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)?
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) 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.)
I don’t know how the figure below can be made of several \(1\times5\) rectangles which do not overlap. I am willing to pay \(1\) pound if you show me a possible way of doing that which I have not seen before. What is the maximal amount of money a person can earn by solving this problem?
One gambler had a pair of dice. Rolling them was something that kept him concentrated. As a result of frequent usage all the numbers were wiped off from both of the dice. In January the gambler went through a rough patch and decided to take a break from gambling. He understood he could not rely only on his luck which has recently failed him. Therefore, our gambler started doing mathematical puzzles to master his mind. The first puzzle is to paint digits on each side of both dice (one digit per one side) in such a way that any natural number between 1 and 31 inclusive can be obtained by putting one dice next to the other. We do not allow the digit “6” to be used as the digit “9” and vice versa. Is there any solution to this problem?