Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

In a company of $6$ sousliks each souslik has $4$ friends. Is it always possible to divide this company into two groups of $3$ sousliks such that in both groups all sousliks are friends?

Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

Tom painted round fence which consists of $2n \ge6$ sections in such way that every section is painted in one of four colours. Then he repeats the following while it is possible: he chooses three neighbouring sections of distinct colours and repaints them into the fourth colour. For which $n$ Tom can repaint the fence in such way infinitely many times?

Prove that there exist infinitely many pairs of real numbers $(x,y)$ such that $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

In a company of 7 sousliks each souslik has 4 friends. Is it always possible to find in this company two non-intersecting groups of 3 sousliks each such that in both groups all sousliks are friends?

Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?