JBMO 2010

The real numbers $a, b, c, d$ satisfy simultaneously the equations

$$a b c-d=1, b c d-a=2, c d a-b=3, d a b-c=-6$$

Prove that $a+b+c+d \neq 0$.

Let $A L$ and $B K$ be angle bisectors in the non-isosceles triangle $A B C$ ( $L$ lies on the side $B C, K$ lies on the side $A C$ ). The perpendicular bisector of $B K$ intersects the line $A L$ at point $M$. Point $N$ lies on the line $B K$ such that $L N$ is parallel to $M K$. Prove that $L N=N A$.

A $9 \times 7$ rectangle is tiled with tiles of the two types shown in the picture below (the tiles are composed by three, respectively four unit squares and the L-shaped tiles can be rotated repeatedly with $90^{\circ}$ ).

Let $n \geqslant 0$ be the number of the $2 \times 2$ tiles which can be used in such a tiling. Find all the values of $n$.