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?
A company board has 4 people: one chair and three ordinary members. Each month they meet to discuss how much they will get paid for serving on the board. This is how the meetings work:
The chair proposes the new pay for all four people.
Only the three ordinary members vote. A member looks at how much their own pay would change (in percentage). They vote YES if their change is at least as large as the change of every other person (ties allowed). Otherwise they vote NO.
If at least two members vote YES, the proposal passes and the new pays take effect. Otherwise nothing changes and they can try again next month.
Is it possible, after some number of meetings, for the chair’s pay to become 10 times larger than it was at the start, while each of the orderinary member’s pay becomes 10 times smaller than their original pay? than at the start?
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?