SILK-ROAD 2006

Found all functions $f: \mathbb{R} \to \mathbb{R}$ , such that for any $x,y \in \mathbb{R}$ , $$f(x^2+xy+f(y))=f^2(x)+xf(y)+y.$$

A subset $S$ of the set $M=\{1,2,.....,p-1\}$ ,where $p$ is a prime number of the kind $12n+11$ ,is essential,if the product ${\Pi}_s$ of all elements of the subset is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The difference $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called the deviation of the subset $S$ .Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is bounding the line $l_2\in L$ ,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$ . Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: 1) The line $l$ is bounding the line $l'$ ; 2) the line $l'$ is not bounding the line $l$ .