Problems

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

I don’t know how the figure below can be made of several $1\times 5$ 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?

There is a counter on the chessboard. Two in turn move the counter to an adjacent on one side cell. It is forbidden to put a counter on a cell, which it has already visited. The one who can not make the next turn loses. Who wins with the right strategy?

There are $100$ people standing in line, and one of them is Arthur. Everyone in the line is either a knight, who always tells the truth, or a liar who always lies. Everyone except Arthur said, "There are exactly two liars between Arthur and me." How many liars are there in this line, if it is known that Arthur is a knight?

We wish to paint the $15$ segments in the picture below in three colours. We want it such that no two segments of the same colour have a common end. For example, you cannot have both $AB$ and $BC$ blue since they share the end $B$. Is such a painting possible?

There are infinitely many couples at a party. Each pair is separated to form two queues of people, where each person is standing next to their partner. Suppose the queue on the left has the property that every nonempty collection of people has a person (from the collection) standing in front of everyone else from that collection. A jester comes into the room and joins the right queue at the back after the two queues are formed.

Each person in the right queue would like to shake hand with a person in the left queue. However, no two of them would like to shake hand with the same person in the left queue. If $p$ is standing behind $q$ in the right queue, $p$ will only shake hand with someone standing behind $q$’s handshake partner. Show that it is impossible to shake hands without leaving out someone from the left queue.

There are $n$ balls labelled 1 to $n$. If there are $m$ boxes labelled 1 to $m$ containing the $n$ balls, a legal position is one in which the box containing the ball $i$ has number at most the number on the box containing the ball $i+1$, for every $i$.

There are two types of legal moves: 1. Add a new empty box labelled $m+1$ and pick a box from box 1 to $m+1$, say the box $j$. Move the balls in each box with (box) number at least $j$ up by one box. 2. Pick a box $j$, shift the balls in the boxes with (box) number strictly greater than $j$ down by one box. Then remove the now empty box $m$.

Prove it is possible to go from an initial position with $n$ boxes with the ball $i$ in the box $i$ to any legal position with $m$ boxes within $n+m$ legal moves.

Let $A=\{1,2,3\}$ and $B=\{2,4\}$ be two sets containing natural numbers. Find the sets: $A\cup B$, $A\cap B$, $A-B$, $B-A$.

Let $A=\{1,2,3,4,5\}$ and $B=\{2,4,5,7\}$ be two sets containing natural numbers. Find the sets: $A\cup B$, $A\cap B$, $A-B$, $B-A$.

Given three sets $A,B,C$. Prove that if we take a union $A\cup B$ and intersect it with the set $C$, we will get the same set as if we took a union of $A\cap C$ and $B\cap C$. Essentially, prove that $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$.