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?
Consider another equilateral triangle. Is it possible to cut it into (a) 9; (b) 16; (c) 28; (d) 2; (e) 42 smaller equilateral triangles (which are not necessarily identical)?
(f) Kyle claims he can cut an equilateral triangle into any number of smaller (not necessarily identical) equilateral triangles if this number is either greater than 8 and divisible by 3, or greater than 3 and has remainder 1 when divided by 3. Prove or disprove Kyle’s statement.
(g)* Let \(n\) be a natural number greater than 5. Is it true one can cut an equilateral triangle into \(n\) smaller equilateral triangles?
a) It seems that the young mathematician was making progress quite fast. On the back side of that piece of paper there are numbers with digits adding up to all natural numbers from 18 to 33. And yet all of them consist of only digits “4” and “7”. Make your own list of that kind.
(b) Is it true that any natural number greater than 17 can be equal to the digit sum of some number written with digits “4” and “7”?
(c) Now let’s try the same question for digits “5” and “8”. What values can you get if you consider the sum of the digits of a number written with the help of digits “5” and “8”?
(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?
a) There are six points on a plane. No matter which five points you choose you can cross them with two lines but one cannot find two lines which cross all six of them. Does such configuration exist?
(b) One extremely successful businesswoman is planning to build a garden in her country house. She wants to have 10 garden beds and several lanes built. She requested her architect to organize the garden in such a way that for every nine beds there are three lanes passing by them (for each garden bed out of these nine beds there is a lane among the three lanes which passes by it). On top of that she demanded that there should not be three lanes which pass by all 10 garden beds. How can the poor architect satisfy this requirement? All lanes have to be straight.
(c) A neighbour of the businesswoman is inspired by her exotic demands. He decides to surpass her on this field. The neighbour plans to build 55 garden beds. They have to be joined by several lanes in such a way that for every 54 garden beds you can find nine lanes crossing them (for each garden bed out of these 54 beds there is a lane among the nine lanes which crosses this bed). Can you help the colleague of the architect? Again all the lanes have to be straight.
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?