Problems

In a pentagon $ABCDE$, diagonal $AD$ is parallel to the side $BC$ and the diagonal $CE$ is parallel to the side $AB$. Show that the areas of the triangles $\mathrm{△}ABE$ and $\mathrm{△}BCD$ are the same.

The figure below is a regular pentagram. What is larger, the black area or the blue area?

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

Let $E$ and $F$ be the midpoints of the sides $BC$ and $AD$ of the parallelogram $ABCD$. Find the area of the quadrilateral formed by the lines $AE,ED,BF$ and $FC$, if it is known that the area $ABCD$ is equal to $S$.