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.

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.

You may remember the game Nim. We will now play a slightly modified version, called Thrim. In Thrim, there are two piles of stones (or any objects of your choosing), one of size $1$ and the other of size $5$.
Whoever takes the last stone wins. The players take it in turns to remove stones - they can only remove stones from one pile at a time, and they can remove at most $3$ stones at a time.
Does the player going first or the player going second have a winning strategy?

We meet a group of people, all of whom are either knights or liars. Knights always tell the truth and liars always lie. Prove that it’s impossible for someone to say “I’m a liar".

We’re told that Leonhard and Carl are knights or liars (the two of them could be the same or one of each). They have the following conversation.

Leonhard says “If $49$ is a prime number, then I am a knight."

Carl says “Leonhard is a liar".
Prove that Carl is a liar.

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 $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$.

$A,B$ and $C$ are three sets. Prove that if we take an intersection $A\cap B$ and unite it with the set $C$, we will get the same set as if we took an intersection of two unions $A\cup C$ and $B\cup C$. Essentially, prove that $(A\cap B)\cup C=(A\cup C)\cap (B\cup C)$. Draw a Venn diagram for the set $(A\cap B)\cup C$.

Let $A,B$ and $C$ be three sets. Prove that if we take an intersection $A\cap B$ and intersect it with the set $C$, we will get the same set as if we took an intersection of $A$ with $B\cap C$. Essentially, prove that it does not matter where to put the brackets in $(A\cap B)\cap C=A\cap (B\cap C)$. Draw a Venn diagram for the set $A\cap B\cap C$.
Prove the same for the union $(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C$.