Problems

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

For three sets $A,B,C$ prove that $A-(B\cup C)=(A-B)\cap (A-C)$. Draw a Venn diagram for this set.

For three sets $A,B,C$ prove that $A-(B\cap C)=(A-B)\cup (A-C)$. Draw a Venn diagram for this set.

How many subsets of $\{1,2,...,n\}$ are there of even size?

In how many ways can $\{1,...,n\}$ be written as the union of two sets? Here, for example, $\{1,2,3,4\}\cup \{4,5\}$ and $\{4,5\}\cup \{1,2,3,4\}$ count as the same way of writing $\{1,2,3,4,5\}$ as a union.

Prove for any natural number $n$ that $(n+1)(n+2)...(2n)$ is divisible by $2^n$.

Between two mirrors $AB$ and $AC$, forming a sharp angle two points $D$ and $E$ are located. In what direction should one shine a ray of light from the point $D$ in such a way that it would reflect off both mirrors and hit the point $E$?
If a ray of light comes towards a surface under a certain angle, it is reflected with the same angle as on the picture.

Consider a set of natural numbers $A$, consisting of all numbers divisible by $6$, let $B$ be the set of all natural numbers divisible by $8$, and $C$ be the set of all natural numbers divisible by $12$. Describe the sets $A\cup B$, $A\cup B\cup C$, $A\cap B\cap C$, $A-(B\cap C)$.

Let $a$, $b$ and $c$ be the three side lengths of a triangle. Does there exist a triangle with side lengths $a+1$, $b+1$ and $c+1$? Does it depend on what $a$, $b$ and $c$ are?

There is a triangle with side lengths $a$, $b$ and $c$. Can you form a triangle with side lengths $\frac{a}{b}$, $\frac{b}{c}$ and $\frac{c}{a}$? Does it depend on what $a$, $b$ and $c$ are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on $a$, $b$ and $c$.
Recall that a triangle can be drawn with side lengths $x$, $y$ and $z$ if and only if $x+y>z$, $y+z>x$ and $z+x>y$.