Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that $|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$, with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.

Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.

Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met: (i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ; (ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$. Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.