Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.

Define a colouring in tha plane the following way: - we pick a positive integer $m$; - let $K_{1}$, $K_{2}$, ..., $K_{m}$ be different circles with nonzero radii such that $K_{i}\subset K_{j}$ or $K_{j}\subset K_{i}$ if $i \neq j$; - the points in the plane that lie outside an arbitrary circle (one that is amongst the circles we pick) are coloured differently than the points that lie inside the circle. There are $2019$ points in the plane such that any $3$ of them are not collinear. Determine the maximum number of colours which we can use to colour the given points.

Let the real numbers $a$, $b$, and $c$ satisfy the equations $(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$. Prove that at least one of $a$, $b$, and $c$ equals $0$.

Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.