but he has a golden chain consisting of 7 links (the chain is not closed). The host agreed on one golden link to be the payment for one day of staying. The traveller wants to stay for the next 7 days. What is the smallest number of links he has to disunite to be able to make the payment every day? (Take into account that the host can give the change “in links” if he already got some from the traveller.)
(b) Assume we have a chain consisting of 23 golden links and now the traveller has to spend 23 days in the motel. Is it enough to disunite 2 links to be able to make the daily payments? As before the host can give the change with the links he gets from the traveller.
(c) Consider 24 links and 24 days now. Can we manage to make daily payments after we disunite some 2 links?
Comment: In all questions above after we disunite the chain at some link in general we obtain three parts: the link itself, the left part of the chain and the right of the chain. Note that there might be no left or no right part.)