SE-SKOLORNAS 2005

Find all integer solutions $x$ , $y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$ .

There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$ , then $AE +CD <AC$ .

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.