BxMO 2016

Find the greatest positive integer $N$ with the following property: there exist integers $x_{1}, \ldots, x_{N}$ such that $x_{i}^{2}-x_{i} x_{j}$ is not divisible by 1111 for any $i \neq j$.

Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{Z}$ such that

$$(f(f(y)-x))^{2}+f(x)^{2}+f(y)^{2}=f(y) \cdot(1+2 f(f(y)))$$

for all $x, y \in \mathbb{R}$.

A circle $\omega$ passes through the two vertices $B$ and $C$ of a triangle $A B C$. Furthermore, $\omega$ intersects segment $A C$ in $D \neq C$ and segment $A B$ in $E \neq B$. On the ray from $B$ through $D$ lies a point $K$ such that $|B K|=|A C|$, and on the ray from $C$ through $E$ lies a point $L$ such that $|C L|=|A B|$. Show that the circumcentre $O$ of triangle $A K L$ lies on $\omega$.