JBMO 2019

Find all prime numbers $p$ for which there exist positive integers $x, y$ and $z$ such that the number

$$x^{p}+y^{p}+z^{p}-x-y-z$$

is a product of exactly three distinct prime numbers.

Let $a, b$ be two distinct real numbers and let $c$ be a positive real number such that

$$a^{4}-2019 a=b^{4}-2019 b=c$$

Prove that $-\sqrt{c}<a b<0$.

Triangle $A B C$ is such that $A B<A C$. The perpendicular bisector of side $B C$ intersects lines $A B$ and $A C$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $A B C$, and let $M$ and $N$ be the midpoints of segments $B C$ and $P Q$, respectively. Prove that lines $H M$ and $A N$ meet on the circumcircle of $A B C$.

A $5 \times 100$ table is divided into 500 unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.