Platonic solids

Show that there exist only five platonic solids (that are objects which are build out of the one sort of regular polygons, like the tetrahedron and the cube). Hint: use that for each solid in three dimensions it holds that the number of vertices + the number of faces is equal to the number of edges + 2.

Long rope

Suppose you have an enormous long rope. You place this rope on the soil along the equator, such that the rope forms a circle around the earth. After you have finished this big job, you increase the length of the rope by one meter. How far from the soil do yo have to attach the rope to make again a circle around the equator? Assume that the earth is a perfect sphere, without mountains etc.

The walled garden

A garden is surrounded by four walls, which form a rectangular trapezoid. The parallel walls have lengths of 34 and 59. An old oak grows in the middle of the garden. The owner of the garden has figured out that the distance from the four walls to the oak are exactly the same. What is the area of the garden?