In a given group of people $\mathcal{F}$, each member has at least two acquaintances from $\mathcal{F}$. Moreover, for each cycle $A_{1} \leftrightarrow A_{2} \leftrightarrow ... \leftrightarrow A_{n} \leftrightarrow A_{1}$ in $\mathcal{F}$ (here '$X \leftrightarrow Y$' means that $X$ and $Y$ are acquaintances), each $A_i$ knows exactly two other members $A_j$ of the cycle. Prove that there exist $X, Y \in \mathcal{F}$ such that each of them has exactly two acquaintances in $\mathcal{F}$, and $X, Y$ have at least one common acquaintance in the group. Proposed by Mirko Petrusevski

Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point. Proposed by Petar Filipovski

Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$. Authored by Nikola Velov

Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$and determine all cases of equality.